Previous Seminars
| London Number Theory Seminar Previous Seminars |
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In Summer 2020, the seminar was hosted by Kings College, and was organised by James Newton. Everything was virtual (on Zoom).
22/4/20 Tiago Jardim Da Fonseca (Oxford)
Title: On Fourier coefficients of Poincaré series
Abstract: Poincaré series are among the first examples of holomorphic and weakly holomorphic modular forms. They are useful in many analytical questions, but their Fourier coefficients seem hard to grasp algebraically. In this talk, I will discuss the arithmetic nature of Fourier coefficients of Poincaré series by characterizing them as cohomological invariants (periods).
29/4/20 Ila Varma (Toronto)
Title: Malle's Conjecture for octic D4-fields.
Abstract: We consider the family of normal octic fields with Galois group D4, ordered by their discriminants. In forthcoming joint work with Arul Shankar, we verify the strong form of Malle's conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove this and related results.
6/5/20 Chris Lazda (Warwick)
Title: A Néron–Ogg–Shafarevich criterion for K3 surfaces
Abstract: The naive analogue of the Néron–Ogg–Shafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if is second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.
13/5/20 Chantal David (Concordia)
Title: Non-vanishing cubic Dirichlet L-functions at $s = 1/2$
Abstract: Joint work with A. Florea and M. Lalin.
A famous conjecture of Chowla predicts that $L(1/2,\chi) \not= 0$ for Dirichlet L-functions attached to primitive characters χ. It was conjectured first in the case where χ is a quadratic
character, which is the most studied case. For quadratic Dirichlet L-functions, Soundararajan then proved that at least 87.5% of the quadratic Dirichlet L-functions do not vanish
at $s = 1/2$, by computing the first two mollified moments. Under GRH, there are slightly
stronger results by Ozlek and Snyder obtained by computing the one-level density.
We consider in this talk cubic Dirichlet L-functions. There are few papers in literature about Dirichlet cubic L-functions, compared to the abundance of papers on Dirichlet qua- dratic L-functions, as this family is more difficult, in part because of the cubic Gauss sums. The first moment for $L(1/2,\chi)$ where $\chi$ is a primitive cubic character was computed by Baier and Young over $\mathbb{Q}$ (the non-Kummer case), by Luo over $\mathbb{Q}(\sqrt{−3})$ (the Kummer case), and by David, Florea and Lalin over function fields, in both the Kummer and non-Kummer case. Bounding the second moment, those authors could obtain lower bounds for the number of non-vanishing cubic twists, but not a positive proportion. Moreover, for the case of Dirichlet cubic L-functions, computing the one-level density under the GRH also gives lower bounds which are weaker than any positive proportion.
We prove in this talk that there is a positive proportion of cubic Dirichlet L-functions non-vanishing at $s = 1/2$ over function fields. This can be achieved by using the recent breakthrough work on sharp upper bounds for moments of Soundararajan and Harper. There is nothing special about function fields in our proof, and our results would transfer over number fields (but we would need to assume GRH in this case).
20/5/20 Rong Zhou (Imperial)
Title: Independence of l for Frobenius conjugacy classes attached to abelian varieties.
Abstract: Let A be an abelian variety over a number field $E\subseteq\mathbb{C}$ and let v be a place of good reduction lying over a prime p. For a prime $l\not=p$, a theorem of Deligne implies that upon making a finite extension of E, the Galois representation on the l-adic Tate module factors as $\rho_l:\Gamma_E\to G_A(\mathbb{Q}_l)$, where $G_A$ is the Mumford-Tate group of A. We prove that the conjugacy class of $\rho_l(Frob_v)$ is defined over $\mathbb{Q}$ and independent of l. This is joint work with Mark Kisin.
27/5/20 Matteo Tamiozzo (Imperial)
Title: Bloch–Kato special value formulas for Hilbert modular forms
Abstract: The Bloch–Kato conjectures predict a relation between arithmetic invariants of a motive and special values of the associated L-function. We will outline a proof of (the p-part of) one inequality in the relevant special value formula for Hilbert modular forms of parallel weight two, in analytic rank at most one.
03/6/20 Yunqing Tang (Paris-Saclay)
Title: Picard ranks of reductions of K3 surfaces over global fields
Abstract: For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces with potentially good reduction everywhere over global function fields. In this talk, I will present the proofs via the (arithmetic) intersection theory on good integral models (and its special fibers) of GSpin Shimura varieties. These results are generalizations of the work of Charles on exceptional isogenies between reductions of a pair of elliptic curves. This talk is based on joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.
12/6/20 Vesselin Dimitrov (Toronto)
Title: p-adic Eisenstein series, arithmetic holonomicity criteria, and irrationality of the 2-adic ζ(5)
Abstract: In this exposition of a joint work in progress with Frank Calegari and Yunqing Tang, I will explain a new arithmetic criterion for a formal function to be holonomic, and how it revives an approach to the arithmetic nature of special values of L-functions. The new consequence to be proved in this talk is the irrationality of the 2-adic version of ζ(5) (of Kubota-Leopoldt). But I will also draw a parallel to a work of Zudilin, and try to leave some additional open ends where the holonomicity theorem could be useful. The ingredients of the irrationality proof are Calegari's p-adic counterpart of the Apery-Beukers method, which is based on the theory of overconvergent p-adic modular forms (IMRN, 2005) taking its key input from Buzzard's theorem on p-adic analytic continuation (JAMS, 2002), and a Diophantine approximation method of Andre enhanced to a power of the modular curve $X_0(2)$. The overall argument, as we shall discuss, turns out to bear a surprising affinity to a recent solution of the Schinzel-Zassenhaus conjecture on the orbits of Galois around the unit circle.
17/6/20 Yifeng Liu (Yale)
Title: Beilinson-Bloch conjecture and arithmetic inner product formula
Abstract: In this talk, we study the Chow group of the motive associated to a tempered global L-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is −1. We show that, under some restrictions on the ramification of π, if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the π-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions (which generalizes the B-SD conjecture). Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain π-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by me a decade ago. This is a joint work with Chao Li.
08/7/20 Jared Weinstein (Boston University)
Title: Partial Frobenius structures, Tate’s conjecture, and BSD over function fields.
Abstract: Tate’s conjecture predicts that Galois-invariant classes in the l-adic cohomology of a variety are explained by algebraic cycles. It is known to imply the conjecture of Birch and Swinnerton-Dyer (BSD) for elliptic curves over function fields. When the variety, now assumed to be in characteristic p, admits a “partial Frobenius structure”, there is a natural extension of Tate’s conjecture. Assuming this conjecture, we get not only BSD, but the following result: the top exterior power of the Mordell-Weil group of an elliptic curve is spanned by a “Drinfeld-Heegner” point. This is a report on work in progress.
In Spring 2020, the seminar was hosted by UCL, and was organised by Chris Birkbeck and Peter Humphries.
15 Jan 2020 - Matthew Bisatt (Bristol)
Title: Tame torsion of Jacobians and the tame inverse Galois problem
Abstract: Fix positive integers g and m. Does there exist a genus g curve, defined over the rationals, such that the mod m representation of its Jacobian is everywhere tamely ramified? I will give an affirmative answer to this question when m is squarefree via the theory of hyperelliptic Mumford curves. I will also and give an application of this to a variant of the inverse Galois problem. This is joint work with Tim Dokchitser.
22 Jan 2020 - Spencer Bloch (University of Chicago)
Title: Gamma Functions, Monodromy, and Apéry Constants.
Abstract:
1) Recall of the theory of periods for local systems on curves.
2) Definition (V. Golyshev) of motivic gamma functions as Mellin transforms of period integrals.
3) Main theorem (joint with M. Vlasenko)
4) Application to the gamma conjecture in mirror symmetry (work of Golyshev + Zagier).
29 Jan 2020- Giada Grossi (UCL)
Title: The p-part of BSD for residually reducible elliptic curves of rank one
Abstract: Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae, allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.
05 Feb 2020 - Efthymios Sofos (University of Glasgow)
Title: Rational points on Châtelet surfaces
Abstract: This talk is on ongoing joint work with Alexei Skorobogatov.
Châtelet surfaces of degree d are surfaces of the form x^2−ay^2=f(t), where f is a fixed integer polynomial of even degree d and a is a fixed non-square integer. When f has degree up to 4 (or when f is a product of integer linear polynomials) it has been shown that the Brauer-Manin obstruction is the only one to the Hasse principle. This is the result of decades of investigations by Swinnerton-Dyer, Colliot-Thélène, Skorobogatov, Browning and Matthiesen, among others.
Going beyond degree 4 for polynomials of general type has been a very popular question which has seen no progress in the last decades. We use techniques from analytic number theory, related to equidistribution of the Möbius function, to prove that for 100% of all polynomials f (ordered by the size of the coefficients) gives Châtelet surfaces that satisfy the Hasse principle.
12 Feb 2020 - Sandro Bettin (Università degli studi di Genova)
Title: The value distribution of quantum modular forms
Abstract: In a joint work with Sary Drappeau, we obtain results on the value distribution of quantum modular forms. As particular examples we consider the distribution of modular symbols and the Estermann function at the central point.
19 Feb 2020 - Sarah Peluse (Oxford University)
Title: Bounds in the polynomial Szemerédi theorem
Abstract: Let P_1,...,P_m be polynomials with integer coefficients and zero constant term. Bergelson and Leibman’s polynomial generalization of Szemerédi’s theorem states that any subset A of {1,...,N} that contains no nontrivial progressions x,x+P_1(y),...,x+P_m(y) must satisfy |A|=o(N). In contrast to Szemerédi's theorem, quantitative bounds for Bergelson and Leibman's theorem (i.e., explicit bounds for this o(N) term) are not known except in very few special cases. In this talk, I will discuss recent progress on this problem.
26 Feb 2020 - Djordje Milicevic (Bryn Mawr/Max Planck)
Title: Extreme values of twisted L-functions
Abstract: Distribution of values of L-functions on the critical line, or more generally central values in families of L-functions, has striking arithmetic implications. One aspect of this problem are upper bounds and the rate of extremal growth. The Lindelof Hypothesis states that zeta(1/2+it)<<(1+|t|)^eps for every eps>0 ; however neither this statement nor the celebrated Riemann Hypothesis (which implies it) by themselves do not provide even a conjecture for the precise extremal sub-power rate of growth. Soundararajan's method of resonators and its recent improvement due to Bondarenko-Seip are flexible first moment methods that unconditionally show that zeta(1/2+it), or central values of other degree one L-functions, achieve very large values.
In this talk, we address large central values L(1/2, f x chi) of a fixed GL(2) L-function twisted by Dirichlet characters chi to a large prime modulus q. We show that many of these twisted L-functions achieve very high central values, not only in modulus but in arbitrary angular sectors modulo pi*Z, and that in fact given any two modular forms f and g, the product L(1/2, f x chi) * L(1/2, g x chi) achieves very high values. To obtain these results, we develop a flexible, ready-to-use variant of Soundararajan's method that uses only a limited amount of information about the arithmetic coefficients in the family. In turn, these conditions involve small moments of various combinations of Hecke eigenvalues over primes, for which we develop the corresponding Prime Number Theorems using functorial lifts of GL(2) forms.
This is part of joint work on moments of twisted L-functions with Blomer, Fouvry, Kowalski, Michel, and Sawin.
04 Mar 2020 - Asbjørn Nordentoft (Copenhagen)
Title: The distribution of modular symbols and additive twists of L-functions
Abstract: Recently Mazur and Rubin, motivated by questions in Diophantine stability, put forth some conjectures regarding the distribution of modular symbols, one of which predicts asymptotic Gaussian behavior. An average version of this conjecture was settled by Petridis and Risager using automorphic methods. Modular symbols are certain line integrals associated to weight two cusp forms and we will in this talk discuss generalizations of the result of Petridis and Risager to higher weight cusp forms. In particular we will explain how to generalize the automorphic methods to show that central values of additive twists of cuspidal L-functions (of arbitrary even weight) are also asymptotically Gaussian.
11 Mar 2020 - Javier Fresán (École polytechnique)
Title: Irregular Hodge filtration and eigenvalues of Frobenius
Abstract: The de Rham cohomology of a connection of exponential type on an algebraic variety carries a filtration, indexed by rational numbers, that generalises the usual Hodge filtration on the cohomology with constant coefficients. I will explain a few results and conjectures relating this filtration to exponential sums over finite fields.
Unfortunately the final two seminars (18/3/20 - Tiago da Fonseca (Oxford University) and 25/3/20 - Chris Lazda (Warwick University)) were cancelled, due to the ongoing COVID-19 situation.
In Autumn 2019, the seminar was hosted by Imperial College, and was organised by Ana Caraiani, Robert Kurinczuk, and Matteo Tamiozzo.
9 October 2019 - Johannes Nicaise (Imperial College)
Title: Convergence of p-adic measures to Berkovich skeleta
Abstract: This talk is based on joint work with Mattias Jonsson (Michigan). The theory of mirror symmetry predicts that the fibers of a maximally unipotent degeneration of polarized complex Calabi-Yau n-folds converge to an n-sphere with respect to the Gromov-Hausdorff metric. Boucksom and Jonsson have shown that, if we choose a family of volume forms on these Calabi-Yau manifolds, then the induced measures converge to a Lebesgue measure on Kontsevich and Soibelman’s essential skeleton of the degeneration, which conjecturally coincides with the Gromov-Hausdorff limit. This convergence takes place in a suitable Berkovich space that contains both the complex fibers and the non-archimedean nearby fiber of the degeneration. In this talk, I will explain a $p$-adic version of this result, answering a question that was raised by Matt Baker.
Friday October 11th - João Lourenço (Bonn)
Title: Integral affine Graßmannians of twisted groups and local models of Shimura varieties.
Abstract: Local models of Shimura varieties are integral models of flag varieties which help in understanding the local geometric behaviour of arithmetic models of Shimura varieties and were first systematically introduced by Rapoport-Zink in EL and PEL cases. More recently, a group-theoretic approach to their definition and study has been possibilitated by the theory of affine Graßmannians, as in the works of Pappas-Rapoport and Pappas-Zhu, where the authors always assume tame ramification.
We generalise the constructions of these last papers, by exhibiting certain smooth affine and connected "parahoric" group models over Z[t] of a given quasi-split Q(t)-group G with absolutely simple simply connected cover splitting over the normal closure of Q(t^{1/e}) with e=2 or 3 (under a mild assumption on the maximal torus). In characteristic e, the group scheme becomes generically pseudo-reductive and we explain in which sense the F_e[t]-model may still be interpreted as parahoric. Then we focus on the affine Graßmannians (both local and global) attached to this group scheme, which are proved to be representable by an ind-projective ind-scheme. We also obtain normality theorems for Schubert varieties in the local and global case (except if G is an odd dimensional unitary group) and an enumeration of the irreducible components of the fibres via the admissible set. Time permitting, we will explain how in the abelian case these global Schubert varieties give rise to the local models conjectured by Scholze.
16 October 2019 - David Hansen (Max Planck Institute, Bonn)
Title: Geometric Eisenstein series and the Fargues-Fontaine curve
Abstract: In the geometric Langlands program, one replaces automorphic forms on a group G with sheaves on the stack of G-bundles over a fixed projective curve. The analogue of Eisenstein series in this setting is the "Eisenstein functor" constructed 20 years ago by Braverman-Gaitsgory, which has many marvelous properties. Recently, Fargues has proposed a completely new kind of geometric Langlands program over the Fargues-Fontaine curve. I'll discuss the prospects for constructing an Eisenstein functor in this setting, and explain an application to the local Langlands correspondence. This is joint work in progress with Linus Hamann.
23 October 2019 - Raphaël Beuzart-Plessis (Marseille)
Title: Recent progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures for unitary groups.
Abstract: In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the non-vanishing of central values of certain Rankin-Selberg L-functions to the non-vanishing of certain explicit integrals of automorphic forms, called 'automorphic periods' on classical groups. These predictions have been subsequently refined by Ichino-Ikeda and Neal Harris into precise conjectural identities relating these two invariants thus generalizing a famous result of Waldspurger for toric periods for GL(2). In the case of unitary groups, those have now been mostly established by Wei Zhang and others using a relative trace formula approach. In this talk, I will review the story of these conjectures and the current state of the art. Finally, time permitting, I will give some glimpse of the proof.
30 October 2019 - Nadir Matringe (Poitiers)
Title: Galois periods vs Whittaker periods for $SL_n$
Abstract: Let $\pi$ be a generic representation of $SL(n)$, either over a $p$-adic or a finite field, or over the ring of adeles of a number field, in which case we assume $\pi$ to be cuspidal automorphic. In all cases one can characterize representations distinguished by the Galois involution inside the $L$-packet of $\pi$ in terms of nonvanishing of "distinguished" Whittaker periods. We will give an idea of the proofs in each case, and if time allows we will give an application in the adelic setting.
6 November 2019 - James Newton (King’s College)
Title: Symmetric power functoriality for modular forms of level 1
Abstract: Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym${}^r$ from automorphic representations of $GL_2$ to automorphic representations of $GL_{r+1}$. I will discuss some joint work with Jack Thorne on the symmetric power lifting for level 1 modular forms.
13 November 2019 - Jaclyn Lang (Paris 13)
Title: The Hodge and Tate Conjectures for self-products of two K3 surfaces
Abstract: There are 16 K3 surfaces (defined over $\mathbb{Q}$) that Livné-Schütt-Yui have shown are modular, in the sense that the transcendental part of their cohomology is given by an algebraic Hecke character. Using this modularity result, we show that for two of these K3 surfaces $X$, the variety $X^n$ satisfies the Hodge and Tate Conjectures for any positive integer $n$. In the talk, we will discuss the details of the Tate Conjecture for $X^2$. This is joint work in progress with Laure Flapan.
20 November 2019 - Arthur-Cesar Le Bras (Paris 13)
Title : Prismatic Dieudonné theory
Abstract : I would like to explain a classification result for
$p$-divisible groups, which unifies many of the existing results in the
literature. The main tool is the theory of prisms and prismatic
cohomology recently developed by Bhatt and Scholze. This is joint work
with Anschütz.
27 November 2019 - Peter Sarnak (Princeton)
Title: Integer points on affine cubic surfaces
Abstract: The level set of a cubic polynomial in four or more variables tend to have many integer solutions, while ones in two variables have a limited number of solutions. Very little is known in case of three variables. For cubics which are character varieties (thus carrying a nonlinear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in, say, a real quadratic field, there is a fundamental change which is closely connected to challenging questions about one-commutators in SL_2 over such rings.
4 December 2019 - Thomas Lanard (Vienna)
Title: On the $l$-blocks of $p$-adic groups
Abstract: We will talk about the category of smooth representations of a
p-adic group. Our main focus will be to decompose it into a product of
subcategories. When the field of coefficients is $\mathbb{C}$, it is
well known thanks to Bernstein decomposition theorem. But when we are
over $\bar{\mathbb{Z}}_l$ it is more mysterious. We will see what can be
done and some links with the local Langlands correspondence.
11 December 2019 - Elena Mantovan (Caltech)
Title: $p$-adic automorphic forms on unitary Shimura varieties
Abstract: We study $p$-adic automorphic forms on unitary Shimura varieties at any unramified prime $p$. When $p$ is not completely split in the reflex field, the ordinary locus is empty and new phenomena arise. We focus in particular on the construct and study of $p$-adic analogues of Maass--Shimura operators on automorphic forms. These are weight raising differential operators which allow us to $p$-adically interpolate classical forms into families. If time permits, we will also discuss an application to the study of mod $p$ Galois representations associated with automorphic forms. This talk is based on joint work with Ellen Eischen.
In Summer 2019, the seminar was hosted by KCL, and was organised by Eran Assaf and James Newton.
24th April 2019 - Jesse Jääsaari (University of Helsinki)
Title: Exponential Sums Involving Fourier Coefficients of higher rank automorphic forms
Abstract: In this talk I will describe various conjectures concerning correlations between Fourier coefficients of higher rank automorphic forms and different exponential phases. I will also discuss recent work (partly in progress) towards some of these conjectures.
01 May 2019 - Kazim Büyükboduk (UC Dublin)
07-08 May 2019 - London-Paris Number Theory Seminar.
08 May 2019 - Ben Heuer (King's College London)
15 May 2019 - No seminar, we're at the p-adic Langlands Programme and Related Topics workshop.
22 May 2019 - Eva Viehmann (Technical University of Munich)
29 May 2019 - Eugenia Rosu (University of Arizona)
05 June 2019 - Paul Ziegler (University of Oxford)
12 June 2019 - Ramla Abdellatif (Université de Picardie Jules Verne)
19 June 2019 - Mikhail Gabdullin (Lomonosov Moscow State University)
26 June 2019 - Daniel Gulotta (University of Oxford)
3 July 2019 - Christopher Frei (University of Manchester)
09 Jan 2019 - Adam Morgan (Glasgow)
16 Jan 2019 - Helene Esnault (Freie Universität Berlin)
23 Jan 2019 - Adam Logan
Abstract: Using the Torelli theorem for K3 surfaces of Pyatetskii-Shapiro and Shafarevich one can describe the automorphism group of a K3 surface over ${\mathbb C}$ up to finite error as the quotient of the orthogonal group of its Picard lattice by the subgroup generated by reflections in classes of square $-2$. We will give a similar description valid over an arbitrary field in which the reflection group is replaced by a certain subgroup. We will then illustrate this description by giving several examples of interesting behaviour of the automorphism group, and by showing that the automorphism groups of two families of K3 surfaces that arise from Diophantine problems are finite. This is joint work with Martin Bright and Ronald van Luijk (University of Leiden).
30 Jan 2019 - Jan Kohlhaase (Universität Duisburg-Essen)
06 Feb 2019 - Mladen Dimitrov (Université de Lille)
13 Feb 2019 - Yiannis Petridis (UCL)
20 Feb 2019 - Pankaj Vishe (Durham)
27 Feb 2019 - Martin Gallauer (Oxford)
06 Mar 2019 - Edgar Assing (Bristol)
Note: This seminar will take place in room 500.
13 Mar 2019 - Jan Vonk (Oxford)
20 Mar 2019 - Alice Pozzi (UCL)
3 Oct 2018 - Adam Harper (Warwick) 10 Oct 2018 - Sarah Zerbes (UCL) 17 Oct 2018 - Netan Dogra (Oxford) 24 Oct 2018 - Andrea Dotto (Imperial) 31 Oct 2018 - Vytas Paskunas (Essen) 7 Nov 2018 - Preston Wake (IAS) 14 Nov 2018 - Victor Rotger (Barcelona) 21 Nov 2018 - Jack Shotton (Durham) 28 Nov 2018 - Yichao Tian (Strasbourg) 5 Dec 2018 - Lucia Mocz (Bonn) 12 Dec 2018 - Jessica Fintzen (Cambridge)
25/04/18 Peter Humphries (UCL) 2/5/18 Alexandra Florea (Bristol) 9/5/18 Steve Lester (QMUL) 16/5/18 Holly Krieger (Cambridge) 23/5/18 Jan Nekovář (Paris) 29/5/18 to 30/5/18 -- the London-Paris Number Theory Seminar.
30/5/18 Matthew Morrow (Paris) 6/6/18 Arno Kret (Amsterdam) 13/6/18 Chris Birkbeck (UCL) 20/6/18 Djordjo Milovic (UCL) 27/6/18 Kęstutis Česnavičius (Paris) (bonus summer talk) 25/7/18 Jeehoon Park (POSTECH)
08/01/18 to 11/01/18: UK-Japan Winter School 2018 on Number Theory -- Galois representations and Automorphic Forms.
10/01/18 Martin Widmer (Royal Holloway) 17/01/18 Kaisa Matomaki (University of Turku) 24/01/18 Andrea Ferraguti (University of Cambridge) 31/01/18 Samir Siksek (University of Warwick) 07/02/18 Sam Chow (University of York) 14/02/18 Valentina Di Proietto (University of Exeter) 21/02/18 Ildar Gaisin (École polytechnique) 28/02/18 Edva Roditty-Gershon (University of Bristol) 07/03/18 Eyal Goren (McGill University) 14/03/18 Daniel Disegni (Université Paris-Sud) 21/03/18 Two seminars: Vincent Pilloni (ENS Lyon) and Peter Schneider (University of Muenster).
Pilloni (1600-1700): Higher p-adic families of Siegel modular forms
4/10/17 Laura Capuano (Oxford) 11/10/17 Abhishek Saha (Queen Mary) 18/10/17 Francis Brown (Oxford/IHES) 25/10/17 Eugen Hellmann (Muenster) 1/11/17 Judith Ludwig (Bonn) 8/11/17 Jean-Francois Dat (Paris VI) 15/11/17 Torsten Wedhorn (Darmstadt) 22/11/17 Jack Thorne (Cambridge) 27-28/11/17 The London-Paris Number Theory Seminar (in Paris) 29/11/17 Chris Williams (Imperial) 6/12/17 Florian Herzig (Toronto) 13/12/17 Cong Xue (Cambridge)
26/04/17 Rebecca Bellovin (Imperial) 03/05/17 Vladimir Dokchitser (KCL) 10/05/17 Chris Hughes (Univ of York) 17/05/17 Nadav Yesha (KCL) 24/05/17 Stefano Vigni (Università di Genova) 31/5/17 Wushi Goldring (Stockholm Uni)
I will report on a project, initiated jointly with J.-S. Koskivirta
and developed further in joint work with Koskivirta, B. Stroh and Y.
Brunebarbe, which contests this common view in two stages: The first
consists in showing that fundamental geometric properties of X are
explained purely by means of \zeta (and its generalizations). The
second is that, while these geometric properties may appear to be
special to Shimura varieties, the G-Zip viewpoint shows that they hold
much more generally, for geometry engendered by G-Zips: Any scheme Z
equipped with a morphism to GZip^{\mu} satisfying some general
scheme-theoretic properties. To illustrate our program concretely, I
will describe results and conjectures regarding two basic geometric
questions about X, Z: (i) Which automorphic vector bundles on X, Z
admit global sections? (ii) Which of these bundles are ample? As a
corollary, we also deduce old and new results over the complex
numbers. Question (i) was inspired by a conjecture of F. Diamond on
Hilbert modular forms mod p.
7/6/17 Jens Marklof (Univ of Bristol) 14/6/17 Beth Romano (Univ of Cambridge) 21/6/17 Samit Dasgupta (UC Santa Cruz) 28/6/17 Morten Risager (Uni of Copenhagen)
11/01/17 Chris Skinner (Princeton) 18/01/17 Jack Lamplugh (UCL) 25/01/17 Rachel Newton (Reading University)
1/2/17 Atsuhira Nagano (KCL/Waseda University)
8/2/17 Christian Johansson (University of Cambridge)
15/2/17 (note room change: Archaeology, Room G6)
22/2/17 (note room change: Archaeology, Room G6)
1/3/17 Giuseppe Ancona (Université de Strasbourg)
8/3/17 Céline Maistret (Warwick)
15/3/17 Aurel Page (Warwick)
22/3/17 Otto Overkamp (Imperial)
The seminar will be preceded by the Study
Groups, and this term there are two of them, one on p-adic integration running 1300-1415 and one on p-adic Hodge theory running 1430-1600.
5/10/16 James Newton (Kings) 12/10/16 Jean-Stefan Koskivirta (Imperial) 19/10/16 Andrea Bandini (Università degli Studi di Parma) 26/10/16 Brian Conrad (Stanford)
Over number fields it is elementary to remove the reductivity
hypothesis from Sansuc's formula, but over function fields that is a
much harder problem; e.g., the Picard group can be infinite. Work in
progress by my PhD student Zev Rosengarten is likely to completely
solve this problem. He has formulated an alternative version, proved
it is always finite, and established the formula in many new cases.
We will discuss some aspects of this result, including one of its key
ingredients: a generalization of Tate local and global duality to the
case of coefficients in any positive-dimensional (possibly non-smooth)
affine algebraic $k$-group scheme and its (typically non-representable)
${\rm{GL}}_1$-dual sheaf for the fppf topology.
2/11/16 Joe Kramer-Miller (UCL) 9/11/16 Carl Wang Erickson (Imperial) 16/11/16 Dimitar Jetchev (EPFL) 23/11/16 Macarena Peche Irissarry (ENS Lyon) 30/11/16 Valentin Hernandez (Paris VI) 7/12/16 Note that this week the seminar is in 130. 14/12/16 Jaclyn Lang (Paris XIII)
27/4/16 Fernando Shao (Oxford) 4/5/16 No seminar because of conference RandomWavesInLondon.
11/5/16 Oleksiy Klurman (Universite de Montreal/UCL) 18/5/16 Giovanni Rosso (Cambridge) 25/5/16 Gergely Zábrádi (Eötvös Loránd) 1/6/16 Trevor Wooley (Bristol) 6th and 7th June -- London-Paris Number Theory Seminar.
8/6/16 Adam Harper (Cambridge) 15/6/16 Min Lee (Bristol) 22/6/16 Maria Valentino (King's) 29/6/16 Nina Snaith (Bristol)
Abstract: Following Ribet's seminal 1976 paper
there have been many results employing congruences between stable
cuspforms and lifted forms to construct non-split extensions of Galois
representations.
This strategy can be extended to construct elements in the Bloch-Kato
Selmer groups of general ±-Asai representations. I will explain how
suitable congruences between automorphic forms over a CM field, whose
associated Galois representations are totally odd polarizable, always
give rise to elements in the Selmer group for exactly the Asai
representation (+ or -) that is critical in the sense of Deligne. In
addition, I will discuss consequences for Fontaine-Mazur style
conjectures.
Abstract:
The theory of modular curves, their integral models and modular forms
on them is well-developed, and had been used in many spectacular
applications. I will recall some of the features that are relevant to my
talk.
Motivated by the state of affairs for curves, we have been studying
unitary Shimura varieties in positive characteristic and, in particular,
the Picard modular surfaces that are associated to a unitary group of
signature (2,1). These are moduli spaces for abelian threefolds equipped
with an action of an imaginary quadratic field. I will explain what we
currently know about their geometry modulo a prime p (building on work
by Bellaiche, Vollaard, Bultel-Wedhorn and borrowing ideas from G. Boxer
and the theory of modular curves). To the extent time allows, I will
discuss Hecke correspondences at p and the rather complicated picture we
face. This is joint work with E. De Shalit (Hebrew University).
Abstract: We give
an asymptotic formula for the even moments of a sum of multiplicative
Steinhaus or Rademacher random variables. This is obtained by expressing
the sum as a multiple contour integral from which the asymptotic
behaviour can be extracted. The result was obtained independently by
Harper, Nikeghbali, and Radziwill by using a result of La Breteche. We
also give an asymptotic relationship between the Steinhaus even moments
and the even moments of a truncated characteristic polynomial of a
unitary matrix, which extends earlier work of Conrey and Gamburd. The
talk is based on joint work with Winston Heap.
Guhan Harikumar (UCL) Abstract: We will first
recall the theory of Darmon cycles due to V. Rotger and M. Seveso. These
are a higher weight analogue of Stark-Heegner points. Then, we will show
how these Darmon cycles are related to (a p-adic family of)
half-integral weight modular forms. The relation follows by the p-adic
interpolation of a well known formula of Waldspurger.
Abstract: Let E be a
quadratic twist of the elliptic curve X_0(49), so that E has complex
multiplication by the ring of integers of Q(sqrt(-7)). Using Iwasawa
theory, Gonzalez-Aviles and Rubin proved that if L(E/Q,1) is nonzero,
then the full Birch--Swinnerton-Dyer conjecture holds for E. We will
consider a more general case: Take p to be any prime which is congruent
to 7 modulo 8, and set K= Q(sqrt(-p)). We will discuss the main
conjecture of Iwasawa theory for an infinite family of elliptic curves
which are defined over the Hilbert class field of K with complex
multiplication by the ring of integers of K. Eugen Hellmann (Bonn)
Abstract: In the theory of p-adic modular forms (or more generally p-adic automorphic forms) the phenomenon occurs that there are non-classical and classical forms that have the same system of Hecke-eigenvalues.
This phenomenon has an explanation in terms of the associated Galois representations. Namely certain p-adic Hodge-structures (so called triangulations) degenerate in the corresponding families of Galois representations. We prove various results about the space of corresponding Galois representations and relate them to questions about companion points.
Abstract: In this talk, we study the problem of counting the
number of varieties in fibrations over projective spaces which contain a
rational point. We obtain geometric conditions that force very few of
the varieties in the family to contain a rational point, in a precise
quantitative sense. This generalises the special case of conic bundles
treated by Serre.
This is joint work with Arne Smeets. Abstract: We explore the idea of Conrey and Li of presenting the Selberg trace formula for Hecke operators, as a Dirichlet series. We enhance their work in few ways and present several applications of our formula. This is a joint work with Andrew Booker.
Abstract:
We shall describe the recent results of Wooley, and of Bourgain
Demeter and Guth, on Vinogradov's mean value, and how they lead to
substantial improvements in estimates for the Riemann Zeta-function.
Abstract: The distribution of divisors of an integer n can be studied
through the distribution of the random variable
D_n := (log d)/log n
where d is chosen uniformly at random from divisors of n. Out of
many interesting aspects of the sequence (D_n), one may ask about its
possible convergence in law, as n tends to infinity along various
sequences of integers. For instance, a classical result of
Deshouillers, Dress and Tenenbaum shows the convergence of the Cesaro
mean of the distribution functions of D_n.
It appears however that for generic n, the situation is very erratic.
In this talk, we will consider the case where n is assumed not to have
any large prime factor (for a certain meaning of "large"). In this
situation, a central limit theorem is expected to hold for D_n, and is
established on average.
This will lead us to various topics in probabilistic number theory. Tim Browning (Bristol) Abstract: I will discuss some recent progress on the Brauer-Manin obstruction for rational points on algebraic varieties, before showing how counting arguments from analytic number theory
can be used to study strong approximation for a special family of varieties defined by norm forms. This allows the resolution of a conjecture of Colliot-Thelene about the sufficiency of the
Brauer-Manin obstruction for varieties admitting a suitable fibration. This is joint work with Damaris Schindler.
Abstract: n broad terms, the polynomial method is the idea of understanding a set of points, for example in euclidean
space, by studying the properties of polynomials vanishing on that set. We'll discuss some applications of this method in
problems related to number theory and combinatorics, including connections
with sieve theory, the study of rational points on curves and the incidence geometry of algebraic varieties.
23/3/16 The seminar this week will be at Kings College, at 2pm, in S4.23 29/3/16 Seminar is on a Tuesday 3pm in S4.36 at Kings
07/10/15 Martin Orr (Imperial)
14/10/15 Stephane Bijakowski (Imperial)
21/10/15 Yiwen Ding (Imperial)
28/10/15 Rob Kurinczuk (Bristol)
04/11/15 Paul Ziegler
9/11/15 London-Paris Number Theory Seminar (in Paris)
11/11/15 Laurent Berger
18/11/15 Julien Hauseux (Kings)
25/11/15 Arne Smeets (Imperial)
2/12/15 Jacques Benatar (Kings)
9/12/15 Jennifer Balakrishnan (Oxford)
16/12/15 Daniel Skodlerack (Imperial)
22/04/15 2.30 (in the usual place S4.23) Jack Shotton (Imperial) 22/04/15 4.00 Christophe Breuil (Paris) 29/04/15 4.00 Victor Beresnevich (York) 06/05/15 4.00 Wansu Kim 13/05/15 4.00 Anne-Maria Ernvall-Hytonen (Helsinki) 20/05/15 4.00 Samuele Anni (Warwick) 27/05/15 4.00 Engeniy Zorin (York) 03/06/15 4.00 Pär Kurlberg (KTH) 10/06/15 4.00 Jon Keating (Bristol) 17/06/15 4.00 Alex Gorodnik (Bristol) 24/06/15 4.00 Daniel Fiorilli (University of Ottawa and Paris 7)
14/01/15 Lucio Guerberoff (UCL)
Abstract: Work of Shimura, Langlands, Milne-Shih, and, more recently, Taylor, have examined the action of complex conjugation on Shimura varieties. We study this topic from the viewpoint of the general theory of these varieties. In this talk, we will concentrate on certain Shimura varieties whose reflex field E is a CM field and stablish descent of these to the maximal totally real subfield E^+. We will discuss a strategy to construct these descent data in more generality, and mention possible applications. This work is joint with Don Blasius (UCLA) and remains in progress.
21/01/15 John Coates (Cambridge)
Abstract: I will discuss joint work with Y. Li, Y. Tian and S. Zhai, which generalizes, for a wide class of elliptic curves defined over Q, the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by discriminants having any prescribed number of prime factors. In addition, we prove stronger results for the family of quadratic twists of the modular elliptic curve X0(49), including showing that there is a large class of explicit quadratic twists whose complex L-series does not vanish at s = 1, and for which the full Birch-Swinnerton-Dyer conjecture is valid.
28/01/15 Fabrizio Andreatta (Milan)
Abstract: In the first part I will recall the approach I, Adrian Iovita, Glenn Stevens and independently Vincent Pilloni have developed in order to define families of overconvergent ellitpic modular forms as sections of suitable line bundles. I will then report on work in progress with Iovita and Pilloni which allows to extend those constructions to the boundary of the weight space, providing characteristic p Banach modules with a compact operator. This confirms an expectation of Robert Coleman.
4/2/15 Marc Masdeu (Warwick)
Abstract: Fifteen years ago Henri Darmon introduced a construction of p-adic points on elliptic curves. These points were conjectured to be algebraic and to behave much like Heegner points, although so far a proof remains inaccessible. Other constructions emerged in the subsequent years, thanks to work of himself and many others. All of these constructions are local (either non-archimedean like the original one, or archimedean), and so far none of these are proven to yield algebraic points, although there is extensive numerical evidence.
In this talk I will present joint work with Xavier Guitart and Haluk Sengun, in which we propose a framework that includes all the above constructions as particular cases, and which allows us to extend the construction of local points to elliptic curves defined over arbitrary number fields. As a by-product, we provide an explicit (though conjectural) construction of the (isogeny class of the) elliptic curve attached to an automorphic form for GL2.
11/2/15 Andrew Booker (Bristol) Abstract: In 1989, Selberg defined what came to be known as the "Selberg class" of L-functions, giving rise to a new subfield of analytic number theory in the intervening quarter century. Despite its popularity, a few things have always bugged me about the definition of the Selberg class. I will discuss these nitpicks and describe some modest attempts at overcoming them, with new applications.
18/2/15 Pierre Colmez (Paris VI)
Abstract: I will explain how one can recover the classical Langlands correspondence for GL(2,Qp) from the p-adic in weight one, and try to explain the relation with a conjecture of Breuil and Strauch.
Also 18/2/15 Wieslawa Niziol (ENS Lyon)
Abstract: I will describe how the theory of (phi-Gamma) modules allows to relate p-adic nearby cycles and syntomic cohomology sheaves. This is a joint work with Pierre Colmez.
25/2/15 Levent Alpoge (Cambridge)
Abstract: It is a theorem of Siegel that the Weierstrass model y^2 = x^3 + Ax + B of an elliptic curve has finitely many integral points. A "random" such curve should have no points at all. I will show that the average number of integral points on such curves (ordered by height) is bounded --- in fact, by 67. The methods combine a Mumford-type gap principle, LP bounds in sphere packing, and results in Diophantine approximation. The same result also holds (though I have not computed an explicit constant) for the families y^2 = x^3 + Ax, y^2 = x^3 + B, and y^2 = x^3 - n^2 x.
4/3/15 Carlos de la Mora (UEA) Abstract: Some of the most important problems in number theory are the Langlands Con- jectures. These conjectures are very general and even the `simplest' cases have very strong applications. The statement of the Langlands conjectures depends on the existence of certain local factors that ought to be attached to smooth representations of reductive groups over local elds. These local factors have not yet been defined in general, but in the case where we have a quasi-split group and a generic representation, they have been defined by the so called Langlands-Shahidi method. One of the main ingredients in the Langlands-Shahidi method is the definition of the local coefficients. The main intent of our research is to find a family of functions that behave like eigenvectors for the intertwining operators such that the `eigenvalues' are the local coefficients. We have developed this construction with the vain of obtaining a generalization of local coefficients of certain representations of an arbitrary reductive group over non-archimedean local field.
11/3/15 Andreas Langer (Exeter)
Abstract: For an artinian local ring R with perfect residue field we define higher displays over the small Witt ring. The second crystalline cohomology of a variety of K3-type X (for example the Hilbert schemes of zero-dimensional subschemes of a K3-surface) is equipped with the additional structure of a 2-display. Then we extend the Grothendieck-Messing lifting theory from p-divisible groups to such varieties: The deformations of X over a nilpotent pd-thickening correspond uniquely to selfdual liftings of the associated Hodge-filtration. For the proof we give an algebraic definition of the Beauville-Bogomolov-Form on the second de Rham cohomology of X and show that for ordinary varieties the deformations of X correspond uniquely to selfdual deformations of the 2-display endowed with its Beauville-Bogomolov form. This is joint work with Thomas Zink.
18/3/15 Simon Wadsley (Cambridge)
Abstract: In this talk I will discuss some recent work with Konstantin Ardakov that seeks to develop a framework for a theory of D-modules for rigid analytic spaces in order to better understand the locally analytic representation theory of p-adic analytic groups. To be more precise, we define and study a canonical completion of the sheaf of classical differential operators on a rigid analytic space (in the sense of Tate) that may be viewed as a quantization of the sheaf of functions on the rigid analytic cotangent bundle. We introduce what we call 'co-admissible modules' for this sheaf of non-commutative rings. When the rigid analytic space is the flag variety of a split semisimple p-adic analytic group then there is an equivalence of categories between the category of "co-admissible D-modules" and co-admissible modules (in the sense of Schneider--Teitelbaum) of a certain canonical completion of the enveloping algebra of the associated Lie algebra with trivial infinitesimal central character. Much of this is written up in recent preprints that can be found on the ArXiv; some of it is not yet written up.
08/10/14 Jack Thorne (Cambridge University)
Abstract: Bhargava, Gross and Wang have studied the group J(k)/2J(k), J the Jacobian of a hyperelliptic curve over a field k, using representation theory and invariant theory. In this talk, I will outline a similar program for smooth plane quartic curves (which are non-hyperelliptic of genus 3) with a marked rational point.
15/10/14 Rene Pannekoek (Imperial)
Abstract: This is joint work with Lisa Berger, Chris Hall, Jennifer Park, Rachel Pries, Shahed Sharif, Alice Silverberg, Douglas Ulmer. For each prime power q and each odd prime r not dividing q, we define a curve C of genus r-1 over F_q(t). We give explicit generators for a subgroup of (Jac(C))(K), where K runs through a tower of extensions of F_q(t), and prove that the rank of this subgroup grows linearly with [K:F_q(t)]. By constructing a proper regular model, we also prove that the subgroup is actually of finite index.
22/10/14 Tony Scholl (Cambridge) Abstract: I will discuss a new, largely conjectural, cohomology theory for a certain class of Shimura varieties, and explain some of its arithmetic applications. This is joint work with Jan Nekovar.
29/10/14 Olivier Taibi (Imperial)
Abstract:
Let G be a Chevalley group which is symplectic or special orthogonal. I
will explain how to explicitly compute the geometric side of Arthur's
trace formula for a function on G(AA) which is a stable
pseudo-coefficient of discrete series at the real place and the unit of
the unramified Hecke algebra at every finite place. Arthur's recent
endoscopic classification of the discrete automorphic spectrum of G
allows to analyse the spectral side in detail. For example one can
deduce dimension formulae for spaces of vector-valued Siegel modular
forms in level one. The computer achieves these computations at least up
to genus 7.
5/11/14 James Maynard (Oxford) Abstract: A 1938 result of Rankin shows that there are consecutive primes less than $x$ whose difference is $\gg (\log{x})(\log\log{x})(\log\log\log\log{x})/(\log\log\log{x})^2$. Over the past 75 years, improvements have only been in the implied constant. We will show how one can use recent progress on small gaps between primes to quantitatively improve this bound. A similar improvement was found independently by Ford, Green, Konyagin and Tao using different techniques.
Monday 10/11/14: London-Paris number theory seminar (in Paris).
12/11/14 Samir Siksek
Abstract: We combine the latest advances in modularity lifting with a
3-5-7 modularity switching argument to deduce modularity of 'most'
elliptic curves over totally real fields. In particular, we show that
all elliptic curves over real quadratic fields are modular. This talk
is based on joint work with Bao Le Hung (Harvard) and Nuno Freitas
(Bonn).
19/11/14 Chris Blake (Cambridge)
Abstract: Scholl and Nekovar conjecture that, in the presence of real multiplication by a totally real field F, certain motives should carry a canonical ("F-plectic") structure. I will talk about a first step towards showing the existence of such "canonical F-plectic models" for Shimura varieties attached to groups of the form G = Res_{F/Q} H. More precisely I will explain how Langlands' construction of the Taniyama group (which played a key role in proving the existence of canonical models for Shimura varieties) can be generalised to the plectic setting.
26/11/14 Judith Ludwig (Imperial) Abstract: In this talk we will study an example of p-adic Langlands functoriality:
Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum.
In this talk we will prove a p-adic version of this transfer. More precisely we will extend the classical transfer to p-adic families of automorphic forms as parametrized by eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.
3/12/14 David Helm (Imperial) Abstract: The integral Bernstein center is an algebra that acts naturally
on all smooth \ell-adic representations of GL_n(F), for F a p-adic field.
If one inverts \ell this algebra has a nice description due to Bernstein
and Deligne, but in the integral context it is much more complicated.
We show that this algebra has a clean description in terms of the
maximal tori of GL_n/F, and use this description to relate the Bernstein
center to the deformation theory of Galois representations via local
Langlands.
10/12/14 Roger Heath-Brown (Oxford)
Absract: We show that there are infinitely many primes of the form a^2+p^4, with p prime, thereby sharpening a result of Friedlander and Iwaniec in which p was not restricted to be prime. The talk will illustrate the general strategy for such problems, and highlight a superficially surprising lemma on the equidistribution of primes in congruence classes.
23/04/14 Jack Lamplugh (Cambridge University) 30/04/14 Yiannis Petridis (UCL) 07/05/14 Avner Ash (Boston College) 14/05/14 David Hansen (Institut de mathématiques de Jussieu) 21/05/14 4.00 Gergely Zabradi (Eötvös Lorànd University, Budapest) 28/05/14 4.00 Pankaj Vishe (University of York) 04/06/14 4.00 Dzmitry Badziahin (Durham University) Monday 9/06/14: London-Paris number theory seminar: details are here.
11/06/14 Minhyong Kim (Oxford University) 18/06/14 Andrew Wiles (Oxford) 25/06/14 Zeev Rudnick (Tel-Aviv University)
15/01/14 Alex Bartel (Warwick)
22/01/14 James Newton (Cambridge)
29/01/14 Chris Williams (Warwick)
05/02/14 Victor Rotger (Univ. Polytècnica de Catalunya)
12/02/14 At 2:30 in 707:
Masato Kurihara (Keio University)
12/02/14 At 4:00 in 707:
Thanasis Bouganis (Durham)
19/02/14 Matthew Morrow (Nottingham)
26/02/14 Guido Kings (Regensburg)
05/03/14 Malte Witte (Heidelberg)
12/03/14 Chris Lazda (Imperial)
19/03/14 Peter Sarnak (Princeton) ( NB: talk will be in room 500)
26/03/14 Jenny Cooley (Warwick)
2/10/13 Toby Gee (Imperial)
9/10/13 Luis Garcia (Imperial)
16/10/13 Tim Dokchitser (Bristol)
23/10/13 Rebecca Bellovin (Imperial)
30/10/13 Martin Taylor (Oxford)
6/11/13 René Pannekoek (Imperial)
13/11/13 David Helm (Imperial)
20/11/13 Bob Hough (Cambridge)
27/11/13 Brandon Levin (IAS)
4/12/13 Martin Orr (UCL)
11/12/13 Lilian Matthiesen (Orsay)
24/4/13 2:30: P. Cassou-Noguès (Bordeaux)
1/5/13 Igor Wigman (KCL)
8/5/13 Vladimir Dokchiter (Cambridge)
15/5/13 Sanju Velani (York)
22/5/13 Stefano Morra (Montpellier)
29/5/13 Dinesh Thakur (Arizona)
3-4/6/13: London-Paris Number Theory Seminar (in London). Speakers: F. Charles (Rennes), Y. Harpaz (Nijmegen), J-L. Colliot-Thelene (Orsay), O. Wittenberg (ENS), D. Testa (Warwick)
5/6/13 P. Xu (UEA)
12/6/13 Luis Dieulefait (Barcelona)
19/6/13 Christopher Daw (UCL)
26/6/13 Ashay Burungale
16/01 Speaker: David Loeffler (Warwick)
23/01 Speaker: Nicola Mazzari (Paris 7)
30/01 Speaker: Laurent Berger (ENS Lyon)
13/02 Speaker: Gebhard Boeckle (Heidelberg)
20/02 Speaker: John Coates (Cambridge)
27/02 Speaker: Amnon Besser (Ben Gurion University/Oxford)
06/03 NO SEMINAR
13/03 Speaker: Massimo Bertolini (Milan)
20/03 Speaker: Damiano Testa (Warwick)
03/10 Speaker: Andrei Yafaev (UCL)
10/10 Speaker: Nuno Freitas (Kings College London)
17/10 Speaker: Toby Gee (Imperial College London)
Monday October 22nd:
London-Paris Number Theory Seminar.
Speakers: Sarah Zerbes (UCL), David Vauclair (Caen), David Burns (Kings College London).
24/10 Speaker: Tom Fisher (Cambridge)
31/10 Speaker: Nicolas Ojeda Bar (Cambridge)
7/11 Speaker: Mehmet Haluk Sengun (Warwick)
14/11 Speaker: Abhishek Saha (Bristol)
21/11 Speaker: Alexei Skorobogatov (Imperial College London)
28/11 Speaker: Paul Ziegler (ETH Zurich)
05/12 Speaker: Bruno Angles (Caen)
12/12 Speaker: Frank Neumann (Leicester)
25 April: Werner Mueller, Bonn
2 May: Stefano Vigni, King's
9 May: David Savitt, Arizona
16 May: Riccardo Brasca, Milan
23 May: Konstantin Ardakov, QMUL
30 May: London-Paris Number Theory Seminar.
Speakers: Joel Rivat (Luminy), Gerald Tenenbaum (Nancy), Adam Harper (Cambridge).
6 June: Jens Marklof, Bristol
13 June: William Conley, UEA
20 June: Alex Paulin, Nottingham
27 June: Christian Johansson, Imperial
Jan 18: Florence Gillibert (Bordeaux and MPI Bonn)
Jan 25 Roger Heath-Brown (Oxford)
Feb 1st: Jan Nekovar (Paris)
Feb 8: Richard Hill (UCL)
Feb 15 Paul-James White (Oxford)
Feb 22nd bonus talk at 2:30: Go Yamashita (Toyota)
Feb 22nd Vytas Paskunas (Bielefeld)
Feb 29th Jonathan Pila (Oxford)
March 7th Rene Pannekoek (Leiden)
March 14th Xavier Caruso
March 21 Anish Ghosh (Norwich)
Speaker: Kevin Buzzard (Imperial College)
Title: Reduction of crystalline representations
Abstract: 2-dimensional crystalline Galois representations of the absolute Galois group of Q_p can be completely determined by linear algebra data. One can also classify 2-dimensional mod p Galois representations easily. This leads us to the following question: given a piece of linear algebra data, what is the reduction of the corresponding p-adic Galois representation? This question turns out to be a little subtle. I'll survey the state of the art and talk about the most recent method of attack -- the p-adic and mod p Langlands Program.
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DATE: 12/10/11
Speaker: Ivan Tomasic (Queen Mary)
Title: Twisted Galois stratification
Abstract: The aim is to present the development of difference algebraic geometry and its applications to counting solutions of difference polynomial equations over fields with powers of Frobenius. We prove a twisted version of Chebotarev's theorem for a Galois covering of difference schemes, and use it to deduce an important direct image theorem: the image of a "twisted Galois formula" by a morphism of difference schemes is again a twisted Galois formula.
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DATE: 19/10/11
Speaker: Jamshid Derakhshan (Oxford)
Title: Model theory of the adeles and connections to number theory
Abstract: This is joint work with Angus Macintyre. Model theory studies definable subsets of a structure in a specific language. For many important structures, definable sets turn out to have a rich geometry in a natural language. Once the family of definable sets has a 'a direct image theorem', the structure of definable sets becomes transparent. This usually implies decidability, but there are also applications to geometry and arithmetic; and structures enjoying such properties can be thought of as 'tame' in some sense.
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DATE: 26/10/11
Speaker: Francois Loeser (Jussieu)
Title: Motivic height zeta functions and the Poisson formula
Abstract: Recently, Chambert-Loir and Tschinkel obtained asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups. In this talk we shall present a motivic version of these results. We show the rationality of the corresponding motivic height zeta functions and determine its leading pole and the residue. Our approach relies on "motivic harmonic analysis". In particular a motivic Poisson formula due to Hrushovski and Kazhdan plays a key role. This is joint work with Antoine Chambert-Loir.
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DATE: 02/11/11
Speaker: Alena Pirutka (Strasbourg)
Title: On some aspects of unramified cohomology
Abstract: During this talk, I would like to explain different interactions of unramified cohomology groups with other problems, such as the study of Chow groups or some local-global principles.
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DATE: 09/11/11
Speaker: Toby Gee (Imperial College)
Title: p-adic Hodge-theoretic properties of etale cohomology with mod p coefficients, and the cohomology of Shimura varieties
Abstract: I will discuss some new results about the etale cohomology of varieties over a number field or a p-adic field with coefficients in a field of characteristic p, and (if time permits) give some applications to the cohomology of unitary Shimura varieties. (Joint with Matthew Emerton.)
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DATE: 16/11/11
Speaker: Bruno Angles (Caen)
Title: On the class group module of a Drinfel'd module
Abstract: Recently L. Taelman has constructed a finite F_q[T]-module attached to a Drinfel'd module which is an analogue of the ideal class group of a number field. Taelman has proved an analytic class number formula for these modules and an analogue of Ribet's Theorem. In this talk, we will consider an analogue of the Kummer-Vandiver problem for these modules and we will present examples which give a negative answer to this problem. This talk is based on a joint work with L. Taelman.
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DATE: 23/11/11
Speaker: Teruyoshi Yoshida (Cambridge)
Title: The Hecke action on the weight spectral sequences
Abstract: We review the question of making algebraic correspondence act on the weight spectral sequence for l-adic cohomology of semistable schemes. To do this we need some intersection theory and cycle classes on regular schemes over the ring of integers. This approach works for the unitary Shimura varieties considered by Harris-Taylor/Shin (as the Hecke correspondences are finite and flat).
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DATE: 30/11/11
Speaker: Peter Scholze (Bonn)
Title: On the cohomology of compact unitary group Shimura varieties at ramified split places
Abstract: Generalizing our previous methods, we give a description of the cohomology of Shimura varieties for which the reductive group G is locally at p a product of general linear groups, allowing arbitrary signature at infinity and arbitrary ramification at p. As applications, we give a complete description of the semisimple local Hasse-Weil zeta function in terms of automorphic L-functions in nonendoscopic cases, and reconstruct the l-adic Galois representations attached to RACSD cuspidal automorphic representations, using endoscopic cases. This is joint work with Sug Woo Shin.
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DATE: 07/12/11
Speaker: Ambrus Pal (Imperial College)
Title: Around de Jong’s conjecture
Abstract: I will talk about one item of my current work in progress, which gives a new proof of de Jong’s conjecture in the rank two case, and a closely related analogue of Serre’s conjecture for function fields.
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DATE: 14/12/11
Speaker: Andreas Langer (Exeter)
Title: An integral structure on rigid cohomology
Abstract: For a quasiprojective smooth variety over a perfect field k of char p we introduce an overconvergent de Rham-Witt complex by imposing a growth condition on the de Rham-Witt complex of Deligne-Illusie using Gauus norms and prove that its hypercohomology defines an integral strcuture on rigid cohomology, i.e. its image in rigid cohomology is a canonical lattice. As a corollary we obtain that the integral Monsky-Washnitzer cohomology (considered before inverting p) of a smooth k-algebra is of finite type modulo torsion. This is joint work with Thomas Zink.
SPEAKER: Alan Lauder (Oxford)
TITLE: "Computations with classical and overconvergent modular forms"
ABSTRACT: We present p-adic algorithms for computing Hecke polynomials
and Hecke eigenforms associated to spaces of classical modular forms
using the theory of overconvergent modular forms.
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DATE: 4/5/11
SPEAKER: Christian Elsholtz (Graz University of Technology)
TITLE: "Sums of Unit Fractions"
ABSTRACT: We study the diophantine equation
$\frac{m}{n}= \frac{1}{x_1}+ \ldots +\frac{1}{x_k}$
(with positive integers $m,n,x_i$) and give a survey of several
questions in this area.
-----------------------------------
DATE: 11/5/11
SPEAKER: Victor Rotger (Universitat Politecnica de Catalunya)
TITLE: "Sato-Tate distributions and Galois endomorphism modules"
ABSTRACT: Let A be an abelian variety of dimension g over a number field k. For any
prime p of k at which A has good reduction let L_p(A,T) be the characteristic
polynomial of a Frobenius at p, unitarized in such way that its trace a_p(A) lies in
[-2g,2g].
One wonders how these traces are distributed in this interval as p
varies. When g=1, the classical results of Deuring and the recent
breakthrough of Clozel, Harris, Shepherd-Barron and Taylor on the
conjecture of Sato-Tate offer a pretty satisfactory answer to this
question. In a recent beautiful paper, Kedlaya and Sutherland explore
in detail next case g=2, proposing a quite precise conjectural
description of the fauna of distributions that can arise.
In this talk we shall review their work, offering a different point of
view and completing their picture at several points. This is a joint
project with F. Fite, K. Kedlaya and A. Sutherland.
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DATE: 18/5/11
SPEAKER: Gunther Cornelissen (Utrecht)
TITLE: "What determines a number field? - A view from quantum
statistical mechanics"
ABSTRACT: I will start with an overview of the history of (not)
reconstructing number field isomorphism from equality/isomorphism of
invariants such as zeta functions, adele rings and abelian/absolute
Galois groups. Then I will discuss joint work with Matilde Marcolli
that reconstructs isomorphism of global fields from isomorphism
of associated quantum statistical mechanical systems (which are
certain dynamical systems derived from class field theory),
and how this implies that abelian L-series determine the isomorphism
type of a global field.
-----------------------------------
DATE: 25/5/11
SPEAKER: Fred Diamond (KCL)
TITLE: TBA
ABSTRACT: TBA
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DATE: 1/6/11
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DATE: 8/6/11
SPEAKER: Burt Totaro (Cambridge)
TITLE: "Pseudo-abelian varieties"
ABSTRACT: Chevalley's theorem states that every smooth connected
algebraic group over a perfect field is an extension of an abelian
variety by a smooth connected affine group. That fails when the base
field is not perfect. We define a class of smooth connected groups over
an arbitrary field k called pseudo-abelian varieties; for k perfect,
these are simply abelian varieties. The definition is arranged so that
every smooth connected group over a field is an extension of a
pseudo-abelian variety by a smooth connected affine group, in a unique way.
We work out much of the structure of pseudo-abelian varieties. These
groups are closely related to unipotent groups in characteristic p and
to pseudo-reductive groups as studied by Tits and Conrad-Gabber-Prasad.
Many properties of abelian varieties such as the Mordell-Weil theorem
extend to pseudo-abelian varieties.
-----------------------------------
DATE: 15/6/11
SPEAKER: Wojciech Gajda (Adam Mickiewicz University, Poznan)
TITLE: "Independence of \ell-adic representations over function fields and abelian
varieties"
ABSTRACT: Let K be a finitely generated field extension of Q. We consider a family of
\ell-adic representations (\ell varies through rational primes) of the absolute Galois
group of K, attached to the \ell-adic cohomology of a fixed separated scheme of finite
type over K.
We prove that the fields cut out from the algebraic closure of K by the
kernels of the representations of the family are linearly disjoint over a finite
extension of K. This gives a positive answer to a question formulated by Serre in
1991.
In addition (if time permits) we will also discuss some recent calculations of images
of \ell-adic Galois representations attached to Tate modules of abelian varieties over
number fields, and applications to geometry and to arithmetic.
-----------------------------------
DATE: 22/6/11
SPEAKER: Daniel Macias Castillo (KCL)
TITLE: "On Higher-Order Stickelberger-Type Theorems"
ABSTRACT: We discuss a conjecture concerning the annihilation, as Galois modules, of
ideal class groups by explicit elements constructed from the values of higher order
derivatives of Dirichlet L-functions. We describe evidence in support of this
conjecture, including a full proof in several important cases.
Jan 26 Berhang Noohi (KCL) "Uniformization of orbifold Riemann surfaces"
Abstract: Riemann's uniformization theory says that there
are exactly three simply connected Riemann surfaces:
Riemann sphere, complex plane, and the hyperbolic plane.
Moreover, the quotient of a free (discrete) group action
on any of these will give rise to a Riemann surface, and
every Riemann surface is obtained this way. Now, if we
allow the action to be properly discontinuous but have
fixed points, the quotient will be an orbifold Riemann
surface. The question is whether all orbifold Riemann
surfaces are obtained this way. The answer is no. In this
talk we establish uniformization theory of orbifold
Riemann surfaces. We show that there is an orbifold
version of the spherical/Euclidean/hyperbolic trichotomy,
and show how to classify orbifold Riemann surfaces by
their uniformization type. There will be lots of examples
in this talk! (This is joint work with Kai Behrend.)
Feb 2nd Lynne Walling (Bristol) "Quadratic forms, representation numbers, and
Siegel-Eisenstein series."
Feb 9th Trevor Wooley (Bristol) "Vinogradov's mean value theorem, function fields,
and moduli spaces"
Feb 16th Martin Bright (Warwick)
Feb 23rd Minhyong Kim (UCL)
March 2nd Vadim Schechtman (Toulouse)
March 9th Jean Gillibert (Bordeaux) "On Tate-Shafarevich groups and class groups"
Abstract:
We discuss here a link between Shafarevich-Tate groups of certain
elliptic curves and class groups of the fields on which these are
defined. The curves we consider are endowed with a rational cyclic
isogeny of prime degree. Basically, the connection is established
via the so-called class group pairing (defined by Mazur and Tate).
We note that similar constructions have already been used by various
authors when performing descent, but with a different approach.
March 16th Christopher Daw (UCL)
March 23rd Alexei Skorobogatov (Imperial)
3 May Alan Lauder (Oxford) 10 May Carlos Castano-Bernard (Cambridge) 17 May Ben Smith (Royal Holloway) 24 May Rob de Jeu (Durham) 31 May Herbert Gangl (Durham) 7 June Guy Henniart (Paris) 14 June Otmar Venjakob (Bonn) 21 June 2.30pm Minhyong Kim (Purdue) 21 June 4.00pm Martin Taylor (Manchester) 26 July 2.00pm Doug Ulmer (U. Arizona, Tucson)
Title: Rank-2 Euler systems for non-ordinary symmetric squares
Title: perfectoid modular forms and a tilting isomorphism at the boundary of weight space
Abstract: Similarly to how complex modular forms are defined as functions on the complex upper half plane, Chojecki--Hansen--Johansson describe p-adic modular forms as functions on Scholze's perfectoid modular curve at infinite level. In this talk, we show that the appearance of perfectoid spaces in this context is not just a technical coincidence, but that this definition gives rise to 'perfectoid phenomena' appearing in the world of p-adic and classical modular forms. As an example of this, we discuss a tilting isomorphism of p-adic modular forms near the boundary of weight space which gives a new perspective on the space of T-adic modular forms defined by Andreatta--Iovita--Pilloni. This isomorphism can be explained by a theory of 'perfectoid modular forms' that we will also discuss in this talk.
Title: Affine Deligne-Lustig varieties
Abstract: Affine Deligne-Lusztig varieties are defined as certain subschemes of affine flag varieties using Frobenius-linear algebra. They are used in arithmetic geometry to describe the reduction of Shimura varieties. Motivated by this relation, I will report on recent geometric results describing affine Deligne-Lusztig varieties, and applications.
Title: Special cycles on orthogonal Shimura varieties
Abstract: Extending on the work of Kudla-Millson and Yuan-Zhang-Zhang, together with Yott we are constructing special cycles for a specific GSpin Shimura variety. We further construct a generating series that has as coefficients the cohomology classes corresponding to the special cycle classes on the GSpin Shimura variety and show the modularity of the generating series in the cohomology group over C.
Title: Geometric stabilization via p-adic integration
Abstract: The fundamental lemma is an identity of integrals playing an important role in the Langlands program. This identity was reformulated into a statement about the cohomology of moduli spaces of Higgs bundles, called the geometric stabilization theorem, and proved in this form by Ngô. I will give an introduction to these results and explain a new proof of the geometric stabilization theorem, which is joint work with Michael Groechenig and Dimitri Wyss, using the technique of p-adic integration.
Title: Restricting p-modular representations of $p$-adic groups to minimal parabolic subgroups
Abstract: Abstract: Given a prime integer $p$, a non-archimedean local field $F$ of residual characteristic $p$ and a standard Borel subgroup $P$ of $GL_2(F)$, Paskunas proved that the restriction to $P$ of (irreducible) smooth representations of $GL_2(F)$ over $\overline{\mathbb F}_p$ encodes a lot of information about the full representation of $GL_2(F)$ and that it may leads to useful statement about $p$-adic representations of $GL_2(F)$. Nevertheless, the methods used by Paskunas at that time heavily rely on the understanding of the action of certain spherical Hecke operator and on some combinatorics specific to the $GL_2(F)$ case. This can be carried to other specific quasi-split groups of rank 1, but this is not very satisfying. In this talk, I will report on a joint work with J. Hauseux. Using an different approach based on Emerton's ordinary parts functor, we get a more uniform context which shed a new light on Paskunas' results and allows us to get a natural generalization of these results for arbitrary rank 1 groups. In particular, we prove that for such groups, the restriction of supersingular representations to a minimal parabolic subgroup is always irreducible.
Title: On the stochasticity parameter of quadratic residues
Abstract: Let $U=\{0\leq u_1<u_2<\cdots<u_k<M\}$ be arbitrary subset of residues modulo $M$; set also $u_{k+1}:=M+u_1$. V.I.Arnold defined the stochasticity parameter of the set $U$ to be the quantity
$\sum_{i=1}^k(u_{i+1}−u_i)^2$ (the sum of squares of the distances between elements of $U$), and it turns out that too small or too large values of $S(U)$ indicate that $U$ is far from a random set: for a fixed $k$, $S(U)$ is minimal when the points of $U$ are equidistributed and $S(U)$ is maximal when $U$ is an interval. M.Z.Garaev, S.V.Konyagin and Yu.V.Malykhin studied the stochasticity parameter of quadratic residues modulo a prime and showed that it is asymptotically equal to the stochasticity parameter of a random set of the same size. We turn to this problem arbitrary modulo $M$ and prove the same asymptotics for a set of moduli of positive lower density; we are also able to show that for these moduli the parameter of quadratic residues is in fact less than the parameter of a random set of the same size. Also we will discuss how (potentially) this result can be extended for almost all moduli.
Title: Vanishing theorems for Shimura varieties at unipotent level
Abstract: We prove a vanishing result for the compactly supported cohomology of certain infinite level Shimura varieties. More specifically, if $X_{K_pK^p}$ is a Shimura variety of Hodge type for a group $G$ that becomes split over $\mathbb{Q}_p$, and $K_p$ is a unipotent subgroup of $G(\mathbb{Q}_p)$, then the compactly supported $p$-adic etale cohomology of $X_{K_pK^p}$ vanishes above the middle degree. We will also give an application to eliminating the nilpotent ideal in the construction of certain Galois representations. This talk is based on joint work with Ana Caraiani and Christian Johansson and on joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih.
Title: Average bounds for l-torsion in class groups.
Abstract: Let l be a positive integer. We discuss average bounds for the l-torsion of the class group for some families of number fields, including degree-d-fields for d between 2 and 5. Refinements of a strategy due to Ellenberg, Pierce and Wood lead to significantly improved upper bounds on average. The case d=2 implies the currently best known upper bounds for the number of D_p - fields of bounded discriminant, for odd primes p. This is joint work with Martin Widmer. (The results presented here are different from those presented by Martin Widmer in his talk with a similar title in Jan 2018.)
Title: Parity of Selmer ranks in quadratic twist families.
Abstract: We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we prove results about the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun–Mazur– Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich–Tate group (if finite) may have order twice a square.
Title: Vanishing theorems for étale sheaves
Abstract: The talk is based on two results: Scholze’s Artin type vanishing theorem for the projective space, which I proved without perfectoid geometry (which implies in particular that it holds in positive characteristic), and a rigidity theorem for subloci of the l-adic character variety stable under the Galois group over a number field (joint work in progress with Moritz Kerz).
Title: Automorphism groups of K3 surfaces over nonclosed fields
Title: Fourier analysis on universal formal covers
Abstract: : The p-adic Fourier transform of Schneider and Teitelbaum has complicated integrality properties which have not yet been fully understood. I will report on an approach to this problem relying on the universal formal cover of a p-divisible group as introduced by Scholze and Weinstein. This has applications to the representation theory of p-adic division algebras.
Title: p-adic L-functions of Hilbert cusp forms and the trivial zero conjecture
Abstract: In a joint work with Daniel Barrera and Andrei Jorza, we prove a strong form of the trivial zero conjecture at the central point for the p-adic L-function of a non-critically refined cohomological cuspidal automorphic representation of GL(2) over a totally real field, which is Iwahori spherical at places above p. We will focus on the novelty of our approach in the case of a multiple trivial zero, where in order to compute higher order derivatives of the p-adic L-function, we study the variation of the root number in partial finite slope families and establish the vanishing of many Taylor coefficients of the p-adic L-function of the family.
Title: Symmetries and spaces [Inaugural lecture]
Abstract: It is a long established idea in mathematics that in order to understand space we need to study its symmetries. This is the centrepoint of the Erlangen program, which, published by Felix Klein in 1872 in Vergleichende Betrachtungen über neuere geometrische Forschungen, is a method of characterizing geometries based on group theory. In a group we can multiply, while on a space we can integrate. I will explore the link between the two starting with the mathematics of the seventeenth century and leading to the arithmetic of elliptic curves.
Title: Rational points over global fields and applications.
Abstract: We present analytic methods for counting rational points on varieties defined over global fields. The main ingredient is obtaining a version of Hardy-Littlewood circle method which incorporates elements of Kloosterman refinement in new settings.
Title: How many real Artin-Tate motives are there?
Abstract: The goals of my talk are 1) to place this question within the framework
of tensor-triangular geometry, and 2) to report on joint work with Paul
Balmer (UCLA) which provides an answer to the question in this
framework.
Title: The sup-norm problem over number fields.
Abstract: In this talk we study the sup-norm of automorphic forms over number fields. This topic sits on the intersection of Quantum chaos, harmonic analysis and number theory and has seen a lot progress lately. We will discuss some of the recent result in the rank one setting.
Title: Singular moduli for real quadratic fields and p-adic mock modular forms
Abstract: The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and the theory of weak harmonic Maass forms.
Title: The eigencurve at Eisenstein weight one points
Abstract: Coleman and Mazur constructed the eigencurve, a rigid analytic space classifying p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is better understood at points corresponding to cuspforms of weight greater than 1, while the weight one case is far more intricate. In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. We focus on the unusual phenomenon of cuspidal Hida families specializing to Eisenstein series at weight one. We discuss the relation between the geometry of the eigencurve and the Gross-Stark Conjecture.
Title: Low moments of character sums
Abstract: Sums of Dirichlet characters $\sum_{n \leq x} \chi(n)$ (where $\chi$ is a character modulo some prime $r$, say) are one of the best studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments $\frac{1}{r-1} \sum_{\chi \text{mod } r} |\sum_{n \leq x} \chi(n)|^{2q}$, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when $0 \leq q \leq 1$. I will focus mainly on the number theoretic issues arising.
Title: Euler systems for Siegel modular forms
Abstract: Euler systems are compatible families of cohomology classes attached to global Galois representations, which play a fundamental role in relating values of L-functions to arithmetic. I will sketch the construction of an Euler system for the spin representation attached to genus 2 Siegel modular forms. The construction reveals a surprising link to branching laws in local representation theory and the Gan-Gross-Prasad conjecture. This is joint work with D. Loeffler and C. Skinner.
(This week the seminar will be in Huxley room 213)
Title: Serre's uniformity question and the Chabauty-Kim method for modular curves
Abstract: Serre's uniformity question asks which Galois representations can arise from the $p$-torsion of an elliptic curve over $\mathbf{Q}$. Equivalently, it can be viewed as a question about rational points on certain modular curves. In this talk, I will explain what is known about the problem, and describe some recent joint work with Samuel Le Fourn and Samir Siksek on understanding these rational points via the Chabauty-Kim method.
Title: Diagrams in the mod $p$ cohomology of Shimura curves.
Abstract: In search of a local mod $p$ Langlands correspondence, one can study globally defined representations that should correspond to a given local Galois representation: for example, those arising from completed cohomology or from spaces of algebraic modular forms. Then there's the issue of proving that these representations are independent of the global context. I will present some recent progress on this problem for mod $p$ representations of the group $\mathrm{GL}(2)$ over finite unramified extensions of $\mathbf{Q}_p$, answering a question of Breuil about an analogue of Colmez's functor. This is joint work with Daniel Le.
Title: On some consequences of a theorem of J. Ludwig
Abstract: We prove some qualitative results about the $p$-adic Jacquet--Langlands
correspondence defined by Scholze, in the $\mathrm{GL}(2,\mathbf{Q}_p)$, residually reducible
case, by using a vanishing theorem proved by Judith Ludwig. In particular, we
show that in the cases under consideration the $p$-adic Jacquet--Langlands
correspondence can also deal with principal series representations in a
non-trivial way, unlike its classical counterpart. The paper is available at
http://arxiv.org/abs/1804.07567.
Title: Variation of Iwasawa invariants in residually reducible Hida families
Abstract: We'll discuss a work in progress describing properties of $p$-adic $L$-functions of a modular form whose Galois representation is residually reducible. As an application, we prove cases of a conjecture of Greenberg about $\mu$-invariants of Selmer groups. This is joint work with Rob Pollack.
Title: Venkatesh's conjecture for modular forms of weight one
Abstract: Akshay Venkatesh and his coauthors (Galatius, Harris, Prasanna) have recently introduced a derived Hecke algebra and a derived Galois deformation ring acting on the homology of an arithmetic group, say with $p$-adic coefficients. These actions account for the presence of the same system of eigenvalues simultaneously in various degrees. They have also formulated a conjecture describing a finer action of a motivic group which should preserve the rational structure $H^i(\Gamma,\mathbf{Q})$. In this lecture we focus in the setting of classical modular forms of weight one, where the same systems of eigenvalues appear both in degree 0 and 1 of coherent cohomology of a modular curve, and the motivic group referred to above is generated by a Stark unit. In joint work with Darmon, Harris and Venkatesh, we exploit the Theta correspondence and higher Eisenstein elements to prove the conjecture for dihedral forms.
Title: Shimura curves and Ihara's lemma
Ihara's lemma is a statement about the structure of the mod l cohomology of modular curves that was the key ingredient in Ribet's results on level raising. I will motivate and explain its statement, and then describe joint work with Jeffrey Manning on its extension to Shimura curves.
Title: Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives
Abstract: In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin-Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof. If time allows, I will also explain some key geometric ingredients in the proof, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.
Title: A New Northcott Property for Faltings Height
Abstract: The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.
Title: Representations of p-adic groups
Abstract: In the 1990s Moy and Prasad revolutionized p-adic representation theory by showing how to use Bruhat-Tits theory to assign invariants to p-adic representations. The tools they introduced resulted in rapid advancements in both representation theory and harmonic analysis -- areas of central importance in the Langlands program. A crucial ingredient for many results is an explicit construction of (types for) representations of p-adic groups. In this talk I will indicate why, survey what constructions are known (no knowledge about p-adic groups assumed) and present recent developments based on a refinement of Moy and Prasad's invariants.
Title: Quantum unique ergodicity in almost every shrinking ball
Abstract: I will discuss the problem of small scale equidistribution of Hecke-Maass eigenforms, namely the problem of the rate at which hyperbolic balls can shrink as the Laplacian eigenvalue tends to infinity for which the Laplacian eigenfunctions still equidistribute on these balls. There is a natural barrier - the Planck scale - for which equidistribution fails, but conditionally equidistribution occurs in almost every shrinking ball at every larger scale. I will also discuss related small scale equidistribution problems for geometric invariants associated to quadratic fields.
Title: Moments of cubic L-functions over function fields
Abstract: I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula with a (maybe a little surprising) main term, which relies on using results from the theory of metaplectic Eisenstein series about cancellation in averages of cubic Gauss sums over functions fields.
Title: Sign changes of Fourier coefficients of half-integral weight modular forms.
Abstract: For a square-free integer n, Waldspurger showed that square of the nth Fourier coefficient of a half-integral weight Hecke cusp form is proportional to the central value of an L-function. It remains to understand the sign of the coefficient. In this talk I will discuss joint work with Maks Radziwill (McGill) in which we study the number of sign changes of coefficients of such forms.
Title: Title: A dynamical approach to common torsion points
Abstract: Bogomolov-Fu-Tschinkel conjectured that there is a uniform upper bound on the number of common torsion points of two nonisomorphic elliptic curves (more precisely, on the number of common images of torsion points when the curve is presented as a double cover of the Riemann sphere). This is an example of the phenomenon of unlikely intersections in arithmetic geometry. I will discuss a dynamical approach to this conjecture via Lattès maps of the Riemann sphere associated to an elliptic curve. I will report on recent progress on this dynamical approach (joint with Laura DeMarco and Hexi Ye) and formulate a more general dynamical conjecture.
Title: Semisimplicity of certain Galois representations occurring in étale cohomology of unitary Shimura varieties
Abstract: Conjecturally, the category of pure motives over a finitely generated field $k$ should be semisimple. Consequently, $\ell$-adic étale cohomology of a smooth projective variety over $k$ should be a semisimple representation of the absolute Galois group of $k$. This was proved by Faltings for $H^1$, as a consequence of his proof of Tate's conjecture. In this talk, which is based on a joint work with K. Fayad, I am going to explain a proof of the semisimplicity of the Galois action on a certain part of étale cohomology of unitary Shimura varieties. The most satisfactory result is obtained for unitary groups of signature $(n,0)^a \times (n-1,1)^b \times (1,n-1)^c \times (0,n)^d$.
Title: (Topological) cyclic homology and p-adic Hodge theory
Abstract: Cyclic homology was introduced by Connes and Feigin—Tsygan in the 1980s as a extension of de Rham cohomology to singular varieties or even to non-commutative spaces in characteristic zero. I will overview the classical theory (in particular, familiarity with cyclic homology is not expected) and then explain how its analogue over the sphere spectrum is similarly related to various p-adic cohomology theories. Joint work with Bhargav Bhatt and Peter Scholze.
Title: Galois representations for the general symplectic group.
Abstract: In a recent preprint with Sug Woo Shin arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain this result and some parts of the construction.
Title: Slopes of Hilbert modular forms near the boundary of weight space.
Abstract: Recent work of Liu–Wan–Xiao has proven in many cases how slopes of modular forms behave near the boundary of weight space, giving us insights into the geometry of the associated eigenvarieties. One can ask if there is similar behaviour in the case of Hilbert modular forms. I will discuss some conjectures on how the slopes should behave near the boundary as well as explaining why the methods of Liu–Wan–Xiao do not appear to extend to the Hilbert case. Lastly, I will discuss some recent examples where it is possible to partially prove these conjectures in the case when chosen prime is inert.
Title: Spins of ideals and arithmetic applications to one-prime-parameter families
Abstract: We will define three similar but different notions of "spin" of an ideal in a number field, and we will show how a number-field version of Vinogradov's method (a sieve involving "sums of type I" and "sums of type II") can be used to prove that spins of prime ideals oscillate. Such equidistribution results have applications to the distribution of 2-parts of class groups of quadratic number fields in thin families parametrized by prime numbers. Parts of this talk are joint work with Peter Koymans.
Title: Purity for the Brauer group
Abstract: A purity conjecture due to Grothendieck and Auslander--Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension \ge 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic (0, p). We will discuss an approach to the mixed characteristic case via the tilting equivalence for perfectoid rings.
Title: Homotopy Lie theory and modular j-invariant.
Abstract: The speaker developed homotopy Lie theory for smooth projective algebraic varieties with his coauthors. Here homotopy Lie theory means that we can explicitly construct DGBV (Differential Gerstenhaber-Batalin-Vilkovisky) algebra, a specific type of homotopy Lie algebra, which computes the cohomologies of algebraic varieties and their Hodge structures. This theory is algebraic and algorithmic and it turns out that Dwork’s p-adic cohomology theory can be captured in the formalism of homotopy Lie theory. Moreover, it also plays a crucial role in the mirror symmetry conjecture. In this talk, we explain this formalism in the case of projective smooth hypersurfaces and give an arithmetic application to periods of elliptic curves. More specifically, for given a complex number N, we give an explicit algorithm to compute the period of the elliptic curve whose j-invariant is N, based on the deformation theory of DGBV algebras. The work on modular j-invariants is a joint work with Kwang Hyun Kim and Yesule Kim.
Title: Average bounds for the $\ell$-torsion in class groups of number fields
Abstract: Let $\ell$ and $d$ be integers >1. By the Cohen-Lenstra-Martinet heuristics the $\ell$-torsion part of the class groups of degree $d$ number fields should be "very small" in terms of the discriminant for "almost all" such fields. However, non-trivial such bounds for all $\ell$ are known only for $d\leq 5$ due to recent work of Ellenberg, Pierce, and Wood. We explain their strategy, how one can improve their bounds for $d=4,5$, and we also present analogous results for certain
families of arbitrarily large degree. (This is joint work with Christopher Frei).
Title: Multiplicative functions in short intervals revisited
Abstract: A couple of years ago Maksym Radziwill and I showed that the average of a multiplicative function in almost all very short intervals is close to its average on long intervals. This result has found many applications. In a work in progress that I will talk about, Radziwill and I revisit the problem and generalise the result to functions which vanish often as well as improve on the upper bound for the number of possible exceptional intervals. This new work has applications for instance to the gaps between numbers that can be represented as a sum of two squares.
Title: Strongly modular models of $\mathbb{Q}$-curves
Abstract: A strongly modular $\mathbb{Q}$-curve is a non-CM elliptic curve over a number field whose L-function is a product of L-functions of classical weight 2 newforms. We address the problem of deciding when an elliptic curve has a strongly modular model, showing that this holds precisely when the curve has a model that is completely defined over an abelian number field. The proof relies on Galois cohomological methods. When the curve is defined over a quadratic or biquadratic field, we show how to find all of its strongly modular twists, using exclusively the arithmetic of the base field. This is joint work with Peter Bruin.
Title: Frey curves, short character sums and a problem of Erdős
Abstract: Consider the following Diophantine problem:
\[
n(n+d)(n+2d)\cdots (n+(k-1)d)=y^\ell, \qquad \gcd(n,d)=1,
\]
where $n$, $d$, $y$ are integers and the exponent $\ell$ is prime.
There are obvious solutions with $y=0$ or $d=0$. A long-standing
conjecture of Erdős states that if $k$ is suitably large then the
only solutions are the obvious ones. We show that if $k$ is suitably
large then either the solution is one of the obvious ones, or $\ell<
\exp(10^k)$.
Our methods include Frey curves and Galois representations,
the prime number theorem for Dirichlet characters, results on
exceptional zeros of Dirichlet L-functions, the large sieve, and
Roth-like theorems on the existence of 3-term arithmetic progressions
in certain sets. This is joint work with Mike Bennett.
Title. Bohr sets and multiplicative diophantine approximation
Abstract. In two dimensions, Gallagher's theorem is a strengthening of the Littlewood conjecture that holds for almost all pairs of real numbers. We prove an inhomogeneous fibre version of Gallagher's theorem, sharpening and making unconditional a result recently obtained conditionally by Beresnevich, Haynes and Velani. The idea is to find large generalised arithmetic progressions within inhomogeneous Bohr sets, extending a construction given by Tao. This precise structure enables us to verify the hypotheses of the Duffin--Schaeffer theorem for the problem at hand, via the geometry of numbers.
Title: A non-abelian algebraic criterion for good reduction of curves
Abstract: For a family of proper hyperbolic complex curves
$f: X \longrightarrow \Delta^*$ over a puntured disc $\Delta^*$ with semistable reduction at the
center, Oda proved, with transcendental methods, that the outer
monodromy action of $\pi_1(\Delta^*) \cong \mathbb{Z}$ on the classical unipotent
fundamental group of the generic fiber of $f$ is trivial if and only if f has
good reduction at the center. In this talk I explain a joint project
with B. Chiarellotto and A. Shiho in which we give a purely algebraic
proof of Oda's result.
Title: Fargues' conjecture in the GL_2-case.
Abstract: Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.
Title: Arithmetic statistics of higher degree L-functions over function fields.
Abstract: A classical problem in number theory is to evaluate the number of primes in an arithmetic progression. This problem can be formulated in terms of the von Mangoldt function. I will introduce some conjectures concerning the fluctuations of the von Mangoldt function in arithmetic progressions. I will also introduce an analogous problem in the function field setting and discuss its generalization to arithmetic functions associated with higher degree L-functions (in the limit of large field size). The main example we will discuss is an elliptic curve L-function and statistics associated with its coefficients. This is a joint work with Chris Hall and Jon Keating.
Title: Theta operators for unitary modular forms.
Abstract: This is joint work with Ehud De Shalit (Hebrew University). We shall consider p-adic modular forms on a unitary Shimura variety associated to a quadratic imaginary field, where p is inert in the field, and its mod p reduction. In this case, theta operators were constructed by Eischen and Mantovan, and by De Shalit-Goren, independently and using different approaches. I will describe our approach that makes heavy use of Igusa varieties. The main two theorems are a formula for the effect of a theta operator on q-expansions and its analytic continuation from the ordinary locus to the whole Shimura variety in characteristic p. Along the way interesting questions about filtrations of automorphic vector bundles arise and, to the extent time allows, I will discuss these questions in light of our work on foliations on unitary Shimura varieties.
Title: The universal $p$-adic Gross-Zagier formula.
Abstract: Around 1986 three great theorems were proved: Gross and Zagier related Heegner points on elliptic curves to derivatives of L-functions; Waldspurger related toric periods of automorphic forms to special L-values; and Hida showed that ordinary modular forms and their L-functions vary in $p$-adic families.
I will explain how the spirits of the second and third results can be infused into the first one. The outcome is a $p$-adic Gross-Zagier formula, valid for Hida families of cuspforms for the group $GL(2)\times U(1)$ over a totally real field. Combined with work of Fouquet, it has applications to the $p$-adic Bloch-Kato conjecture for the Selmer groups of such forms.
Schneider (1530-1530): Character varieties and $(\varphi_L,\Gamma_L)$-modules.
Abstract: After reviewing old work with Teitelbaum, in which we constructed the character variety $X$ of the additive group $o_L$ in a finite extension $L/Q_p$ and established the Fourier isomorphism for the distribution algebra of $o_L$, I will briefly report on more recent work with Berger and Xie, in which we establish the theory of $(\varphi_L,\Gamma_L)$-modules over $X$ and related it to Galois representations. Then I will discuss an ongoing project with Venjakob. Our goal is to use this theory over $X$ for Iwasawa theory.
Abstract: We will describe a theory of p-adic families of higher coherent cohomology classes for the group GSp_4/Q together with some arithmetic applications to the Hasse-Weil Zeta function of genus 2 curves (joint with G. Boxer, F. Calegari, T. Gee) or to the construction of Galois representations.
Title: Unlikely intersections in families of abelian varieties and some polynomial Diophantine equations
Abstract: What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension $r$ and $s$ are non "likely" to intersect if $r <\,$codim $s$, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. After a small survey on these problems, I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of curves in families of abelian varieties. This gives also applications to the study of the solvability of the so called "almost-Pell" equations, generalising some results due to Masser and Zannier.
Title: Integral representation and critical $L$-values for the standard $L$-function of a Siegel modular form
Abstract: I will talk about some of my recent work with Pitale and Schmidt where we prove an explicit pullback formula that gives an integral representation for the twisted standard $L$-function for a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. Further, by specializing our integral representation to the case $n = 2$, we prove an algebraicity result for the critical $L$-values in that case (generalizing previously proved critical-value results for the full level case).
Title: Non-holomorphic modular forms for $\mathrm{SL}_2(\mathbb{Z})$
Abstract: I will discuss elementary properties of a class of non-holomorphic modular forms for the full modular group $\mathrm{SL}_2(\mathbb{Z})$. It contains the classical holomorphic theory, but is completely distinct from Maass' theory of harmonic modular forms. This class is related to the theory of mixed motives, mapping class groups, mock modular forms, and string theory. I may mention one or two of these connections.
Title: A local model for the triangulate variety and applications to $p$-adic automorphic forms
Abstract: In joint work with C. Breuil and B. Schraen we prove (under mild additional hypothesis) that $p$-adic automorphic forms on eigenvarieties for definite unitary groups are classical if their associated $p$-adic Galois representation is crystalline at places dividing $p$.
In the same setup we further determine all non-classical overconvergent forms of finite slope that give rise to the same Galois representation as a classical automorphic form.
These results rely on a close analysis of the local geometry of a space parametrizing $p$-adic Galois representations with a certain prescribed behaviour at $p$ - so called triangulate representations. We study this geometry by proving that the space is smoothly equivalent to a certain variety showing up in geometric representation theory.
Title: A quotient of the Lubin-Tate tower
Abstract: In this talk I will report on joint work with Christian Johansson. The aim of our project is to construct a quotient of an infinite level Lubin-Tate space by a certain parabolic subgroup of $\mathrm{GL}(n,F)$ ($F/ \mathbb{Q}_p$ finite) as a perfectoid space.
The motivation for constructing this quotient is as follows. As I will explain in the talk, Scholze recently constructed a candidate for the mod $p$ Jacquet-Langlands correspondence and the mod $p$ local Langlands correspondence for $\mathrm{GL}(n, F)$. Given a smooth admissible representation $\pi$ of $\mathrm{GL}(n, F)$, the candidate for these correspondences is given by the etale cohomology groups of the adic projective space $\mathbb{P}^{n-1}$ with coefficients in a sheaf $F_\pi$ that one constructs from $\pi$.
The finer properties of this candidate remain mysterious.
As an application of the quotient construction one can show a vanishing result for some of these cohomology groups $\mathrm{H}^i_{et}(\mathbb{P}^{n-1},F_\pi)$.
Title: Jordan decomposition for $\ell$-blocks of $p$-adic groups
Abstract: Not much is known on the $\ell$-modular or $\ell$-integral representation theory of $p$-adic groups, beyond the case of $GL_n$. Even worse, the main property used by Vigneras in her treatment of the $GL_n$ case is now known to generally fail for other groups. Inspired by the theory of ``Jordan decomposition'' for $\ell$-blocks of finite reductive groups, we have conjectured the existence of a decomposition of $\ell$-integral representations into factors parametrized by Langlands parameters with source the prime-to-$\ell$ inertia subgroup, and which obeys some version of the Langlands functoriality principle associated to a morphism of $L$-groups. I will discuss recent progress on this conjectural picture, both for depth 0 and for positive depth representations.
Title: Variations of invariants of reductive group actions
Abstract: Whenever a reductive group acts on a variety there are certain fundamental invariants of this action: complexity,
weight lattice, valuation cone. I will explain these notions
and report on recent work how these invariants vary
in families.
Title: Life after the Langlands--Tunnell theorem
Abstract: The theorem of the title states that if $K$ is a number field, then any representation $\rho : G_K \to \mathrm{GL}_2(\mathbb{C})$ with projective image $S_4$ arises from automorphic forms.
Wiles famously used this theorem in the case $K = \mathbb{Q}$, together with a group-theoretic coincidence, to establish the automorphy of the mod 3 representations attached to elliptic curves over $\mathbb{Q}$. The rest of the story you know.
We will discuss what goes wrong with this argument when studying elliptic curves over more general number fields, and what one can do instead.
Title: $p$-adic Asai L-functions of Bianchi modular forms
Abstract: The Asai (or twisted tensor) L-function attached to a Bianchi modular form is the 'restriction to the rationals' of the standard L-function. Introduced by Asai in 1977, subsequent study has linked its special values to the arithmetic of the corresponding form. In this talk, I will discuss joint work with David Loeffler in which we construct a $p$-adic Asai L-function -- that is, a measure on $\mathbb{Z}_p^{*}$ that interpolates the critical values $L^{As}(f,\chi,1)$ -- for ordinary weight 2 Bianchi modular forms. The method makes use of techniques from the theory of Euler systems, namely Kato's system of Siegel units, building on the rationality results of Ghate.
Title: Ordinary representations and locally analytic socle for $\mathrm{GL}_n(\mathbb{Q}_p)$
Abstract: Suppose that $\rho$ is an irreducible automorphic $n$-dimensional global $p$-adic Galois representation that is upper-triangular locally at $p$. In previous work with Breuil we constructed a unitary representation of $\mathrm{GL}_n(\mathbb{Q}_p)$ on a $p$-adic Banach space (depending only on $\rho$ locally at $p$) that is an extension of finitely many principal series, and we conjectured that this representation occurs globally in a space of $p$-adic automorphic forms cut out by $\rho$. In work in progress we prove many new cases of this conjecture, assuming that $\rho$ is moreover crystalline.
Title: Cuspidal cohomology of stacks of shtukas
Abstract: We will talk about the $\ell$-adic cohomology of the classifying stacks of shtukas for a constant split reductive group over a function field. We will construct the constant term morphisms on the cohomology groups. And we will show that the cuspidal cohomology, defined as the intersection of the kernels of these constant term morphisms, is of finite dimension and equals to the Hecke-finite cohomology defined by V. Lafforgue. The essential ingredients are the compatibility of the geometric Satake equivalence with the constant term functor and the relative contractibility of deep enough Harder-Narasimhan strata in the stacks of shtukas.
Title: Local $\varepsilon$-isomorphisms in families
Abstract: Given a representation of $Gal_{Q_p}$ with coefficients in a $p$-adically complete local ring $R$, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex. When $R=Z_p$ and the representation is a lattice in a de Rham representation, this trivialization should be related to the $\varepsilon$-factor of the corresponding Weil--Deligne representation. Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations. This is joint work with Otmar Venjakob.
Title: Arithmetic of hyperelliptic curves over local fields
Abstract: Let $C:y^2 = f(x)$ be a hyperelliptic curve over a local field $K$ of odd residue characteristic. I will explain how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of $f(x)$. This is joint work with Tim Dokchitser, Celine Maistret and Adam Morgan.
Title: A new upper bound on Skewes' number
Abstract: The Prime Number Theorem tells us that the logarithmic integral, $li(x)$, is a good approximation to $\pi(x)$, the number of primes up to x. Numerically it always seems to be an overestimate, so $\pi(x)-li(x)$ is negative. The first point where this ceases to be the case is known as Skewes' number whose true value is as yet unknown. I will report on joint work with Chris Smith and Dave Platt, where we improve the best upper bound on Skewes' number.
Title: Pair correlation for quadratic polynomials mod 1.
Abstract: It is an open question whether the fractional parts of nonlinear polynomials at integers have the same fine-scale statistics as a Poisson point process. We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.
Title: A Gross-Zagier formula for a certain anticyclotomic p-adic L-function of a rational elliptic curve.
Abstract: Let E be a (semistable) rational elliptic curve of conductor N, let K be an imaginary quadratic field satisfying a "Heegner hypothesis" relative to N and let p be a prime of split multiplicative reduction for E that splits in K. Following a recipe proposed by Bertolini and Darmon, I will define a p-adic L-function L_p(E/K) in terms of distributions of Heegner points on Shimura curves that are rational over the anticyclotomic Z_p-extension of K. The "special value" of L_p(E/K) is 0, and I will sketch a proof of a Gross-Zagier formula for the first derivative of L_p(E/K) involving a Heegner point over K and a p-adic L-invariant of E à la Mazur-Tate-Teitelbaum. The strategy is based on level raising arguments and Jochnowitz-type congruences. This is joint work (in progress) with Rodolfo Venerucci.
Title: Geometry engendered by G-Zips: Shimura varieties and beyond.
Abstract: Moonen, Pink, Wedhorn and Ziegler initiated a theory of
G-Zips, which is modeled on the de Rham cohomology of varieties in
characteristic p>0 "with G-structure", where G is a connected
reductive F_p-group. Building on their work, when X is a good
reduction special fiber of a Hodge-type Shimura variety, it has been
shown that there exists a smooth, surjective morphism \zeta from X to
a quotient stack G-Zip^{\mu}. When X is of PEL type, the fibers of
this morphism recover the Ekedahl-Oort stratification defined earlier
in terms of flags by Moonen. It is commonly believed that much of the
geometry of X lies beyond the structure of \zeta.
Title: Higher dimensional Steinhaus problems.
Abstract: The three gap theorem (or Steinhaus conjecture) asserts that there are
at most three distinct gap lengths in the fractional parts of the
sequence $\alpha, 2\alpha,\ldots, N\alpha$, for any integer $N$ and real number $\alpha$. This
statement was proved in the 1950s independently by various authors. In
this talk I will explain a different approach, which is based on the
geometry of the space of two-dimensional Euclidean lattices (with
Andreas Strombergsson, Amer. Math. Monthly, in press). This approach
can in fact be generalised to deal with analogous higher dimensional
Steinhaus problems for gaps in the fractional parts of linear forms.
Here we are able to shed new light on a question of Erdos, Geelen and
Simpson, proving the existence of parameters for which the number of
distinct gaps is unbounded (joint work with Alan Haynes).
Title: On the arithmetic of simple singularities of type E.
Abstract: Given a simply laced Dynkin diagram, one can use Vinberg theory of graded Lie algebras to construct a family algebraic curves. In the case when the diagram is of type E7 or E8, Jack Thorne and I have used the relationship between these families of curves and their associated Vinberg representations to gain information about integral points on the curves. In my talk, I'll focus on the role Lie theory plays in the construction of the curves and in our proofs.
Title: On the characteristic polynomial of Gross's regulator matrix.
Abstract: Let $F$ be a totally real field and $\chi$ a totally odd character of $F$. Gross conjectured that the leading term of the Deligne-Ribet $p$-adic $L$-function associated to $\chi$ at $s=0$ is equal to a $p$-adic regulator of $p$-units in the extension of $F$ cut out by $\chi$. I recently proved this result in joint work with Mahesh Kakde and Kevin Ventullo. The topic of this talk is a refinement of Gross's conjecture. I will propose an analytic formula for the characteristic polynomial of Gross's regulator matrix, rather than just its determinant. The formula is given in terms of the Eisenstein cocycle and in fact applies (conjecturally) to give all the principal minors of Gross's matrix. For the diagonal entries, the conjecture overlaps with the conjectural formula presented in prior work. This is joint work with Michael Spiess.
Title: Arithmetic statistics of modular symbols.
Abstract: Mazur, Rubin, and Stein have recently formulated a series of
conjectures about statistical properties of modular symbols in order
to understand central values of twists of elliptic curve L-functions.
Two of these conjectures relate to the asymptotic growth of the first
and second moments of the modular symbols. We prove these on average
by using analytic properties of Eisenstein series twisted by modular
symbols. Another of their conjectures predicts the Gaussian
distribution of normalized modular symbols. We prove a refined version
of this conjecture.
This is joint work with Yiannis Petridis.
Title: Recent progress on the Iwasawa theory of elliptic curves and modular forms.
Abstract: This talk will describe some of the recent work on the Iwasawa theory of modular forms (at both ordinary and non-ordinary primes) with an emphasize on the strategy of proof, which involves two different main conjectures.
Title: An Euler system for a pair of CM modular forms.
Abstract: Given a pair of modular forms and a prime p, Lei-Loeffler-Zerbes have constructed an Euler system for the tensor product of the p-adic Galois representations attached to each of the forms. When the forms have CM by distinct imaginary quadratic fields, this representation is induced from a character $\chi$ over an imaginary biquadratic field F. I will explain how one can use this Euler system to obtain upper bounds for Selmer groups associated to $\chi$ over the $\mathbf{Z}_p^3$-extension of F.
Title: The Hasse norm principle for abelian extensions
Abstract: Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^\times$ and $K^\times$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^\times\to K^\times$ on $L^\times$. Thus, if an element of $K^\times$ is a norm from $L^\times$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^\times$ which is a norm from $J_L$ is in fact a norm from $L^\times$. The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis. This is joint work with Christopher Frei and Daniel Loughran.
Title: K3 surfaces and a construction of a Shimura variety
Abstract: In old times, elliptic modular functions appeared in the study of elliptic curves. They are applied to the construction of class fields. (This is classically called Kronecker's Jugendtraum.) K3 surfaces are 2-dimensional analogy of elliptic curves. In this talk, the speaker will present an extension of the classical result by using K3 surfaces. Namely, we will obtain Hilbert modular functions via the periods of K3 surfaces and construct a certain model of a Shimura variety explicitly.
Title: Integral models for eigenvarieties
Abstract: I will discuss a construction of integral models of eigenvarieties using a generalization of the overconvergent distribution modules of Ash and Stevens, and their relation to recent work of Andreatta-Iovita-Pilloni and Liu-Wan-Xiao on the geometry of the Coleman-Mazur eigencurve near the boundary of weight space. This is joint work with James Newton.
Alan Lauder (Oxford)
Title: Stark points on elliptic curves and modular forms of weight one
Abstract: I shall discuss some work with Henri Darmon and Victor Rotger on the explicit construction of points on elliptic curves. The elliptic curves are defined over $\mathbb{Q}$, and the points over fields cut out by Artin representations attached to modular forms of weight one.
Erick Knight (Harvard/Bonn)
Title: A p-adic Jacquet-Langlands correspondence
Abstract: In this talk, I will construct a p-adic Jacquet-Langlands correspondence, which is a correspondence between Banach space representations of GL2(Qp) and Banach space representations of the unit group of the quaternion algebra D over Qp. The correspondence satisfies local-global compatibility with the completed cohomology of Shimura curves, as well as a compatibility with the classical Jacquet-Langlands correspondence, in the sense that the $D^\times$ representations can often be shown to have the expected locally algebraic vectors.
Title: Standard conjectures for abelian fourfolds
Abstract: Let X be a smooth projective variety and V be the finite dimensional Q vector space of algebraic cycles on X modulo numerical equivalence. Grothendieck defined a quadratic form on V (basically using the intersection product) and conjectured that it is positive definite. This conjecture is a formal consequence of Hodge Theory in characteristic zero, but almost nothing is known in positive characteristic. Instead of studying this quadratic form at the non archimedean place (the signature) we will study it at the p-adic places. It turns out that this question is more treatable. Moreover, using a product formula formula, the p-adic information will give us non trivial informations on the non archimedean place. For instance we will show the original conjecture when X is an abelian variety of dimension 4.
Title: Parity of ranks of abelian surfaces
Abstract: Let K be a number field and A/K an abelian surface (dimension 2 analogue of an elliptic curve). By the Mordell-Weil theorem, the group of K-rational points on A is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the L-series determines the parity of the rank of A/K. Under suitable local constraints and finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces. We also prove analogous unconditional results for Selmer groups.
Title: Computing the homotopy type of compact arithmetic manifolds
Abstract: Cohomology of arithmetic manifolds equipped with the action of Hecke operators provides concrete realisations of automorphic representations. I will present joint work with Michael Lipnowski where we describe and analyse an algorithm to compute such objects in the compact case. I will give a gentle introduction to the known case of dimension 0, sketch ideas and limitations of previous algorithms in small dimensions, and then explain some details and ideas from the new algorithm.
Title: Finite descent obstruction and non-abelian reciprocity
Abstract: For a nice algebraic variety X over a number field F, one of the central problems of Diophantine Geometry is to locate precisely the set X(F) inside X(A_F), where A_F denotes the ring of adeles of F. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for finite étale group schemes over F on X. More recently, Kim proposed an iterative construction of another subset of X(A_F) which contains the set of rational points. In this paper, we compare the two constructions. Our main result shows that the two approaches are equivalent.
Title: Patching and the completed homology of locally symmetric spaces.
Abstract: I will explain a variant of Taylor--Wiles patching which applies to
the completed homology of locally symmetric spaces for $\mathrm{PGL}(n)$ over a
CM field. Assuming some natural conjectures about completed homology,
I will describe some applications of our construction to the study of
Galois representations and (p-adic) automorphic forms. This is joint
work with Toby Gee.
Title: Generalized Hasse invariants and some applications
Abstract: This talk is a report on a paper with Wushi Goldring. If A is an abelian variety over a scheme S of characteristic p, the isomorphism class of the p-torsion gives rise to a stratification on S. When it is nonempty, the ordinary stratum is open and the classical Hasse invariant is a section of the p-1 power of the Hodge bundle which vanishes exactly on its complement. In this talk, we will explain a group-theoretical construction of generalized Hasse invariants based on the stack of G-zips introduced by Pink, Wedhorn, Ziegler Moonen. When S is the good reduction special fiber of a Shimura variety of Hodge-type, we show that the Ekedahl-Oort stratification is principally pure. We apply Hasse invariants to attach Galois representations to certain automorphic representations whose archimedean part is a limit of discrete series, and to study systems of Hecke-eigenvalues that appear in coherent cohomology.
Title: Stickelberger series and Iwasawa Main Conjecture for $\mathbb{Z}_p^\infty$-extensions of function fields
Abstract: Let $F:=\mathbb{F}_q(\theta)$ and let $\mathfrak{p}$
be a prime of $A:=\mathbb{F}_q[\theta]$ ($q=p^r$ and $p$ a prime).
Let $\mathcal{F}_{\mathfrak{p}}/F$ be the $\mathfrak{p}$-cyclotomic
$\mathbb{Z}_p^\infty$-extension of $F$ generated by the $\mathfrak{p}^\infty$-torsion of the Carlitz module and let $\Lambda$
be the associated Iwasawa algebra. We give an overview of the Iwasawa
theory for the $\Lambda$-module of divisor class groups and then define a Stickelberger series in $\Lambda[[u]]$, whose specializations
enable us to prove an Iwasawa Main Conjecture for this setting.
As an application we obtain a close analogue of the Ferrero-Washington theorem for $\mathcal{F}_{\mathfrak{p}}$. (Joint work with Bruno
Anglès, Francesc Bars and Ignazio Longhi)
Title: Sansuc's formula and Tate global duality (d'apr\`es Rosengarten).
Abstract: Tamagawa numbers are canonical (finite) volumes attached to smooth
connected affine groups $G$ over global fields $k$; they arise in mass
formulas and local-global formulas for adelic integrals. A conjecture
of Weil (proved long ago for number fields, and recently by Lurie and
Gaitsgory for function fields) asserts that the Tamagawa number of a
simply connected semisimple group is equal to 1; for special orthogonal
groups this expresses the Siegel Mass Formula. Sansuc pushed this
further (using a lot of class field theory) to give a formula for the
Tamagawa number of any connected reductive $G$ in terms of two finite
arithmetic invariants: its Picard group and degree-1 Tate-Shafarevich
group.
Title: F-isocrystals with infinite monodromy
Abstract: Let $U$ be a smooth geometrically connected affine curve over $\mathbb{F}_p$ with compactification $X$. Following Dwork and Katz, a $p$-adic representation $\rho$ of $\pi_1(U)$ corresponds to an etale F-isocrystal. By work of Tsuzuki and Crew an F-isocrystal is overconvergent precisely when $\rho$ has finite monodromy. However, in practice most F-isocrystals arising geometrically are not overconvergent and instead have logarithmic decay at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log decay F-isocrystals in terms of asymptotic properties of higher ramification groups.
Title: Pseudorepresentations and the Eisenstein ideal
Abstract: In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed a number of questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients?
In joint work with Preston Wake, we give an answer to these questions using the deformation theory of Galois pseudorepresentations. The answer is intimately related to the algebraic number theoretic interactions between the primes N and p, and is given in terms of cup products (and Massey products) in Galois cohomology.
Title: The p-part of the Birch and Swinnerton-Dyer Conjecture for
Elliptic Curves of Analytic Rank One
Abstract: I will explain the recent proof of the p-part of the Birch
and Swinnerton-Dyer Conjectural formula for elliptic curves over Q of
analytic rank one. The proof is based on choosing a suitable
parametrization of the elliptic curve with a Shimura curve, using
Kolyvagin's Euler system method to get the upper bounds and an
anticyclotomic Iwasawa main conjecture as well as a control theorem to
get the lower bounds. This is joint work with Chris Skinner and Xin Wan.
Title: Reduction of $G$-ordinary crystalline representations with $G$-structure
Abstract: Fontaine's $D_{\mathrm{cris}}$ functor allows us to associate an isocrystal to any crystalline representation. For a reductive group $G$, we study the reduction of lattices inside a germ of crystalline representations with $G$-structure $V$ to lattices (which are crystals) with $G$-structure inside $D_{\mathrm{cris}}(V)$. Using Kisin modules theory, we give a description of this reduction in terms of $G$, in the case where the representation $V$ is ($G$-)ordinary. In order to do that, first we need to generalize Fargues' construction of the Harder-Narasimhan filtration for $p$-divisible groups to Kisin modules.
Title: $\mu$-ordinary Hasse invariants and the canonical filtration of a p-divisible group.
Abstract: In his 1973 paper, Katz constructed overconvergent modular forms on the modular curve geometrically, using the Hasse invariant and Lubin’s Theorem on the canonical subgroup of an elliptic curve. Many improvements have since been made on these constructions on many Shimura varieties, but this approach is now well known only when the ordinary locus is non-empty.
I will try to explain how to get rid of this assumption, and detail the construction of a replacement for the Hasse invariant and the construction of the canonical filtration focusing on the local analogue of the Picard modular surface.
Joaquin Rodrigues (UCL)
Title: p-adic Galois representations and p-adic L functions
Abstract: Let p be a prime number. We will discuss how to associate, to a modular
form f of level N, a (partial) p-adic L-function interpolating special
values of the complex L-function of f. This construction is based upon
Kato's Euler system and the theory of $(\varphi, \Gamma)$-modules. We
will also discuss a functional equation on the Iwasawa theory for Galois
representations of dimension 2 and how this gives, on the one hand a
functional equation for our p-adic L-function, and on the other hand
results on Kato's local epsilon conjecture 2-dimensional representations.
Title: Images of Galois representations associated to Hida families
Abstract: We explain a sense in which Galois representations associated to non-CM Hida families have large images. This is analogous to results of Ribet and Momose for Galois representations associated to classical modular forms. In particular, we show how extra twists of the Hida family decreases the size of the image.
Title: Vinogradov's three primes theorem with almost twin primes
Abstract: The general theme of this talk is about solving linear equations in sets of number theoretic interest.
Specifically I will discuss the problem with the linear equations being N = x+y+z (for a fixed large N) and the set being "almost twin primes".
The focus will be on the underlying ideas coming from both additive combinatorics and sieve theory. This is joint work with Kaisa Matomaki.
Title: Correlations of multiplicative functions and applications.
Abstract: A deep problem in analytic number theory is to understand correlations of general multiplicative functions. In this talk, we derive correlations formulas for so-called bounded "pretentious" multiplicative functions. This has a number or desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\to \{-1,1\}$ with bounded partial sums. This answers a question of Erd\H{o}s from 1957 in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. If time permits, we discuss some further applications to the related problems.
Title: Trivial zero for a p-adic L-function associated with Siegel forms
Abstract: We shall begin with an introduction to L-functions in arithmetic, their p-adic interpolation and trivial zero. We shall state a conjecture of Greenberg and Benois which predicts the order and the leading coefficient of p-adic L-functions when trivial zeros appear. We shall then explain how one calculates the first derivative of the standard p-adic L-function of an ordinary Siegel form with level at p.
Title: Smooth mod p^n representations and direct powers of Galois groups.
Abstract: Let G be a Qp-split reductive group with connected centre and Borel subgroup B=TN.
We construct a right exact functor D from the category of smooth modulo p^n representations of B to the category of projective limits of continuous mod p^n representations of a direct power of the absolute Galois group Gal(Qpbar/Qp) of Qp indexed by the set of simple roots. The objects connecting the two sides are (phi,Gamma)-modules over a multivariable (commutative) Laurent series ring which correspond to the Galois side via an equivalence
of categories. Parabolic induction from a subgroup P = L_P N_P amounts to the extension of the representation on the Galois side to the copies of Gal(Qpbar/Qp) indexed by the simple roots alpha not contained in the Levi component L_P using the action of the image of the cocharacter dual to alpha and local class field theory. D is exact and yields finite dimensional representations on the category SP of finite length representations with subquotients of principal series as Jordan-Hölder factors. Using the G-equivariant sheaf of Schneider, Vigneras, and the author on the flag variety G/B corresponding to the Galois representation we show that D is fully faithful on the full subcategory of SP with Jordan-Hölder factors isomorphic to irreducible principal series. Breuil has (preliminary) conjectures for
the values of D at certain representations of GL_n(Qp) built out from some mod p Hecke isotypic subspaces of global automorphic representations.
Title: Sub-convexity in certain Diophantine problems via the circle method.
Abstract: The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.
Title: Gaussian and non-Gaussian behaviour of character sums
Abstract: Davenport and Erdos, and more recently Lamzouri, have investigated the distribution of short character sums $\sum_{x < n \leq x+H} \chi(n)$ as $x$ varies, for a fixed non-principal character $\chi$ modulo $q$. In particular, Lamzouri conjectured that these sums should have a Gaussian limit distribution (real or complex according as $\chi$ is real or complex) provided $H=H(q)$ satisfies $H \rightarrow\infty$ but $H = o(q/\log q)$.
I will describe some work in progress in connection with this conjecture. In particular, I will try to explain that the conjecture cannot quite be correct (one need not have Gaussian behaviour for $H$ as large as $q/\log q$), but on the other hand one should see Gaussian behaviour for even larger $H$ for most characters.
Title: Effective equidistribution of primitive rational points on expanding horospheres
Abstract:
The limit distribution of primitive rational points on expanding horospheres on SL(n,Z)\SL(n, R) has been derived in the recent work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. For n=3, in our joint project with Jens Marklof, we prove the effective equidistribution of q-primitive points on expanding horospheres as q tends to infinity.
Title: On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms
Abstract: In this talk we shall begin with an introduction to Drinfeld cusp forms for arithmetic subgroups using Teitelbaum's interpretation as harmonic cocycles.
We shall then address the problem of the diagonalizability of the function field analogous of the Atkin U-operator carrying the Hecke action over harmonic cocycles.
Title: Unearthing random matrix theory in the statistics L-functions: the story of Beauty and the Beast
Abstract: There has been very convincing numerical evidence since the 1970s that the positions of zeros of the Riemann zeta function and other L-functions show the same statistical distribution (in the appropriate limit) as eigenvalues of random matrices. Proving this connection, even in restricted cases, is difficult, but if one accepts the connection then random matrix theory can provide unique insight into long-standing questions in number theory. I will give some history of the attempt to prove the connection, as well as propose that the way forward may be to forgo the enticing beauty of the determinantal formulae available in random matrix theory in favour of something a little less elegant (work with Brian Conrey and Amy Mason)
Title: Oddness of Galois representations and criticality of
L-values
Title:
Picard modular surfaces in positive characteristic
Title: Even
moments of random multiplicative functions
Title: Darmon cycles
and the Kohnen-Shintani lifting
Title: On the main conjecture of Iwasawa theory for certain elliptic
curves with complex multiplication
Title: On companion points on eigenvareties
Title: Fibrations with few rational points
Title: The Selberg trace formula as a
Dirichlet series
Title: Vinogradov's Mean Value and the Riemann Zeta-Function
Title: Divisors, additive functions and smooth numbers
Title:
Rational points on varieties via counting
Title: On the polynomial method in number theory
Samit Dasgupta (UCSC)
Title: On the Gross-Stark conjecture
Abstract: In 1980, Gross conjectured a formula for the expected leading term at $s=0$ of the Deligne--Ribet $p$-adic $L$-function associated to a totally even character $\psi$ of a totally real field $F$. The conjecture states that this value is equal to a $p$-adic regulator of units in the abelian extension of $F$ cut out by $\psi \omega^{-1}$. In this talk I will describe a proof of Gross's conjecture. Our methods build on our previous joint work with Darmon and Pollack and work of Ventullo, which together prove this conjecture in the rank 1 case. The current work is joint with Mahesh Kakde and Kevin Ventullo.
Anish Ghosh (Tata Institute)
Title: Values of quadratic forms at integer points
Abstract: I will discuss Margulis' resolution of a long open conjecture of Oppenheim on values of irrational indefinite quadratic forms and some recent quantitative analogues. The latter is joint work with Alexander Gorodnik and Amos Nevo.
Title: A bound for rational representations of isogenies in the fundamental set
Abstract: Let z and z' be two points in the standard fundamental set in the upper half-plane.
If the corresponding elliptic curves are related by an isogeny of degree N, then there is a 2x2 matrix with integer coefficients and determinant N which maps z to z'.
As an ingredient in their work on unlikely intersections, Habegger and Pila proved that the entries of this matrix are bounded by a uniform polynomial in N.
I will discuss the generalisation of this result to moduli of abelian varieties and beyond, to Riemannian symmetric spaces of non-compact type.
Title: The partial degrees of the canonical subgroup
Abstract: If the Hasse invariant of a p-divisible group is small enough, then one can construct a canonical subgroup inside its p-torsion. I will first present an alternative approach to this problem, assuming the existence of a subgroup satisfying some simple conditions.
A key property is the relation between the Hasse invariant and the degree of the canonical subgroup. When one considers a p-divisible group with extra structures, more information is available. I will define the partial Hasse invariants, the partial degrees, and relate them for the canonical subgroup.
Title: L-invariants and local-global compatibility for GL2
Abstract: Let F be a totally real number field, w a prime of F above p, V a 2-dimensional p-adic representation of the absolute Galois group G_F of F which appears in etale cohomology of quaternion Shimura curves. When the restriction Vw of V to the decomposition group of G_F at w is semi-stable non-crystalline in Fontaine's sense, we can associate to Vw the so-called Fontaine-Mazur L-invariants, which are invisible in classical local Langlands correspondance. We show that these L-invariants can be found in the completed cohomology group of Shimura curves.
Title: Cuspidal l-modular representations of classical p-adic groups
Abstract: For a classical groups (unitary, special orthogonal, symplectic) over locally compact non-archimedean fields of odd residual characteristic p, Shaun Stevens has developed an approach to studying their (smooth) complex representations based on the theory of types of Bushnell and Kutzko. I will describe some joint work with Shaun Stevens, in which we relate positive level cuspidal representations in Stevens' construction to level zero cuspidal representations in certain associated groups and consider a generalisation to modular representations in characteristic prime to p.
Title: p-kernels occurring in isogeny classes of p-divisible groups
Abstract: I will give a criterion which allows to determine, in terms of the combinatorics of the root system A_n, which p-kernels occur in a given isogeny class of p-divisible groups over an algebraically closed field of positive characteristic. This question is related to the relationship between Newton and Ekedahl-Oort strata on reductions of Shimura varieties as well as the non-emptiness of affine Deligne-Lusztig varieties.
Title: Iterated extensions and relative Lubin-Tate groups
Abstract: An important construction in p-adic Hodge theory is the 'field of norms' corresponding to an infinite extension K_infty/K. For the cyclotomic extension, it is possible to lift the field of norms to characteristic zero, and we can ask for which other extensions K_infty/K this is possible. The goal of this talk is to explain this question and discuss some partial answers. This involves p-adic dynamical systems, Coleman power series and relative Lubin-Tate groups.
Title: Parabolic induction and extensions
Abstract: Let G be a p-adic reductive group. We describe the extensions between admissible smooth mod p representations of G which are parabolically induced from supersingular representations of Levi subgroups. More precisely, we determine which extensions do not come from parabolic induction. In order to do so, we partially compute Emerton's delta-functor of derived ordinary parts on any parabolically induced representation of G. These computations work with mod p^n coefficients, thus some of the results on extensions can be lifted in characteristic 0 for admissible unitary continuous p-adic representations of G.
Title: Logarithmic good reduction and cohomological tameness
Abstract: I will discuss two notions of tameness for varieties defined over a field equipped with a discrete valuation, which are only interesting if the residual characteristic is positive: cohomological tameness, and logarithmic good reduction. The first notion is weaker than the second one (Nakayama). I will explain why these notions are equivalent in the case of abelian varieties; this can be seen as a logarithmic version of the Néron-Ogg-Shafarevich criterion (joint work with A. Bellardini). I will also discuss a cohomological trace formula for the tame monodromy operator, conjectured by Nicaise for cohomologically tame varieties, proven by the speaker for varieties with logarithmic good reduction.
Title: Goldbach versus de Polignac numbers
Abstract: We discuss the following statement, connecting two well-known conjectures. Either consecutive Goldbach numbers lie within a finite distance from one another or else the set of de Polignac numbers has full density in 2N.
Title: Rational points and quadratic Chabauty
Abstract: Let C be a curve over the rationals of genus g at least 2. By
Faltings' theorem, we know that C has finitely many rational points.
When the Mordell-Weil rank of the Jacobian of C is less than g, the
Chabauty-Coleman method can often be used to find these rational
points through the construction of certain p-adic integrals.
When the rank is equal to g, we can use the theory of p-adic height
pairings to produce p-adic double integrals that allow us to find
integral points on curves. In particular, I will discuss how to carry
out this "quadratic Chabauty" method on hyperelliptic curves over
number fields (joint work with Amnon Besser and Steffen Mueller) and
present related ideas to find rational points on bielliptic genus 2
curves (joint work with Netan Dogra).
Title: Cuspidal representations for inner forms of classical groups
Abstract: In this talk I present the strategy to classify the cuspidal irreducible representations of inner forms of classical groups, and I will give some remarks on the role of Endo-classes for classical groups.
Title: Local Galois deformation rings when l != p.
Abstract: Given a mod p representation of the absolute Galois group of Q_l, consider the universal framed deformation ring R parametrising its lifts. When l and p are distinct I will explain a relation between the mod p geometry of R and the mod p representation theory of GL_n(Z_l), that is parallel to the Breuil-Mézard conjecture in the l = p case. I will give examples and say something about the proof, which uses automorphy lifting techniques.
Title: Classicality on Eigenvarieties
Abstract: Let p be a prime number. It is expected that a
p-adic overconvergent automorphic eigenform of classical
weight (on a definite unitary group of arbitrary dimension)
such that its associated Galois representation is crystalline
at p should be classical. I will sketch a proof of many new
cases of this conjecture. This is joint work with E. Hellmann
and B. Schraen.
Title: Badly approximable numbers
Abstract: Real numbers badly approximable by rational numbers have been known for well over a century, thanks to continued fractions.
In this talk I will discuss recent results in Diophantine approximation on manifolds that led to the proof of the existence of transcendental
real numbers badly approximable by algebraic numbers of arbitrary fixed degree.
Title: Rapoport-Zink spaces of Hodge type and applications to Shimura varieties
Abstract: Rapoport-Zink spaces of (P)EL type are local analogues of Shimura varieties of PEL type. Examples include Lubin-Tate spaces and Drinfeld upper half spaces.
In this talk, we construct the "Hodge-type generalisation" of Rapoport-Zink spaces under the unramifiedness hypothesis, and apply it to the integral models of Hodge-type Shimura varieties. The new examples are Spin and orthogonal Rapoport-Zink spaces (of arbitrary rank) -- local analogues of Spin and orthogonal Shimura varieties.
We will start with the description of geometric points of "Hodge-type Rapoport-Zink spaces" and the completed local rings thereof. Some applications to the study of Hodge-type Shimura varieties will be given.
Title: On Baker type bounds and generalised transcendence measure
(joint work with K. Leppälä and Tapani Matala-aho)
Abstract: If $\alpha$ is transcendental, then $P(\alpha)\ne 0$ for all
polynomials $P$ with integer coefficients. The transcendence measure
tells how far from zero these values of polynomials must be (at
least). During my talk, I will first give a sketch of the current
knowledge about the transcendence measure of $e$, and I will also
briefly explain how these results can be obtained. I will then move to
explaining how the transcendence measure can be generalised, and what
is known and what is believed about the generalised transcendence
measure.
Title: Residual modular Galois representations: images and applications.
Abstract: Let l be a prime number. To any mod l modular form, which is an eigenform for all Hecke operators, it is associated a 2-dimensional residual representation of the absolute Galois group of the rationals. Two different mod l modular forms can give rise to the same Galois representation. Analogously, a residual modular Galois representation can arise as twist of a representation of lower conductor. In this talk, after a brief introduction on residual modular Galois representations and mod l modular forms, I will address these problems and outline an algorithm for computing the image of such representations. I will also describe two applications of this algorithm (both still work in progress): solving Diophantine equations and graphs of modular forms.
Title: Diophantine properties of Mahler numbers
Abstract: In this talk, I will explain what the class of Mahler Numbers is, present their Diophantine approximation properties and their link with Computer Science.
Title: Ergodicity for point scatterers on arithmetic tori
Abstract: The Seba billiard was introduced to study the transition between
integrability and chaos in quantum systems. The model seem to exhibit
intermediate level statistics (i.e., repulsion between nearby
eigenvalues, though not as strong as predicted by random matrix theory),
as well as Gaussian value distribution of eigenfunctions ("wave
chaos"). We investigate the very closely related "toral point
scatterer"-model, namely eigenfunctions of the Laplacian perturbed by a
delta-potential, on arithmetic 2D-tori. For a full density subsequence
of "new" eigenfunctions we prove decay of matrix coefficients associated
with pure momentum observables. This, together with previous work by
Rudnick-Ueberschaer, allows us to conclude that quantum ergodicity holds
for the set of "new" eigenfunctions. In particular, almost all new
eigenfunction are equidistributed in both the position and the momentum
representation. Time permitting we will discuss some recent "scar"
constructions (i.e., sequences of eigenfuntions that do not
equidistribute.)
Title: Divisor Sums in Function Fields
Abstract - I will review some classical problems in number theory concerning sums of the (generalised) divisor function. I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals.
Title: Randomness in geometry of numbers
Abstract: We discuss the problem of counting solutions of Diophantine inequalities.
While a general asymptotic formula for the counting function has been established
by W. Schmidt, finer statistical properties of this function are still not well understood.
We investigate its limiting distribution and establish the central limit theorem
in this context. This is a joint work with Anish Ghosh.
Title: Chebyshev's bias for elliptic curves over function fields
Abstract: Since Chebyshev's observation that there seems to be more primes of the form 4n+3 than of the form 4n+1, many other types of 'arithmetical biases' have been found. For example, such a bias appears in the count of points on reductions of a fixed elliptic curve E; this bias is mainly created by the analytic rank. In this talk we will discuss the analogous question for elliptic curves over function fields. We will first discuss the occurrence of extreme biases, which originate from very different source than in the number field case. Secondly, we will discuss what happens to a 'typical curve', and discuss results of linear independence of the zeros of the associated L-functions. This is joint work with Byungchul Cha and Florent Jouve.
Title: Shimura varieties and complex conjugation.
Title: Quadratic twists of elliptic curves.
Title: The eigencurve and its characteristic p special fibre.
Title: Darmon points for number fields of mixed signature.
Title: L-functions as distributions.
Title: p-adic local Langlands in weight 1.
Title: Syntomic cohomology and p-adic nearby cycles.
Title: The average elliptic curve has few integral points.
Title: On a generalisation of local coefficients
Title: Grothendieck-Messing deformation theory for varieties of K3-type.
Title: D-modules on rigid analytic spaces
Title: Arithmetic of plane quartic curves.
Title: Explicit unbounded ranks of Jacobians in towers of function fields.
Title: Plectic cohomology of Shimura varieties
Title:
Computing dimensions of spaces of automorphic/modular forms for
classical groups using the trace formula
Title: Large gaps between primes
Title: Modularity of elliptic curves over totally real fields
Title: The F-plectic Taniyama group
Title: p-adic Langlands functoriality.
Title: Maximal tori and the integral Bernstein center for GL_n
Title: Primes of the form a^2+p^4
Title: Class numbers in some non-cyclotomic $\mathbb{Z}_p$-extensions.
Abstract: The class numbers of the finite layers of the cyclotomic $\mathbb{Z}_p$-extensions of the rationals have been the subject of much study. By studying the absolute height of cyclotomic units, Horie effectively proved that 100% of prime numbers $q$ never divide the class numbers of the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of the rationals. I will talk about how similar methods can be applied to $\mathbb{Z}_p$-extensions of an imaginary quadratic field $K$ which are unramified outside of a split prime above $p$.
Title: On the hyperbolic lattice-point problem
Abstract:
Title: Reducible Galois representations and homology of $GL(n,\mathbb{Z})$.
Abstract: In joint work with Darrin Doud, I have connected some reducible 3-dimensional representations of the absolute Galois group of $\mathbb{Q}$ with homology Hecke eigenclasses for congruence subgroups of $GL(3,\mathbb{Z})$. I will explain what we have done and how it fits into the picture of generalized reciprocity laws for $GL(n,\mathbb{Z})$.
Title: Eigenvarieties beyond the case of discrete series
Abstract: Given a connected reductive group $G$ over a number field $F$, I'll explain a construction of eigenvarieties parametrizing "Betti-cohomological overconvergent $p$-adic modular forms" on $G$ under a very mild hypothesis (namely that $G$ splits over $F_v$ for each $v|p$). This construction generalizes previous works of a great number of authors, which have largely focused on case where $G(F \otimes R)$ has a discrete series. Beyond the setting of discrete series, the qualitative geometric properties of eigenvarieties change drastically. I'll discuss the known and expected properties of these spaces, and their (largely conjectural) relationship with Galois representations.
Title: Algebraic functional equations and completely faithful Selmer groups
Abstract: Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p\geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_{\infty}/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group. This gives an algebraic functional equation of the conjectured $p$-adic $L$-function. As an application we construct completely faithful Selmer groups in case the $p$-adic Lie extension is obtained by adjoining the $p$-power division points of another non-CM elliptic curve $A$. This is joint work with T. Backhausz.
Title: Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings
Abstract: We use the dynamics on $SL(3,\mathbb{R})/SL(3,\mathbb{Z})$ to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.
Title: On potential counterexamples to mixed Littlewood conjecture.
Abstract: Mixed Littlewood conjecture proposed by de Mathan and Teulie in
2004 states that for every real number $x$ one has
$$
\liminf_{q\to\infty} q\cdot |q|_D\cdot ||qx|| = 0.
$$
where $|*|_D$ is a s called pseudo norm which generalises the standard $p$-adic norm.
In the talk we'll consider the set $\mad$ of potential
counterexamples to this conjecture. Thanks to the results of
Einsiedler and Kleinbock we already know that the Haudorff dimension
of $\mad$ is zero, so this set is very tiny. During the talk we'll
see that the continued fraction expansion of every element in $\mad$
should satisfy some quite restrictive conditions. As one of them
we'll see that for these expansions, considered as infinite words,
the complexity function can neither grow too fast nor too slow.
Title: "Non-abelian reciprocity laws and Diophantine geometry"
Title: On class groups of imaginary quadratic fields
Title: "Some problems in analytic number theory for polynomials over a finite field".
Abstract: The lecture explores several problems of analytic number theory
in the context of function fields over a finite field, where they can
be approached by methods different than those of traditional analytic
number theory.
The resulting theorems can be used to check existing conjectures
over the integers, and to generate new ones. Among the problems
discussed are: Counting primes in short intervals and in arithmetic
progressions; Chowla's conjecture on the autocorrelation of the
Mobius function; the distribution of squarefree numbers.
The tools used are from analytic number theory, algebraic geometry,
field arithmetic and beyond.
Title: The Cohen-Lenstra heuristic revisited
Title: Local-global compatibility for Galois representations associated to Hilbert modular forms of low weight
Title: Overconvergent modular symbols over imaginary quadratic fields
Title: The Birch and Swinnerton-Dyer conjecture for non-abelian twists of elliptic curves.
Title: Higher Fitting ideals of arithmetic objects
Title: p-adic measures for Hermitian modular forms and the Rankin-Selberg method
Title: Deformation of algebraic cycles in characteristic p
Title: Eisenstein classes in syntomic cohomology and applications
Title: Grothendieck trace formulas and Iwasawa main conjectures in characteristic p
Title: Rigid rational homotopy theory and mixedness
Title: Families of L-functions and their symmetry
Title: Mordell-Weil point generation on cubic surfaces over finite fields
Title: The p-adic local Langlands correspondence beyond GL_2(Q_p).
Title: Periods of modular forms on Shimura curves and singular theta lifts
Title: L-functions of curves.
Title: p-adic Hodge theory in rigid analytic families
Title: Second Chern class and SK1
Title: The Brauer-Manin obstruction and ranks of quadratic twists
Title: A derived local Langlands correspondence for GL_n
Title: The minimum modulus of a covering system is bounded.
Title: G-valued flat deformations and local models.
Title: Unlikely intersections in abelian varieties and Shimura varieties.
Title: Generalised Fourier coefficients of multiplicative functions and applications.
Title: On simple rational polynomials.
24/4/13 4:00 (same day): Ph. Cassou-Noguès (Bordeaux)
Title: Cohomological invariants of quadratic forms
Title: Nodal length fluctuations for arithmetic random waves.
Title: Galois equivariance of L-values and the Birch-Swinnerton-Dyer conjecture
Title: Multiplicative and Inhomogeneous Diophantine Approximation
The action of compact subgroups on some p-modular automorphic representations of GL(2,Q_p)
Title: Higher multiplicities in number theory
Title: On the irreducible mod p representations of unramfied U(2,1)
New results on Langlands functoriality: non-solvable base change, symmetric powers and tensor products.
Title: Degrees of strongly special subvarieties and the André-Oort conjecture.
Title: P-rigidity and Iwasawa mu-invariants.
Title: Euler systems for the Rankin-Selberg convolution of modular forms.
Title: On the syntomic regulator
Title: The p-adic local Langlands correspondence and Lubin-Tate groups
06/02 Speaker: Judith Ludwig (Imperial)
Title: p-adic functoriality for inner forms of unitary groups
Title: Independence of ell-adic Galois representations over finitely generated fields
Title: Congruent numbers.
Title: Toric regulators for varieties with totally degenerate reduction over p-adic fields
Title: Kato's reciprocity law and a p-adic Beilinson formula
Title: The Buechi K3 surface and its rational points
Title: Hyperbolic Ax-Lindemann theorem in the cocompact case
Abstract: This is a joint work with E. Ullmo. The classical Ax-Lindemann theorem is a statement in transcendence theory of the exponential function. We prove an analogous statement for the map uniformising compact Shimura varieties.
Title: Fermat-type equations of signature (13,13,p) via Hilbert cuspforms
Abstract: In this talk I am going to discuss how a modular approach via Hilbert cuspforms can be used to show that equations of the form x^13 +y^13 = Cz^p have no non-trivial primitive solutions (a,b,c) such that 13 does not divide c if p > 4992539. We will first relate a putative solution of the previous equation to the solution of another Diophantine equation with coefficients in the real quadratic extension of Q with discriminant 13. Then we attach Frey curves E over this field to solutions of the latter equation. Finally, by proving modularity of E and irreducibility of certain Galois representations attached to E we are able to apply a modular approach via Hilbert modular forms.
Title: Patching functors and the cohomology of Shimura curves
Abstract: I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti-Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure. The key technique is the Taylor-Wiles-Kisin patching argument, which, when combined with a new, geometric perspective on the Breuil-Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).
Title: Visibility of Tate-Shafarevich groups in abelian surfaces and abelian threefolds
Abstract: Let E be an elliptic curve over the rationals. The Tate-Shafarevich group of E measures the failure of the Hasse Principle for principal homogeneous spaces under E. Mazur suggested that one should attempt to ``visualize'' elements of this group as cosets of E inside some larger abelian variety A. I will discuss what is known about the visibility dimension (i.e. the least possible dimension of A) and give some examples of elements of order 7 that are visible in abelian surfaces and abelian threefolds.
Title: Towards the cohomological construction of Breuil-Kisin modules.
Abstract: We will present an approach to the construction of Breuil-Kisin modules using crystalline cohomology. Along the way, we will define a new PD-base for crystalline cohomology that improves in some respects upon the ring S used by Breuil and Faltings in their appraoch to integral p-adic Hodge theory.
Title: Cohomology of Bianchi Groups and Arithmetic
Abstract: Bianchi groups are groups of the form SL(2,R) where R is the ring of an imaginary quadratic field. They arise naturally in the study of hyperbolic 3-manifolds and of certain generalisations of the classical modular forms (called Bianchi modular forms) for which they assume the role of the classical modular group SL(2,Z). After giving the necessary background, I will start with a discussion of the problem of understanding the behaviour of the dimensions of the cohomology of Bianchi groups and their congruence subgroups. Next, I will focus on the amount of the torsion that one encounters in the cohomology . Finally, I will discuss the arithmetic significance of these torsion classes.
Title: Quantum Unique Ergodicity for holomorphic newforms
Abstract: Let f be a classical holomorphic newform of level q and even weight k. I will describe recent joint work with Paul Nelson and Ameya Pitale where we prove that the pushforward to the full level modular curve of the mass of f equidistributes as qk goes to infinity. This generalizes previous work by Holowinsky-Soundararajan (the case q=1, k-> infinity) and Nelson (the case qk -> infinity over squarefree integers q). Thus we settle the holomorphic quantum unique ergodicity
conjecture in all aspects (for classical modular forms of trivial nebentypus). A potentially surprising aspect of our work is that we obtain a power savings in the rate of equidistribution as q becomes sufficiently ``powerful'' (far away from being squarefree), and in particular in the ``depth aspect'' as q traverses the powers of a fixed prime.
Title: Rational points on conic and quadric bundles with many degenerate fibres
Abstract: This is a talk on the joint work with Tim Browning and Lilian Matthiesen. We apply additive combinatorics of Green and Tao to prove that the Brauer-Manin obstruction controls weak approximation for rational points on pencils of conics or quadrics over Q, provided that all singular fibres are defined over Q.
Title: Mordell-Lang in positive characteristic
Abstract: I will talk about the Mordell-Lang conjecture in positive characteristic and sketch a new algebro-geometric proof for the case that the subgroup involved is finitely generated.
Title: The Herbrand-Ribet theorem for function fields revisited
Abstract: We will present a new proof of Taelman's Herbrand-Ribet Theorem based on an equivariant class number formula for cyclotomic function fields recently obtained in a joint work with L. Taelman. If we have time, we will also show how this equivariant formula implies that a conjecture "à la Vandiver" made by G. Anderson in 1996 is false.
Title: Etale Homotopy of Deligne-Mumford stacks.
Abstract: We will give an overview of etale homotopy theory a la Artin-Mazur of Deligne-Mumford stacks and discuss several examples including moduli stacks of algebraic curves and principally polarised abelian varieties.
LNTS Schedule
"Applications of the trace formula to spectral theory of automorphic forms."
"Darmon points on Shimura curves and abelian varieties"
"The Buzzard-Diamond-Jarvis conjecture for unitary groups"
"p-adic modular forms of non-integral weight over Shimura curves"
"Quillen's Lemma for affinoid enveloping algebras"
"Circulant graphs, Frobenius numbers and ergodic theory"
"Inertial types for automorphic representations"
"The p-adic Geometric Langlands Correspondence"
"Classicality for overconvergent automorphic forms on some quaternionic Shimura varieties"
LNTS Schedule
Title: Rational points on Atkin-Lehner quotients of Shimura curves
Title: The method of Bertolini and Darmon
Title: Some unlikely intersections beyond
Andre-Oort.
Title: On the p-adic density of the rational points on K3 surfaces.
Title: An algorithm to compute the semi-simplification modulo p of a
semi-stable representation
Title: Effective density of rational points on homogeneous varieties
LNTS Schedule
DATE: 05/10/11
LNTS Schedule
DATE: 27/4/11
Spring 2011
This term, the seminar was held at Kings College and was organized
by Andrei Yafaev.
The seminar program for Spring 2011
Jan 19: Laurent Fargues (Orsay, Paris-11) "Curves and vector bundles in p-adic Hodge theory"
Autumn 2010
This term, the seminar was held at Imperial College and was
organised by Ambrus Pal.
The seminar program for Autumn 2010
6 October
Ambrus Pal (Imperial)
Crystalline Chebotarev density theorems
13 October
Ivan Fesenko (Nottingham)
Two adelic structures on arithmetic surfaces and the Tate version of the BSD
conjecture.
20 October
Lenny Taelman (Leiden)
27 October
Daniel Caro (Caen)
3 November
Lorenzo Ramero (Lille)
Cohomological epsilon factors and p-adic analytic geometry.
8 November
The London-Paris Number Theory Seminar
speakers: F.Loeser (ENS, Paris), J.Pila (Bristol), B.Zilber (Oxford).
10 November
Jean-Louis Colliot-Thelene (Orsay)
Unramified cohomology in degree 3.
17 November
Pierre Lochak (Jussieu)
Topological methods in Grothendieck-Teichmueller theory.
24 November
Jon Pridham (Cambridge)
1 December
Tomer Schlank (Jerusalem)
8 December
Tony Scholl (Cambridge)
Hypersurfaces and the Weil conjectures
15 December
TBA
Summer 2010
This term, the seminar was held at Kings College and was organised by Payman Kassaei and Manuel Breuning.
The seminar program for Summer 2010
12 May
Matthias Flach (Caltech)
"Weil-etale cohomology of regular arithmetic schemes"
19 May
Jared Weinstein (UCLA)
"Resolution of singularities on the Lubin-Tate tower"
Abstract: A fundamental result in local class field theory is the
1965 paper of Lubin and Tate, which classifies the abelian extensions
of a nonarchimedean local field in terms of an algebraic structure
known as a one-dimensional formal module. We'll review this result,
and show how the question of constructing nonabelian extensions leads
to the study of the Lubin-Tate tower, which can be viewed as an
infinitesimal version of the classical tower of modular curves
$X(p^n)$.
By results of Harris-Taylor and Boyer, the
cohomology of the Lubin-Tate tower encodes precise information about
non-abelian extensions of the local field (namely, it realizes the
local Langlands correspondence). The Lubin-Tate tower has a horribly
singular special fiber, which hinders any direct study of its
cohomology, but we will show that after blowing up a singularity there is a
model for the tower whose reduction contains a very curious
nonsingular hypersurface defined over a finite field -- curious
because it seems to have the maximum number of rational points
relative to its topology. We will write
down the equation for this hypersurface and formulate a conjecture
(alas, still unproved)
regarding its zeta function.
26 May
Antonio Lei (Cambridge)
"Wach modules and Iwasawa theory for modular forms II"
Abstract: This is a follow-up of Sarah Zerbes' talk from last term. In
Sarah's talk, she talked about the construction of Coleman maps using the
theory Wach modules in order to reformulate Kato's main conjecture for
modular forms at supersingular primes under some technical conditions. In
this talk, I will introduce a new result on elementary divisors for Wach
modules and Dieudonne modules which enables us to remove many of the
assumptions in our previous works.
2 June
The London-Paris Number Theory Seminar
speakers: F. Brown, M. Hindry, M. Kakde
9 June
Peter Swinnerton-Dyer (Cambridge)
"Density of rational points on certain K3 surfaces"
16 June
Shu Sasaki (King's College London)
"On Artin representations and nearly ordinary Hecke algebras
over totally real fields"
Abstract:
I will explain how to prove an analogue in the "completely split"
Hilbert case of a result of Buzzard and Taylor about two-dimensional Artin
representations and weight one forms, and prove new cases of the strong
Artin conjecture for totally odd, two-dimensional "icosahedral"
representations of the absolute Galois group of a totally real field.
23 June
Kazim Buyukboduk (Istanbul)
"Euler systems of rank $r$ and Kolyvagin systems"
Abstract:
For a $p$-adic Galois representation $T$, I will devise an Euler
system / Kolyvagin system machinery which as an input takes an Euler
system of rank $r$ (in the sense of Perrin-Riou), and gives a bound on
the Bloch-Kato Selmer group in terms of an r x r determinant. I will
give two fundamental applications of this refinement: The first with
the (conjectural) Rubin-Stark elements; and the second with
Perrin-Riou's (conjectural) $p$-adic $L$-functions.
Spring 2010
This term, the seminar was held at University College and was organised by Andrei Yafaev.
The seminar program for Spring 2010
13 January
Michael Schein
"On families of irreducible supersingular mod p representations of GL_2(F)"
20 January
Mathieu Florence (Paris)
"Equivariant birational geometry of Grassmannians"
Abstract:
Let k be a field, and A a finite-dimensional k-algebra. Let d be an integer. Denote by Gr(d,A)
the Grassmannian of d-subspaces of A (viewed as a k-vector space), and by GL_1(A) the algebraic
k-group whose points are invertible elements of A. The group GL_1(A) acts naturally on Gr(d,A)
(by the formula g.E=gE). My aim is to study some birational properties of this action. More
precisely, let r be the gcd of d and dim(A). Under some hypothesis on A (satisfied if A/k is
etale), I will show that the variety Gr(d,A) is birationally and GL_1(A)-equivariantly isomorphic
to the product of Gr(r,A) by an affine space (on which GL_1(A) acts trivially). By twisting,
this result has a few corollaries in the theory of central simple algebras. For instance, let B
and C be two central simple algebras over k, of coprime degrees. Then the Severi-Brauer variety
SB(B \otimes C) is birational to the product of SB(B) \times SB(C) by an affine space of the
correct dimension. These corollaries are in the spirit of Krashen's generalized version of
Amitsur's conjecture.
27 January
Fernando Villegas (Texas)
"Hypergeometric motives"
3 February
Toby Gee (Harvard)
"The Sato-Tate conjecture for Hilbert modular forms"
Abstract:
I will discuss the Sato-Tate conjecture for Hilbert modular forms,
which I recently proved in collaboration with Thomas Barnet-Lamb and
David Geraghty.
10 February
Jonathan Pila (Bristol)
"A model-theoretic approach to problems of Manin-Mumford-Andre-Oort-type"
Abstract:
I will describe a result, joint with Alex Wilkie, about the
distribution of rational points on certain non-algebraic sets in real space.
The natural setting is an 'o-minimal structure over the real numbers',
a notion from model-theory. A surprising strategy, proposed by Umberto
Zannier, uses this result to approach diophantine problems in the
Manin-Mumford-Andre-Oort circle of conjectures. I will describe some
implementations of this strategy, including an unconditional proof of
the Andre-Oort conjecture for products of modular curves.
17 February
Frank Neumann (Leicester)
"Moduli stacks of vector bundles on curves and Frobenius morphisms"
Abstract: After giving a brief introduction into moduli problems and moduli stacks I will
indicate how to calculate the l-adic cohomology ring of the moduli stack of vector
bundles on an algebraic curve in positive characteristic and explicitly describe the
actions of the various geometric and arithmetic Frobenius morphisms on the cohomology
ring. It turns out that using the language of algebraic stacks instead of geometric
invariant theory this becomes surprisingly easy. If time permits I will indicate how
to prove some analogues of the classical Weil conjectures for the moduli stack.
This is work in progress with Ulrich Stuhler (Goettingen).
24 February
Wansu Kim (Imperial)
"Galois deformation theory for norm fields"
3 March
Lawrence Breen (Paris)
"Non-abelian and partly abelian cohomology theories"
10 March
Sarah Zerbes (Exeter)
"Wach modules and Iwasawa theory for modular forms"
17 March
Cecile Armana (Paris/Barcelona)
"Coefficients of Drinfeld modular forms and Hecke operators"
Abstract: We will talk about Drinfeld modular forms, which are analogues, over the
function field $\mathbf{F}_{q}(T)$, of classical modular forms. Given a
classical cusp form $f$, there exists a simple formula relating the
$n$-th Fourier coefficient of $f$ to the first coefficient of $T_{n}(f)$
($T_n$ denotes the $n$-th Hecke operator). This property has several
consequences, for instance the multiplicity one theorem. Drinfeld
modular forms possess series expansion and Hecke operators acting on
them. The aim of the talk is to present a formula giving, for any Hecke
Drinfeld eigenform, some of its coefficients in terms of its eigenvalues.
Autumn 2009
This term, the seminar was held at Imperial College and was organised by Alexei Skorobogatov.
The seminar program for Autumn 2009
7 October
Samir Siksek (Warwick)
"Explicit Chabauty over Number Fields"
Abstract:
Let $C$ be a curve of genus at least $2$ over a number field
$K$ of degree $d$. Let $J$ be the Jacobian of $C$ and $r$ the rank of
the Mordell-Weil group $J(K)$. Chabauty is a practical method for
explicitly computing $C(K)$ provided $r \leq g-1$. In unpublished
work, Wetherell suggested that Chabauty's method should still be
applicable provided the weaker bound $r \leq d(g-1)$ is satisfied. We
give details of this and use it to solve the Diophantine equation
$x^2+y^3=z^{10}$ by reducing the problem to determining the
$K$-rational points on several genus $2$ curves over
$K=\Q(\sqrt[3]{2})$.
14 October
Florian Pop (University of Pennsylvania and the Newton Institute)
"On the Ihara/Oda-Matsumoto Conjecture"
Abstract:
In his "Esquisse d'un programme", Grothendieck
suggested that one should be able to give a non-tautological
description of the absolute Galois group of the rationals via
its action on the geometric fundamental group of "interesting"
varieties. Similar was suggested/asked by Ihara, and a precise
conjecture was made by Oda-Matsumoto. In my talk I plan to
report on the status of the art of this problem.
21 October
Imperial Commemoration day (no seminar)
28 October
Lassina Dembele (Warwick)
"Nonsolvable Galois number fields ramified at 2, 3 and 5 only"
Abstract:
In the mid 90s, Dick Gross proposed the following conjecture.
Conjecture: For every prime p, there is a nonsolvable Galois number
field K ramified at p only.
For p>=11, this conjecture is a consequence of results of Serre and
Deligne (using classical modular forms). In this talk, we will show that
the conjecture is true for p=2, 3 and 5. The extensions K we constructed
in those cases are obtained by using Galois representations attached to
Hilbert modular forms. We will also outline a strategy to tackle the case
p=7 using automorphic forms on U(3).
4 November
Roger Heath-Brown (Oxford)
"Counting points on cubic curves"
Abstract:
Given a smooth plane cubic curve C defined over the rationals, we are
interested in upper bounds for the number of rational points of height at
most B, say, which are uniform in the curve C. Two previous approaches
will be described, along with a new hybrid version.
11 November
Don Blasius (UCLA)
"Asymptotic Fullness of Automorphic Galois Representations"
Abstract:
On a reductive group G over a number field, limit
multiplicity theorems give the growth rate, as a function of
suitably growing level, for the number of cusp forms $\pi$ which
have given discrete series type at infinity. In this talk we look at
some finer structure arising from the existence of Galois
representations attached to such forms. Specifically, we ask whether
the subset of those with largest Zariski closure has density one
among all the forms. For some simple cases we prove the conjecture,
or provide a positive density result. One proof of the latter uses a
result about the asymptotic distribution of Hecke eigenvalues at a
fixed unramified finite place, namely that this distribution is
Plancherel measure.
16 November
The London-Paris Number Theory Seminar
speakers: M. Emerton, A. Skorobogatov, S. David
18 November
Herbert Gangl (Durham)
"Double zeta values and periods of modular forms"
Abstract:
We give new relations among double zeta values \zeta(r,s)=\sum_{m>n>0} m^{-r} n^{-s}
and show that the structure of the Q-vector space of all relations among double
zeta values of weight k is connected in several different ways with the structure
of the space of modular forms of weight k on the full modular group.
(Joint work with M.Kaneko and D.Zagier.)
25 November
Fabien Trihan (Nottingham)
"On the p-parity conjecture in the function field case"
Abstract:
Let F be a function field in one variable with field of constants
a finite field of characteristic p>0. Let E/F be an elliptic curve over F.
We show that the order of the Hasse-Weil L-function of E/F at s=1 and
the corank of the p-Selmer group of E/F have the same parity
(joint work with C. Wuthrich).
2 December
Behrang Noohi (King's)
"Galois cohomology of crossed-modules and cohomology of reductive groups"
Abstract:
A 2-group (or a crossed-module) is a categorified version of a group.
Line bundles over a scheme, for instance, form the Picard 2-group.
Galois cohomology of 2-groups can be used to give information about
Galois cohomology of ordinary groups (via, say, certain long exact
sequences). We discuss the basics of the theory and give some simple
examples involving Picard and Brauer groups. We then explain Borovoi's
application of these ideas to the study of Galois cohomology of
reductive groups.
9 December
Javier Lopez (Queen Mary)
"Torified schemes and geometry over the field with one element"
Abstract:
In this talk we introduce the notion of torified variety as a
reduced scheme X of finite type that admits a decomposition
$T = \{T_i\}_{i\in I}$ by split tori. This is a general concept that
includes toric varieties, homogeneous spaces and Chevalley group schemes
among others. We will show some of the main
properties of torified varieties, show how the torifications define
geometries over the field with one element. We also show how a
torification provides an easy way to compute the counting function of
$X$, which can be immediately applied to compute the corresponding zeta
functions over $\mathbb{F}_1$.
Summer 2009
This term, the seminar was held at King's College and was organised by Fred Diamond.
The seminar program for Summer 2009
8 April
Mehmet Haluk Sengun (Duisburg-Essen)
"Computing With Bianchi Modular Forms"
Abstract: Bianchi modular forms are modular forms over imaginary quadratic fields.
In this talk,
we present an algorithm to compute these forms and the
Hecke action on them. Then we discuss
their conjectural connections with
mod p Galois representations, presenting certain results and calculations.
29 April
Francis Brown (Paris-Jussieu)
"Feynman graphs, moduli spaces and multiple zeta values"
Abstract: I will begin by explaining how Feynman graphs in perturbative quantum
field theory define interesting periods in the sense of algebraic
geometry. Extensive computations by physicists suggest that these evaluate
numerically to multiple zeta values in all known cases, but recent work of
Belkale and Brosnan leads one to expect that the underlying motives may be
of general type.
After giving an overview of recent work on the subject, I will try to give
a geometric and combinatorial explanation for these observations.
6 May
Alexander Stasinski (Southampton)
"Unramified and Regular Representations"
Abstract: The talk will be about two rather different constructions of
smooth (complex) representations of certain compact $p$-adic groups. The
first is a cohomological construction of so called unramified
representations of reductive groups over finite local rings, and is a
generalization of the classical construction of Deligne and Lusztig.
This gives in particular a family of representations of any compact
$p$-adic group of the form $G(\mathfrak{o})$, where $G$ is a reductive
group over the ring of integers $\mathfrak{o}$ in a local
non-Archimedean field. The second construction is a purely algebraic
approach to the regular representations of $GL_N(\mathfrak{o})$, which
is formally similar to the Bushnell-Kutzko construction of supercuspidal
representations. We shall describe the main features of the
constructions, and discuss some open questions regarding their overlap,
that is, to what extent representations given by one construction are
also given by the other.
13 May
Alex Bartel (Cambridge)
"On class number relations in dihedral extensions of number fields"
Abstract: In 1950, Brauer and Kuroda independently considered relations of class
numbers and of regulators of intermediate fields in Galois extensions.
These relations arise from the analytic class number formula and Artin
formalism for L-functions and allow one to express certain quotients of
class numbers in terms of corresponding quotients of regulators and of
numbers of roots of unity. In some special cases, the regulator quotient
can then be interpreted as a unit index. For extensions with dihedral
Galois group of order 2p for p an odd prime, this was first done by
Halter-Koch over Q and more recently by Lemmermeyer over arbitrary fields
but under a very restrictive assumption. I will show how to derive a
formula for arbitrary dihedral extensions of order 2p^n. The technique,
which is purely representation theoretic, comes from the theory of elliptic
curves, where one can consider a similar compatibility of the Birch and
Swinnerton-Dyer conjecture with Artin formalism.
20 May
Andreas Langer (Exeter)
"Torsion zero cycles and p-adic integration theory"
Abstract: We study the Chow-group of zero-cycles on the self-product
of a CM-elliptic curve over the field of p-adic numbers and prove that its
p-primary torsion subgroup is finite, provided that p is an ordinary good
reduction prime and the p-adic L-function L_p(E,s) does not vanish at s=0.
In the course of the proof we construct a new indecomposable element in K_1
which is integral at p, by using Coleman's p-adic integration theory and Besser's
computation of syntomic regulators for K_2 of curves and K_1 of surfaces.
27 May
two talks
room 2B08
2:00-3:00, room 2B08
Nigel Boston (UC-Dublin, Wisconsin)
"The fewest primes ramifying in a G-extension of Q"
Abstract:
If G is a finite group, what is the smallest number of primes ramifying in a G-extension
of the rationals? We give evidence for a conjectural answer, together with a conjectural
density for such n-tuples. [Parts are joint work with Ellenberg-Venkatesh and Markin.]
3:30-4:30, room 2B08
Jean-Pierre Serre
"Variation with p of the number of solutions mod p of a given family of equations"
3 June
- 4 June
The London-Paris Number Theory Seminar
at King's College London, room 2B08
theme: p-adic modular forms
speakers: Buzzard, Fargues, Loeffler, Colmez, Mokrane, Dimitrov, Panchishkin
10 June
cancelled
cancelled
The previously announced talk by Lassina Dembele is cancelled because of delays
in getting his visa.
17 June
Christopher Deninger (Muenster)
"Vector bundles on p-adic curves and p-adic representations"
Abstract: The classical Narasimhan-Seshadri correspondence gives a
bijection between stable vector bundles of degree zero on a compact
Riemannian surface and irreducible unitary representations of its
fundamental group. In joint work with Annette Werner we have transferred
this correspondence to some extent to a p-adic setting. We will
report on recent progress and the main open questions in this area.
There is related work of Faltings on p-adic Higgs bundles.
Spring 2009
This term, the seminar was held at University College and was organised by Andrei Yafaev.
The seminar program for Spring 2009
14 January
Victor Abrashkin (Durham)
"p-adic semistable representations and
generalization of the Shafarevich Conjecture"
Abstract: Breuil's theory of semistable p-adic representations is applied to prove the following
property: if X is a projective variety over Q with semistable reduction modulo 3 and good reduction at
all other primes then its Hodge number h^{2,0} = 0.
21 January
no seminar
(Minhyong Kim's inaugural lecture "On numbers and figures" in room 505,
Mathematics Department, UCL at 4.30pm)
28 January
Anna Cadoret (Bordeaux)
"A uniform open image theorem for p-adic representations of etale
fundamental groups of curves"
4 February
Tim Browning (Bristol)
"Rational points on cubic hypersurfaces"
Abstract:
Given a cubic hypersurface X defined over Q, the circle method furnishes
a method for establishing the existence of Q-rational points on the
hypersurface, provided that the dimension is sufficiently large.
Thanks to work of Davenport, and more recently of Heath-Brown, we can
now treat cubic hypersurfaces of dimension at least 12. In this talk I
show how this can be improved to dimension 11 when the underlying cubic
form can be written as the sum of two forms without any variables in common.
11 February
Pierre Parent (Bordeaux)
"Method of Runge and modular curves"
18 February
Adrian Diaconu (Nottingham)
"Trace formulas and moments of automorphic L-functions"
25 February
TWO TALKS
2:00-3:00
Hershy Kisilevsky (Concordia)
"Critical values of derivatives of (twisted) elliptic L-functions"
Abstract: Let $L(E/\Q,s) be the $L$-function of an elliptic curve $E$ defined over the rational
field $\Q.$ If $\chi$ is a Dirichlet character of odd prime order such that $L(E,1,\chi)=0,$ we
examine the special values of the derivative. If $L'(E,1,\chi)$ is non-zero, we provide
computational evidence for an "explicit formula" for its value. We also have some cases of
higher order special values in the case that $\ord_{s=1}L(E,s,\chi)>1.$
4:00-5:00
Christian Wuthrich (Nottingham)
"Self-points on Elliptic Curves"
Abstract: Let $E$ be an elliptic curve of conductor $N$. Given a cyclic subgroup
$C$ of order $N$ in $E$, we construct a modular point $P_C$ on $E$,
called self-point, as the image of $(E,C)$ on $X_0(N)$ under the
modular parametrisation $X_0(N)\to E$. In many cases (e.g. when E is semi-stable), one can prove
that the point is of infinite order in the Mordell-Weil group of $E$ over the field of definition
of $C$. The study of these points in the $PGL_2(\mathbb{Z}_p)$-tower inside $\mathbb{Q}(E[p^\infty])$
continues earlier work of Harris. It is also possible to construct ``derivatives'' \`a la Kolyvagin.
4 March
TWO TALKS
2:00-3:00
Pierre Debes (Lille)
"Specializations of Galois covers"
Abstract:
The motivation is to investigate the specializations
of a Galois cover over some field. The rational points on some
twisted cover provide a key to the problem. Good behaviour of
these twists with respect to reduction leads to some concrete
answer over "big" fields. Our results relate to some questions
in inverse Galois theory, to some works of Fried, Colliot-Thelene,
Ekhedal on Hilbert's irreducibility theorem and to some classical
theorems of Grunwald and Neukirch.
(This is a joint work with Nour Ghazi).
4:00-5:00
Gunter Harder (Bonn)
"Denominators of Eisenstein classes"
9 March
(Monday)
London Number Theory Seminar, Special Lectures
Bao Chau Ngo (Institute for Advanced Study)
Lecture 1: "Fundamental lemma and Hitchin fibration"
at 3:00 pm, room 706
Lecture 2: "Symmetry of Hitchin fibration and endoscopy"
at 4:30 pm, room 706
11 March
Ambrus Pal (Imperial)
"Rational points on genus one curves"
13 March
(Friday)
London Number Theory Seminar, Special Lectures
Bao Chau Ngo (Institute for Advanced Study)
Lecture 3: "Decomposition theorem in the case of the Hitchin fibration"
at 4:00 pm, room 500
18 March
Richard Hill (UCL)
"Residually infinite extensions of arithmetic groups"
25 March
Victor Snaith (Sheffield)
"Computing the Borel regulator"
in room 505
Autumn 2008
This term, the seminar was held at Imperial College and was organised by Kevin Buzzard.
The seminar program for Autumn 2008
1 October
14.00-15.00
room 342
Thomas Zink (Bielefeld)
"p-divisible groups over regular local rings of mixed characteristics"
8 October
Go Yamashita
"Upper bounds for the dimensions of p-adic multiple
zeta values"
15 October
Gihan Marasingha (Bristol)
"A degree 4 del Pezzo surface: Manin's
conjecture and almost primes"
22 October
Cecilia Busuioc (Imperial)
"Milnor K-theory and Modular Symbols"
29 October
Tejaswi Navilarekallu (Vrije Universiteit Amsterdam)
"Equivariant p-adic L-values"
5 November
Owen Jones (Imperial)
"Analytically induced representations and generalised
Verma modules"
12 November
Seidai Yasuda (Kyoto)
"Diagonal periods of GL(n) over the rational function field"
17 November
(Monday)
The London-Paris Number Theory Seminar
at the Insitut Henri Poincaré in Paris
19 November
Jeanine van Order (Cambridge)
"Analogues of Rohrlich's theorem"
26 November
Kevin McGerty (Imperial)
"A gentle introduction to the geometric Langlands program"
3 December
Mohamed Saidi (Exeter)
"On Grothendieck's anabelian section conjecture for curves"
10 December
Gaetan Chenevier (ENS Paris)
"The infinite fern of Galois representations of type U(3)"
17 December
Payman Kassaei (King's)
"Geometry of Hilbert modular varieties and canonical subgroups of abelian varieties with
real multiplication"
Summer 2008
This term, the seminar was held at King's College and was organised by David Solomon.
The seminar program for Summer 2008
23 April
Henri Johnston (Oxford)
"Non-existence and splitting theorems for normal integral bases"
Abstract: This is joint work with Cornelius Greither. We establish new
conditions that prevent the existence of (weak) normal integral bases in
tame Galois extensions of number fields. This leads to the following
result: under appropriate technical hypotheses, the existence of a
normal integral basis in the upper layer of an abelian tower
Q \subset K \subset L
forces the tower to be split in a very strong sense.
30 April
Richard Hill (UCL and Heilbronn Institute)
"Vanishing theorems for p-adic automorphic forms"
7 May
- 8 May
The London-Paris Number Theory Seminar
13 May
Tuesday
3:30-4:30
room 436
Cristian Popescu (UCSD)
"On the Coates-Sinnott Conjectures"
14 May
Andrew Booker (Bristol)
"Computing automorphic forms on GL(3)"
Abstract:
My student, Ce Bian, announced the computation of a few
"generic" rank 3 automorphic forms (meaning they are not lifts from
lower rank examples) at the AIM workshop "Computing arithmetic
spectra" in March. I will give a brief introduction to the theme of
the workshop and describe Bian's computations. I'll also say a few
words about the bewildering amount of attention that the work received
subsequently.
21 May
Ambrus Pal (Imperial College)
"The Manin constant of elliptic curves over function fields"
Abstract:
We study the p-adic valuation of the values of normalized
Hecke eigenforms attached to non-isotrivial elliptic curves defined over
function fields of transcendence degree one over finite fields of
characteristic p. Under certain assumptions we derive lower and upper
bounds on the smallest attained valuation in terms of the minimal
discriminant. As a consequence we show that the former can be
arbitrarily small. We also use our results to prove for the first time
the analogue of the degree conjecture unconditionally for infinite
families of strong Weil curves defined over rational function fields.
28 May
Gautam Chinta (City College of New York)
"Sums of two squares and sums of three squares"
Abstract:
I will begin by describing three results -- Gauss's
three squares theorem, Hamburger's converse theorem, and Maass's evaluation
of a sum over Heegner points of the Eisenstein series for the modular group.
I then will describe conjectural generalizations of these results to GL(3).
The unifying theme is a conjecture of Jacquet on orthogonal periods of automorphic
forms on GL(r). This is a joint work with Omer Offen.
4 June
Doug Ulmer (U. Arizona at Tucson, and Paris)
"On Mordell-Weil groups of abelian varieties over function fields"
Abstract:
I will sketch a construction which, among other things,
relates CM of certain abelian varieties over a field k to Mordell-
Weil groups of certain abelian varieties over K=k(t). The
construction yields completely explicit Mordell-Weil groups of
arbitrarily large rank for finite k and a less explicit, but new,
construction of abelian varieties over K of moderately large rank when
k is the field of algebraic numbers.
11 June
Steven Galbraith (Royal Holloway)
"Applications of the Frobenius map in elliptic curve cryptography"
Abstract:
Elliptic curves over finite fields provide groups for which
the discrete logarithm problem seems to be hard. Hence, elliptic curves
have applications in public key cryptography. The Frobenius map has been
used to speed up arithmetic on elliptic curves. The talk will survey
some of these ideas. We will also discuss security implications of using
Frobenius maps and present a new algorithm for solving the "Frobenius
expansion discrete logarithm problem"
18 June
Mahesh Kakde (Cambridge)
"On the non-commutative Main Conjecture for totally real number fields"
25 June
Jonathan Pila (Bristol)
"Rational points of definable sets and the Manin-Mumford conjecture"
Abstract:
I will discuss problems and results concerning the distribution of
rational points on certain non-algebraic sets.
More specifically, definable sets in o-minimal structures. I will
describe a result, joint with Wilkie, that such a set X can have only
``few'' rational points in a suitable sense, that do not lie on some
connected semi-algebraic subset of X of positive dimension. I will
describe some further results and conjectures, connections with
transcendence theory, and a new proof (with Zannier) of the
Manin-Mumford conjecture by combining these ideas with a result of
Masser.
Spring 2008
This term, the seminar was held at University College and was organised by Minhyong Kim.
The seminar program for Spring 2008
9 January
Pierre Debes (Lille)
"Inverse Galois theory, Abelian varieties and modular towers"
Abstract: This is a joint work with Anna Cadoret. We show a new
constraint in constructing Galois covers of $\P^1$ over $\Q$ with
a given Galois group $G$. If for some prime $p$, the order of the
abelianization $P^{ab}$ of the $p$-Sylow subgroups $P$ of $G$
is suitably large, compared to the index $[G:P]$ and the number
$r$ of branch points, then the branch points must coalesce
modulo small primes. This is related to some rationality
questions on the torsion of abelian varieties. This
connection also provides a new viewpoint and
new results on the Modular Tower program.
16 January
Makis Dousmanis (Paris 13)
"Reductions of some families of
two-dimensional crystalline representations"
23 January
Konstantin Ardakov (Nottingham)
"Reflexive ideals in Iwasawa algebras"
30 January
Ben Green (Cambridge)
"Distribution of Polynomials over finite fields"
Abstract: Let F be a finite field and consider polynomials P : F^n -> F in
n variables. What can we say about polynomials which are not
equi-distributed, i.e. for which the proportion of x for which P(x) = c is
not roughly 1/|F|? We introduce a notion of "rank" for multi-variable
polynomials and show that such polynomials must have low rank.
We apply this result to the study of so-called Gowers norms. Let P : F^n
-> F be any function. It is well-known that P is a polynomial of degree d
if the polynomial Q obtained by differencing d+1 times is identically
zero. What if Q is not identically zero, but merely biased towards zero?
The inverse conjecture for the Gowers norms predicted that, in this case,
P correlates with a degree d polynomial. Using the result described above
we establish this in certain cases. We will also discuss an example which
shows that the conjecture can fail in very low characteristic.
Joint work with T. Tao.
6 February
two talks
4 pm-6 pm
4 pm-5 pm:
Takako Fukaya (Keio and Cambridge)
"Root numbers, Selmer groups, and non-commutative Iwasawa theory"
5 pm-6 pm:
Kazuya Kato (Kyoto and Cambridge)
"Classifying spaces of mixed Hodge (resp. p-adic Hodge) structures"
13 February
Nick Shepherd-Barron (Cambridge)
"Thomae's formula for non-hyperelliptic curves"
Abstract: In 1857 Thomae gave formulae for the theta-constants of a
hyperelliptic curve in terms of projective data of the curve. In this
talk we explain what this means in terms of moduli spaces and extend it
to non-hyperelliptic curves.
20 February
Fabien Trihan (Nottingham)
"Crystalline representations and F-crystals"
27 February
Andreas Doering (Imperial)
"Topos theory in the foundations of physics"
5 March
Burt Totaro (Cambridge)
"Moving codimension-one subvarieties over finite fields"
12 March
Tamas Hausel (Oxford)
"Arithmetic harmonic analysis on character and quiver varieties"
Autumn 2007
This term, the seminar was held at Imperial College and was organised by Ambrus Pal.
The seminar program for Autumn 2007
3 October
David Loeffler (ICL)
"Overconvergent p-adic automorphic forms and
eigenvarieties for compact reductive groups"
Abstract:
I shall describe a construction of an eigenvariety parametrising p-adic automorphic
forms for any reductive group G over Q that is split at p and compact at infinity.
The construction generalises the work of Chenevier for compact forms of GL_n and
Buzzard for quaternion algebras. The method gives a space of automorphic forms for
each standard parabolic subgroup P of G; in this gives a hierarchy of "semi-classical"
automorphic forms intermediate between the space of classical forms (corresponding to P = G)
and the spaces constructed by Chenevier in the unitary case (which correspond to P = Borel).
If there is time, I shall also mention ongoing work on classicality criteria and connections
to Galois representations.
10 October
Yiannis Petridis (UCL)
"On the distribution of modular symbols"
17 October
Andreas Schweizer (University of Exeter)
"On the torsion of elliptic curves
over sufficiently general function fields"
Abstract:
If K varies over all complex function fields and E varies over all elliptic curves over K
with j(E) not in C, it is known that the size of the torsion group E(K)_{tors} can be uniformly
bounded by a number depending only on the genus of K. Moreover, if one restricts to function
fields K that are ``special'', for example hyperelliptic, one can even give absolute bounds
(not depending on the genus of K) for the size of E(K)_{tors}. We will discuss what happens
if K varies over all function fields that are ``sufficiently general''.
24 October
David Solomon (King's College)
"Stickelberger's Theorem Revisited"
Abstract: Stickelberger's Theorem (from 1890) gives an explicit ideal in the Galois group-ring
which annihilates the imaginary part of the class group of an abelian field. In the 1980s Tate
and Brumer proposed a generalisation (the "Brumer-Stark conjecture" ) for in any abelian
extension of number fields K/k with K CM and k totally real. Both the theorem and the conjecture
leave certain questions unanswered: Is the (generalised) Stickelberger ideal the full annihilator,
the Fitting ideal or what? And, at a more basic level, what can we say in the plus part, eg for a
real abelian field? (In the latter case, Stickelberger's theorem amounts to little more than 0=0!)
I shall discuss possible answers, some still conjectural, to pieces of these puzzles, using two
new p-adic ideals of the group ring. There are interesting connections with Iwasawa Theory,
the Equivariant Tamagawa Number Conjecture etc.
31 October
Samir Siksek (University of Warwick)
"Chabauty for Symmetric Powers of Curves"
Abstract: Chabauty is a classical method for computing the rational points of curves of higher
genus. In this talk, we explain an adaptation of Chabauty which allows us in many cases to
compute all rational points on the d-th symmetric power of a curve provided the rank of the
Mordell-Weil group of the Jacobian is at most g-d (where g is the genus). We illustrate this
by giving two examples of genus 3, one hyperelliptic and the other plane quartic.
7 November
Urs Hartl (University of Münster)
"Period Spaces for Hodge-Structures in Equal Characteristic"
Abstract: We construct period spaces for Hodge structures in equal characteristic. These Hodge
structures were invented by Pink. The period spaces are analogues of the Rapoport-Zink period
spaces for Fontaine's filtered isocrystals in mixed characteristic. For our period spaces we
determine the image of the period morphism as a Berkovich open subspace. We prove the analogue
of a conjecture of Rapoport Zink stating the existence of interesting local systems on this
image. Moreover, we prove the analogue of the Colmez-Fontaine Theorem that "weakly admissible
implies admissible". As a consequence the Berkovich open subspace mentioned above contains
every classical rigid analytic point of the period space.
12 November
The London-Paris Number Theory Seminar
14 November
Alberto Minguez (University of East Anglia)
"On the Howe correspondence"
Abstract: The aim of this talk is to introduce the audience to the theory of local Howe
correspondence. For the dual pair of type (Gl(n), Gl(m)) we will show a new proof which
allows us to describe the correspondence in terms of Langlands parameters. At the end, we
will discus about the possibility of having such a correspondence for l-modular representations.
21 November
4 pm-5:30 pm
Eyal Goren (McGill University, Montréal)
"Class invariants for CM fields of degree 4"
Abstract: The problem of effective construction of units in abelian extensions of number fields is
at present out of reach. Notwithstanding conjectural constructions, the only exceptions are the
constructions for abelian extensions of Q and of a quadratic imaginary field, where the units are
the cyclotomic and elliptic units respectively. One reason one seeks such constructions is to find
Stark units which appear in Stark's conjectures on special values of L functions.
In this talk, after explaining what is the source of the difficulty, I shall survey what we know
at present about the case of CM fields of degree 4, focusing on my work with Ehud de Shalit,
Kristin Lauter and Daniel Vallieres. Time allowing, I shall try and put the results in the
perspective of the work of Jan Bruinier and Tonghai Yang, indicate some proofs of our results
and discuss work in progress.
28 November
Carlos Castano-Bernard (ICTP, Trieste)
"On the subgroup generated by the traces of
Heegner points on elliptic curves"
Abstract: Consider an elliptic curve E over Q and assume its L-function has a simple zero at
s = 1. In particular, there is a non-constant morphism X0(N)/wN --> E defined over Q, where
wN is the Fricke involution and N is the conductor of E. So the trace of each Heegner point
on E is Q-rational. Moreover, it is well-known that E(Q)/E(Q)tors is isomorphic to Z, and in
fact the images of the traces in E(Q)/E(Q)tors generate a subgroup of finite index I. In this
talk we shall discuss a conjecture that predicts that whenever N is prime and the index I > 1,
then the real locus (X0(N)/wN)(R) has more than one connected component or--less likely--the
Tate-Shafarevich group of E is non-trivial.
5 December
Laurent Fargues (Université Paris-Sud, Orsay)
"Ramification of Lubin-Tate groups
and the Bruhat-Tits building"
Abstract: One of the purposes of this talk is to give a description of the
isomorphism between the p-adic Lubin-Tate and Drinfeld towers at
the level of their skeletons. For the Drinfeld space, its skeleton
is the Bruhat-Tits building of the linear group. For example, we
can describe explicitly the pull-back of this simplicial structure on
the open p-adic ball associated to the Lubin-Tate space. We also
study in detail the different ramification filtrations (upper
and lower) associated to Lubin-Tate groups. We give applications
to generalized canonical subgroups and fundamental domains for
Hecke correspondences.
12 December
Sarah Zerbes (ICL)
"Formulae for the higher Hilbert pairing"
Summer 2007
This term, the seminar was held at University College and was organised by Richard Hill.
The seminar program for Summer 2007
25 April
Shaun Stevens (UEA)
"Supercuspidal representations of p-adic classical groups"
2 May
The London-Paris Number Theory Seminar
11 am - 4:30 pm, Imperial College London
9 May
Ben Smith (Royal Holloway)
"Computing Explicit Isogenies"
Abstract:
Isogenies --- surjective homomorphisms of algebraic groups
with finite kernel --- are basic objects in number theory.
Algorithms for computing with isogenies of elliptic curves
are well-known; in higher dimensions, however, the situation
is more complicated, and few explicit non-trivial examples
of isogenies are known. We will describe some interesting
examples of explicit isogenies of Jacobians of low-genus
curves, discuss some of the computational issues, and give
some applications in modern cryptography.
16 May
Matthias Strauch
"Potentially crystalline representations
and associated p-adic representations of GL_2"
Abstract:
This is about joint work in progress with C. Breuil.
For a certain family of potentially crystalline but not semistable
two-dimensional representations of the absolute Galois group
of Q_p we construct locally analytic representations of GL_2(Q_p),
naturally parameterised by the Galois representations.
23 May
Manuel Breuning (KCL)
"Determinant functors and Euler characteristics"
30 May
Jayanta Manoharmayum (Sheffield)
"Lifting Galois representations"
Spring 2007
This term, the seminar was held at King's College and was organised by David Burns.
The seminar program for Spring 2007
17 January
Kevin Buzzard (Imperial)
"Mod p Galois representations and modular forms"
24 January
Richard Hill (UCL)
"Singular cohomology of modular curves"
Abstract: Let \Gamma be an arithmetic group acting on the upper half-plane H, either with cusps or
cocompact. Let \Gamma' be a normal subgroup of \Gamma. Then the quotient group G=\Gamma/\Gamma' acts
on the cohomology of \Gamma'. I'll describe the structure of H^1(\Gamma',\Z) as a \ZG-module.
31 January
David Burns (King's)
"Iwasawa theory of elliptic curves in p-adic Lie extensions"
7 February
Alex Paulin (Imperial)
"Local to Global Compatibility on the Eigencurve"
Abstract:
To any classical cuspidal eigenform one can attach both a smooth irreducible representation of GL_2(Q_l)
and a two dimensional Frob-semisimple Weil-Deligne representation. Classical Local-Global compatibility
ensures that these agree under the (correctly normalised) local langlands correspondence. I will discuss
ways of attaching such objects to overconvergent p-adic eigenforms across the eigencurve and to what
extent local-global compatibility remains valid.
14 February
Tom Fisher (Cambridge)
"Finding rational points on elliptic curves using 6-descent and 12-descent"
Abstract:
Descent on an elliptic curve E is used to obtain partial
information about both the group of rational points (the Mordell-Weil
group) and the failure of the Hasse principle for certain twists of E (the
Tate-Shafarevich group). The Selmer group elements computed may be
represented as n-coverings of E. Traditionally one takes n to be a prime
power. Breaking with this tradition, I explain how to combine the data of
an m-covering and an n-covering, for m and n coprime, to obtain an
mn-covering. This technique improves the search for rational points on E.
In particular using 6-descent and 12-descent, I was recently able to find
all the "missing" generators for the elliptic curves of analytic rank 2 in
the Stein-Watkins database.
21 February
Daniel Delbourgo (Nottingham)
"Non-abelian congruences between L-values of elliptic curves"
28 February
Olivier Brinon (Paris 13)
"Overconvergence of $p$-adic representations: the relative case (joint
work with F. Andreatta)"
7 March
Jan Kohlhaase (Muenster)
"The Cohomology of locally analytic Representations"
Abstract:
Starting with smooth representations of a p-adic reductive group, we will recall what is meant by
a supercuspidal representation and the role such representations play in the local Langlands
correspondence. We will then pass to locally analytic representations in the sense of
Schneider/Teitelbaum, sketch the construction of locally analytic cohomology and generalize the
above notion of supercuspidality to locally analytic representations. In the end, we will compute
the (higher) Jacquet modules of locally analytic principal series representations and indicate why
this is of importance for the p-adic Langlands correspondence.
14 March
Colin Bushnell (King's)
"Characters and constants"
21 March
Tobias Berger (Cambridge)
"Congruences between modular forms over imaginary quadratic fields"
Abstract:
We present two applications of congruences involving Harder's Eisenstein cohomology classes. We
first prove a lower bound for the size of the Selmer group of certain Galois characters of
imaginary quadratic fields coinciding with the value given by the Bloch-Kato conjecture. We
further show how to obtain instances of the Fontaine-Mazur conjecture for imaginary quadratic
fields in the residually reducible case. The latter is joint work in progress with Kris Klosin.
Autumn 2006
This term, the seminar was held at Imperial and was organised by
Alexei Skorobogatov.
The seminar program for Autumn 2006
4 October
Andrei Yafaev (UCL)
"On the triviality of rational points on certain Atkin-Lehner
quotients of Shimura curves"
Abstract:
This is a joint work with Pierre Parent. We use a modification of Mazur's method
to prove that the only possible rational points on certain Atkin-Lehner quotients of
Shimura curves come from CM points.
11 October
Alexei Skorobogatov (Imperial College)
"A finiteness theorem for the Brauer group of K3 surfaces"
Abstract:
Let k be a field finitely generated over the rationals, and let X be a K3 surface over k.
We prove that Br(X)/Br(k) is finite.
18 October
Tim Dokchitser (Cambridge)
"Parity of ranks for elliptic curves with a cyclic isogeny"
Abstract:
This is a joint work with Vladimir Dokchitser. Let E be an elliptic curve over a number field K
which admits a cyclic p-isogeny and semistable at primes above p. Then one can determine the root
number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for
such curves (with an extra mild assumption for p=2).
25 October
no seminar (Imperial's Commemoration Day)
1 November
Jan Nekovar (Paris 7)
"Parity of ranks of Selmer groups in p-adic families"
8 November
Sarah Zerbes (Imperial)
"Higher-dimensional logarithmic derivatives"
Abstract:
In my talk, I will explain how to construct logarithmic derivative maps for n-dimensional local
fields of mixed characteristic (0,p). The main ingredients for this construction are higher-dimensional
rings of overconvergent series and Tony Scholl's work on general fields of norms. As an application of
the logarithmic derivative, I will give a new construction of Kato's dual exponential map for K_n.
13 November
Séminaire de théorie des nombres
Londres-Paris à l'Institut Henri Poincaré
programme
Orateurs: Fred Diamond,
Toby Gee,
Florian Herzig.
15 November
Andres Helfgott (Bristol)
"How small must ill-distributed sets be?" (joint with A Venkatesh)
Abstract:
Consider a set $S\subset \mathbb{Z}^n$. Suppose that, for many primes $p$,
the distribution of $S$ in congruence classes $\mo p$ is far from uniform.
How sparse is $S$ forced to be thereby?
A clear dichotomy appears: it seems that $S$ must either be very small or
possess much algebraic structure. We show that, if $S\subset
\mathbb{Z}^2 \cap \lbrack 0, N\rbrack^2$
occupies few congruence classes $\mo p$ for many $p$, then either
$S$ has fewer than $N^{\epsilon}$ elements or most of $S$ is contained in
an algebraic curve of degree $O_{\epsilon}(1)$. Similar statements are
conjectured for $S\subset \mathbb{Z}^n$, $n\neq 2$.
We follow an approach that combines ideas from the larger sieve of
Gallagher \cite{Ga} and from the work of Bombieri and Pila \cite{BP}. All
techniques used are elementary.
22 November
Fred Diamond (King's)
"The weight part of Serre's conjecture for Hilbert modular forms"
Abstract:
I will explain the statement of a generalization of
Serre's conjecture on mod p Galois representations
to the context of Hilbert modular forms. The emphasis
will be on the recipe for the set of possible weights
(formulated by Buzzard, Jarvis and myself, and partly
proved by Gee) and its behavior in some special cases.
29 November
Ambrus Pal (Imperial)
"On a conjecture about the cohomology of arithmetic groups"
6 December
Jean Gillibert (Manchester)
"Geometric Galois module structure and abelian varieties of higher dimension"
Abstract:
The so-called class-invariant homomorphism $\psi_n$, introduced by M. J.
Taylor, measures the Galois module structure of (rings of integers of)
extensions of the form $K(\frac{1}{n}P)/K$, where $K$ is a number field,
$P$ is a $K$-rational point on an abelian variety $A$, and $n>1$ is an integer.
When $A$ is an elliptic curve and $n$ is coprime to 6, then $\psi_n$
vanishes on torsion points. We explain here how, using Weil restrictions
of elliptic curves, it is possible to construct abelian varieties of higher
dimension for which this vanishing result is no longer true.
13 December
Payman Kassaei (King's)
"A ``subgroup-free" approach to Canonical Subgroups"
Abstract:
I will discuss joint work with E. Goren in which we present a
``subgroup-free'' approach to canonical subgroups, which in particular
extends all aspects of the classical theory of canonical subgroups of
elliptic curves to many various Shimura curves of interest.
Summer 2006
This term, the seminar was held at KCL and was organised by
David Burns.
The seminar program for Summer 2006
"Zeta functions of surfaces over finite fields"
"Two approaches in the study of $X_0^+(N)(Q)$"
Abstract: In this talk we discuss two approaches in the study of the
rational points of $X_0^+(N)$. The first was suggested by certain geometric
properties of rational points of the canonical embedding of $X_0^+(N)$ in
projective space, while the second approach was suggested by questions
raised by Shimura on the connected components of the real locus of a given
variety defined over the rational numbers.
"Examples of explicit isogenies"
Abstract: In "Factorisations explicites de g(y) - h(z)", Cassou-Nogues and
Couveignes give a classification of the pairs of irreducible polynomials g and h such that g(y) - h(z) is
reducible. We describe their results, and show how they may be used to derive families of hyperelliptic
Jacobians with an explicit isogeny.
"The Beilinson conjectures for $K_2$ of curves"
Abstract: The Birch-Swinnerton-Dyer conjecture relates the value of the L-function of an elliptic
curve defined over a number field at 1 with its codimension 1 Chow group. The latter is part of $K_0$ of the
curve. Beilinson formulated conjectures of a similar flavour for all K-groups of any smooth projective variety
defined over number fields, and we'll discuss those conjectures for $K_2$ of curves.
"Double zeta values and modular forms"
Abstract: We give new relations among double zeta values \zeta(r,s) = \sum_{m>n>0}
m^{-r}n^{-s} and show that the structure of the Q-vector space of all
relations among double zeta values of weight k is connected in several
different ways with the structure of the space of modular forms of
weight k on the full modular group.
"Representations of p-adic GL(2), old and new"
Abstract: Classical modular forms have their local counterpart, which are irreducible representations
of GL(2,Q_p) on complex vector spaces. Congruences between modular forms lead to study continuous representations
of GL(2,F), where F is a p-adic field, on non-Archimedean Banach spaces,or on fields of positive characteristic.
I shall review recent work and conjectures, with special emphasis on the case of mod-p representations.
"On the leading terms of p-adic L-functions in
non-commutative Iwasawa theory"
Abstract: In the $GL_2$-main conjecture for elliptic curves without complex multiplication formulated
by Coates, Fukaya, Kato, Sujatha and the speaker the existence of a (non-commutative) $p$-adic $L$-function
$\mathcal{L}$ is crucial satisfying an interpolation formula related to the value at $s=1$ of certain twists of
the Hasse-Weil $L$-function of $E$ twisted by certain Artin representations. By the work of Fukaya and Kato
the interpolation property and existence of $\mathcal{L}$ is a consequence of their non-commutative Tamagawa
number conjecture as well as a local analogue of the non-commutative Tamagawa number conjecture. The aim of
this talk is to report on recent joint work with David Burns: We introduce leading terms of the $p$-adic
$L$-function $\mathcal{L}$ at Artin representations and extend Fukaya and Kato's interpolation formula now
also involving the leading terms of the twisted complex Hasse-Weil $L$-functions.
"Fundamental groups and Diophantine Geometry"
"Galois structure and Riemann-Roch theorems"
Abstract: The talk will begin with a general discussion of the importance of Euler
characteristics the study of arithmetic modules. We shall then consider the use of Riemann Roch theorems in the
determination of Euler characteristics: firstly, in a fairly general situation of varieties over fields; then
for various torsors over arithmetic varieties.
"Abelian varieties of large rank over function
fields"
Abstract: For every prime p and every integer g>0 we construct explicit
abelian varieties of dimension g over F_p(t) for which the Birch and
Swinnerton-Dyer conjecture holds and with arbitrarily large Mordell-Weil rank.Spring 2006
This term, the seminar was held at KCL and was
organised by David Solomon.
The seminar programme for Spring 2006
18 January
Tim Dokchitser (Cambridge)
"Ranks of elliptic curves in cubic extensions"
Abstract:
For an elliptic curve E over a number field K,
I shall prove that the algebraic rank of E goes up in infinitely many
extensions of K obtained by adjoining a cube root of an element of K.
I shall also discuss how this relates to root numbers and Iwasawa theory,
with E=X_1(11) over Q as a specific example.
25 January
Andrei Yafaev (UCL)
"The Andre-Oort conjecture"
Abstract:
This is a joint work with Bruno Klingler. We explain a proof of the Andre-Oort
conjecture under the assumption of the generalised Riemann Hypothesis.
1 February
David Burns (KCl)
"Algebraic p-adic L-functions in non-commutative Iwasawa theory"
Abstract:
There have been several important developments
in non-commutative Iwasawa theory over the last few years. We discuss a natural
construction of algebraic p-adic L-functions in this setting and discuss some
interesting consequences for the main conjectures of non-commutative Iwasawa
theory formulated by Coates, Fukaya, Kato, Sujatha and Venjakob and by Fukaya
and Kato.
8 February
Daniel Caro (Durham)
"Towards a good p-adic cohomology"
Abstract:
First, we will trace the history of the search for a good p-adic
cohomology over schemes in characteristic p. We arrive at the construction
of Berthelot's arithmetical D-modules. We will explain why these objects
now represent the only possibility of obtaining a good p-adic cohomology.
An important result which inspires trust in this theory is the following:
for every overholonomic F-complex E of arithmetical D-modules over a
variety X of characteristic p, there exists a splitting of X into locally closed
subvarieties X _i such that the restrictions of E on X _i become much simpler
(i.e., come from overconvergent F-isocrystals). In this talk, we will recall basic
definitions and explain the meaning of this splitting.
15 February
Ivan Horozov (Durham)
"Euler characteristics of arithmetic groups"
Abstract:
The talk will be about Euler characterisitcs of the
general linear group, the special linear group, and the symplectic
group over rings of algebraic integers. I will present a method for
computing the homological Euler characteristic, as well as some
applications to values of Dedekind zeta function at -1 and at -3
and to Kummer and Greenberg criteria for divisibility of certain
class numbers by a prime.
22 February
Avner Ash (Boston College)
"Symmetries of Algebraic Numbers"
Abstract:
The Absolute Galois Group of Q, notated G_Q acts on
the solution sets of systems of polynomial equations with rational
coefficients. Linear representations of G_Q play a leading role in
getting information about the solution sets, e.g. in Wiles's proof
of Fermat's Last Theorem. One influential conjecture in this area
is that of Serre, concerning 2-dimensional representations of G_Q
over finite fields. I will briefly review Serre's conjecture and
discuss generalizations to higher dimensional representations.
1 March
Emmanuel Kowalski (Bordeaux)
"The algebraic principle of the large sieve"
Abstract:
Linnik's original large sieve gives upper bounds for the
number of integers in an interval with reductions modulo primes
restricted to fall in fairly small sets. The talk will describe an
abstract sieve setting which can lead to such results in more general
situations. Then applications to the average distribution of Frobenius
elements in families of algebraic varieties over finite fields will be
discussed, and some work in progress concerning arithmetic properties of
integral unimodular matrices. In one case the Riemann Hypothesis of
Deligne is the crucial ingredient, in the other the spectral theory of
automorphic forms appears naturally.
8 March
Misha Gavrilovich (Oxford)
"Model theory, Z-extensions of C*, the exponential function,
and a homotopy-theory viewpoint on some arithmetic issues"
15 March
James McKee (Royal Holloway)
"Salem numbers, Pisot numbers, graphs, and Mahler measure"
Abstract:
This is joint work with Chris Smyth.We use graphs to define
sets of Salem and Pisot numbers, and prove that the union of these
sets is closed, supporting a conjecture of Boyd that the set of all
Salem and Pisot numbers is closed. We find all trees that define Salem
numbers. We show that for all integers n the smallest known element
of the n-th derived set of the set of Pisot numbers comes from a graph.
We define the Mahler measure of a graph, and find all graphs of Mahler
measure less than (1+sqrt5)/2. We start the task of extending this work
from graph adjacency matrices to all integer symmetric matrices by classifying
all such matrices having all eigenvalues in the interval [-2, 2].
22 March
Graham Everest (UEA)
"Descent and Divisibility"
Abstract:
In 2001, Bilu, Hanrot and Voutier proved
that for every n>30, the nth term of a Lucas or Lehmer
sequence must have a primitive divisor. This is
a remarkable result because of its uniform
nature and the smallness of the bound 30. I will
report on an elliptic analogue of their theorem.
Also, I will report on the apparently harder statement
about prime values.
5 April
Werner Bley (University of Kassel)
"Computation of class groups"
12 April
Andrew Jones (KCL)
"Dirichlet $L$-functions at $s=1$ and Fitting invariants of ideal class groups"
Abstract:
We show that a special case of the equivariant Tamagawa number
conjecture (ETNC) of Burns and Flach implies a refinement of a
`$p$-adic integrality conjecture' of Solomon concerning leading
terms at $s = 0$ of certain `Twisted Zeta-functions'. In fact, the
ETNC implies that an ideal defined by Solomon belongs to the initial
Fitting invariant of a certain ideal class group. Since the relevant
case of the ETNC is known to be valid for absolutely abelian fields
we thereby obtain new results about the structure of ideal class
groups. In particular, we obtain an analogue of Stickelberger's
Theorem.
Autumn 2005
This term, the seminar was held at Imperial and was
organised by Toby Gee.
The seminar programme for Autumn 2005
5 October
Toby Gee (Imperial)
"On the weights of mod p modular forms"
12 October
Manuel Breuning (KCL)
"A reinterpretation of the Chinburg
conjectures"
19 October
Adam Joyce (Imperial)
"Stable models of modular curves"
Abstract:
Using the Drinfeldian notion of level structures, one can obtain integral
moduli spaces over rings of cyclotomic integers, whose generic fibres are
the well-known algebraic curves over Q, X_0(N) and X_1(N). These moduli
spaces are not smooth in characteristics dividing the level. For applications
to arithmetic geometry, it is advantageous to have stable models (in the sense
of Deligne-Mumford) for algebraic curves. The moduli spaces above are often not
stable models for their generic fibres. We describe certain stable models, using
tools from algebraic geometry.
26 October
David Solomon (KCL)
"Stark Units, Hilbert Symbols and a Stickelberger Ideal at s=1"
Abstract:
Let M/k be an abelian extension of totally real number fields. For
appropriately chosen finite sets of places S of k, Stark's conjecture
predicts the existence of (an element of an exterior power of the) S
-units of M whose equivariant regulator gives the leading term at s=0
of Artin L-functions L_S(s,\chi) for all characters \chi of
G=Gal(M/k). These `Stark Units' are cyclotomic units if k=Q but
otherwise are not known to exist. If instead M is of CM type (and k is
still totally real) the Brumer-Stark conjecture predicts that the values
at s=0 of the L_S(s,\chi) now define an ideal of ZG that generalises
the Stickelberger ideal (the case k=Q). In particular, it annihilates
the `odd' part of Cl(M). I shall discuss two p-adic conjectures that,
in a sense, tie these two other conjectures together. For all odd \chi,
the functional equation relates L_S(0,\chi) to L_S(1,\chi) . For each
odd prime p, the latter values allow us to define a map the exterior
power of the p-semilocal units of M into Q_pG . I conjecture firstly
that the image of this map lies in Z_pG and secondly (when M contains
p^nth roots of unity) that the map is congruent modulo p^n to one
defined using Hilbert symbols and the (conjectural) Stark units coming
from the maximal real subextension M^+/k.
2 November
Cornelius Greither (Munich)
"Fitting ideals of class groups via the
Equivariant Tamagawa Number conjecture"
9 November
Samir Siksek (Warwick)
"Classical and modular approaches to
exponential Diophantine problems"
16 November
Denis Benois (Besan n)
"Iwasawa theory of crystalline representations and
$(\phi,\Gamma)$-modules"
23 November
Tony Scholl (Cambridge)
"Higher fields of norms and
(phi,Gamma)-modules"
30 November 2.30pm
Don Zagier (Bonn)
"Double zeta values and modular forms"
30 November 4.00pm
Richard Pinch
"The distribution of Carmichael numbers"
7 December
David Whitehouse (Caltech)
"The twisted weighted fundamental lemma for the transfer of automorphic forms from GSp(4) to GL(4)"
14 December
Tim Browning (Bristol)
"Density of rational points on smooth
hypersurfaces"
Abstract:
Let $X$ be a non-singular projective hypersurface of degree $d>1$ and
dimension $k$. It has been conjectured that the number of rational points
on $X$, which have height at most $B$, should be $O(B^{k+\eps})$ for any
$\varepsilon>0$. The implied constant here should be allowed to depend at
most upon $d,k$ and the choice of $\eps$. In this talk, which comprises
joint work with Heath-Brown, we discuss the final resolution of this
conjecture.
Summer 2005
This term, the seminar was held at Imperial and was
organised by Alexei Skorobogatov.
The seminar programme for Summer 2005
20 April
Toby Gee (Imperial)
"New results for
companion forms over totally real fields"
27 April
Kazuhiro Fujiwara (Nagoya and Cambridge)
"Galois representations over cyclotomic towers"
4 May
Peter Swinnerton-Dyer (Cambridge)
"Counting rational points and the Manin conjecture"
11 May
Vassily Golyshev (Independent Moscow University)
"Differential equations of quantum cohomology and higher Apery recurrences."
Abstract:
We say that a linear polynomial recurrence with integer coefficients is
of
Apery type if it has two solutions, the quotient of which b_n/a_n tends
to
an irrational zeta (or L-function) value at an integer point. We
consider
recurrences that are Mellin transforms of differential equations of
quantum cohomology for Grassmannians. We conjecture
that these recurrences are of Apery type. We prove certain cases of this
conjecture for Grassmannians of classical groups by reducing the DEs in
question to modular ones.
18 May
Daniel Delbourgo (Nottingham)
"Euler products over C_p"
Abstract:
Standard Euler products converge in some right half-plane
Re(s)>constant. If one tries the same trick replacing the complex
numbers with the p-adics, things quickly go wrong. We first explain a
way of making Euler products converge over C_p, the Tate field.
Fortunately, these products converge to the values of classical
L-functions with appropriate modifications to the Euler factor at p.
The proof uses fractional calculus and something resembling an explicit
reciprocity law. If we've got time, we'll mention some convergence
calculations for p-adic L-functions of elliptic curves.
25 May
Chris Skinner (Michigan)
"L-values, congruences, and Selmer groups"
1 June
Alexei Skorobogatov (Imperial)
"Global points on Shimura curves"
Abstract:
It is an open question whether all counterexamples to the Hasse
principle
on smooth projective curves over number fields are due to the Manin
obstruction.
In the 1980-s Bruce Jordan proved that global points don't exist on
certain Shimura
curves, producing counterexamples to the Hasse principle. I'll show how
these and
other known counterexamples are explained by the Manin obstruction.
8 June
Andrei Yafaev (UCL)
"Recent progress on the Andre-Oort conjecture"
15 June
Teruyoshi Yoshida (Harvard and Nottingham)
Compatibility of local
and global Langlands correspondences" (with R.Taylor)
Abstract:
The work of Harris-Taylor, which proved the local Langlands
correspondence for GLn, included the construction of l-adic Galois
representations attached to certain class of automorphic representations
of GLn over CM fields, compatible with the local Langlands
correspondence up to semisimplification at all places outside l. By
studying the semistable reduction of the relevant Shimura varieties, we
strengthen this result to show that the local monodromies are also the
correct ones. The irreducibility of global Galois representations
follows.
22 June
Tom Fisher (Cambridge)
"Computing models for visible elements of
the Tate-Shafarevich group"
29 June
Bjorn Poonen (Berkeley and Cambridge)
"Multiples of subvarieties in
algebraic groups over finite fields"
Spring 2005
This term, the seminar was held at KCL and was
organised by David Solomon.
The seminar programme for Spring 2005
12 January
Michael Harris (Universite Paris VII)
"Deformations of automorphic Galois representations"
Abstract:
The l-adic cohomology of Shimura varieties attached to certain
unitary groups provide compatible systems of n-dimensional
l-adic representations of the absolute Galois group of
a CM field. These representations are necessarily polarized
(self-dual, more or less) and their Hodge-Tate weights have
multiplicity one. In joint work with Taylor we have proved,
under the usual restrictions, that any representation with
these properties arises in this way, provided (a) the reduction
mod l has at least one modular lifting and (b) the
representation is a minimal lifting of its reduction
mod l. Work of R. Mann allows us to remove the minimality
condition (b) under a precise conjecture on mod l
modular forms known as Ihara's Lemma.
19 January
Prof E.V. Flynn (Liverpool)
"The Brauer-Manin Obstruction on Curves"
Abstract:
When a variety violates the Hasse principle, this can
be due to an obstruction known as the Brauer-Manin obstruction.
It is an unsolved problem whether all violations of the
Hasse principle on curves are due to this obstruction.
I shall describe work in progress which tests
a wide selection of curves, and tries to decide for these examples
whether the Brauer-Manin obstruction is the cause
of all violations of the Hasse principle. This has involved
the development of several new techniques, exploiting
the embedding of a curve in its Jacobians via a rational
divisor class of degree 1, and has produced examples of certain
new types (in response to a request of Alexei Skorobogatov).
If there is time, I shall also discuss the loosely related question
of annihilation and visualisation of members of the
Shafarevich-Tate group of Jacobians.
26 January
Toby Gee (Imperial)
"A new proof of an old theorem on companion forms"
Abstract:
Results on companion forms over Q were obtained in the early 1990s
by Gross and Coleman-Voloch. I gave a generalisation to totally real fields
last year. In this talk I will discuss a new and much more conceptual proof
of the results of Gross and Coleman-Voloch, and indicate the possibilities
for further generalisations to totally real fields.
2 February
Prof Francis Johnson (UCL)
"Orders in quaternion algebras and
recent developments in algebraic homotopy theory"
Abstract:
It is a fundamental question in non-simply connected
homotopy theory (and thereby also in combinatorial group theory) to
decide whether, over a given fundamental group G, every algebraic
2-complex is geometrically realizable. Although for most finite G
this problem is increasingly well understood, the quaternion groups
Q_4n have proved to be exceptional.
Using Swan's work on non-cancellation phenomena for modules over
orders in quaternion algebras over number fields, we describe
families of algebraic 2-complexes over Q_4n for which no geometric
realisations are currently known (or, in terms of combinatorial
group theory, which do not correspond to any known group
presentation)
9 February
Vladimir Dokchitser (Cambridge)
"Root numbers and the rank of elliptic curves"
Abstract:
Fix an elliptic curve E over Q. I will discuss the
behaviour of the sign in the functional equation of the L-function
L(E/K,s), where K varies over different number fields. When the sign is -1,
the L-function (assuming it is exists and is analytic) has a zero at s=1,
and the Birch-Swinnerton-Dyer conjecture predicts that E should have a
point of infinite order over the number field K. It is then
possible to obtain examples of elliptic curves over Q which,
while not having any rational points of infinite order,
must conjecturally have points of infinite order over all the
fields Q(m^{1/3}) for every (non-cube) m>1. I will discuss this,
and similar phenomena.
16 February
Andreas Langer (Exeter)
"Gauss-Manin connection via Witt-differentials"
Abstract:
For a scheme X that is smooth over a p-adic base we show an
equivalence of categories between the category of locally free crystals
and the category of Witt-connections on X. The proof uses the relative
de Rham-Witt complex and generalizes a recent result of Bloch. As
an application we realize the Gauss-Manin connection in the
de Rham-Witt complex.
23 February
Kyu-Hwan Lee (U. Toronto)
"Spherical Hecke algebras of GL_n over 2-dimensional local fields"
Abstract:
At the beginning of the talk, we will briefly review the classical
Satake isomorphism, which plays an important role in the Langlands
program. Then we will try to generalize the theory to the 2-dimensional
local field case. More precisely, we will construct spherical Hecke
algebras of GL_n over 2-dimensional local fields and prove the Satake
somorphism
for the algebras. We will use Fesenko's measure to define the Satake
isomorphism.
A connection to Kac-Moody groups will also briefly discussed. This is
a joint work with Henry Kim.
2 March
Jayanta Manoharmayum (Sheffield)
"Minimal deformations
of Galois representations"
Abstract:
I will describe how one can get, in some cases, lifts of residual
representation which are minimally ramified.
9 March
Ivan Fesenko (Nottingham)
"Poles of the Hasse zeta function"
Abstract:
The talk will try to discuss some of the applications of the study of the
Hasse zeta function of elliptic curves over global fields via 2d zeta
integrals to:
(a) Riemann hypothesis for the zeta function;
(b)
location of poles of the zeta function on the critical line;
(c) an
extension of the class of zeta functions all whose motivic L-factors are
automorphic, using Laplace-Carleman transforms of odd mean periodic
functions, and importance of this for the Langlands programme;
(d) the
rank part of the BSD conjecture.
16 March
Jan Nekovar (Jussieu)
"The Euler system of CM points"
Abstract: We shall discuss a generalization of Kolyvagin's
results on Heegner points.
23 March
Steven Galbraith (Royal Holloway College)
"Pairings on
abelian varieties and cryptography"
Abstract:
The talk will survey some applications of elliptic curves over
finite fields to cryptography. In particular, applications of
the Weil and Tate pairings will be described and some new results
on efficient computation of these pairings will be presented.
Autumn 2004
This term the seminar was held at Imperial College and was organised by Toby Gee
The seminar programme for Autumn 2004
27 October
No seminar
3 November
Pierre
Parent (Bordeaux)
"On the triviality of
X0+ (pr) (Q), r>1"
Let E be an elliptic curve over Q, without
complex multiplication over \overline{Q}. For
p a prime number, consider the representation
Gal(\overline{Q} /Q )-->
GL2 (Fp) induced by the Galois
action on the group of p-torsion points of E. A theorem of
Serre, published in 1972, asserts that there exists an integer
BE such that the above representation is surjective if
p is larger than BE. Serre then asked the
following question: can BE be chosen independently of
E? This boils down to proving the triviality, for large enough
p, of the sets of rational points of four families of modular
curves, namely X0 (p),
Xsplit(p), Xnon-split(p)
and XA4(p) (we say that a point of one of
these curves is trivial if it is either a cusp, or the underlying
isomorphism class of elliptic curves has complex multiplication over
\overline{Q}). The (so-called exceptional) case of
XA4(p) was ruled out by Serre. The fact that
X0 (p)(Q ) is made of only cusps for
p>163 is a well-known theorem of Mazur. In this talk we will
discuss the case of Xsplit (p)(Q).
Slightly more generally (because one has a
Q-isomorphism between Xsplit
(p) and X0+ (p2 )), we will in
fact give a criterion for the triviality of X0+
(pr ) (Q) (with r>1), and show it
is verified by a positive density of primes (satisfying explicit
congruences).
10 November
Nick
Shepherd-Baron (Cambridge)
"Perfect forms and moduli of abelian
varieties"
Perfect quadratic forms lead to a compactification
of Ag whose geometry is particularly accessible. The ample
classes are characterized by one inequality, generalizing the existence of
the discriminant when g = 1. Over a field of char. zero, it is canonical
(in the sense of Reid and Mori, not Shimura...).
17 November
Toby Gee (Imperial)
"Companion Forms"
This will be a different talk to the one I gave last year - it will
hopefully be a very relaxed introduction to the Fontaine-Mazur and Serre conjectures,
and a discussion of some issues arising from these conjectures.
24 November
Ian Grojnowski (Cambridge)
"Geometric Satake for local fields of dimension 2"
I'll
explain the usual Satake isomorphism, its central role in the Langlands
programme, and a generalisation of all this to fields like Qp((t)), C((s))((t)).
Should be of interest to geometers also--- this can be respelled as
theorems about the moduli of G-bundles on an algebraic surface---
"Donaldson theory".
1 December
Shaun Stevens (UEA)
"Supercuspidal representations of p-adic classical groups"
The
Local Langlands Correspondence relates the representations of the
Weil-Deligne group of a locally compact non-archimedean local field to
the irreducible smooth representations of general linear groups. Mostly
conjecturally, there are also such correspondences for other p-adic
groups.
In this talk I will try to describe what this means,
what's known and also some explicit constructions of representations
for p-adic symplectic, orthogonal and unitary groups.
8 December
Jens Marklof (Bristol)
"Number Theory and Quantum Chaos"
I
will review recent developments in some fundamental problems in quantum
chaos that have attracted the interest of number theorists. No
knowledge of quantum mechanics is necessary.
15 December
Denis Charles(Wisconsin)
"Computing Modular Polynomials" (joint work with Kristin Lauter)
We
present a new probabilistic algorithm to compute modular polynomials
modulo a prime. Modular polynomials parameterize pairs of isogenous
elliptic curves and are useful in many aspects of computational number
theory and cryptography. Our algorithm has the distinguishing feature
that it does not involve the computation of Fourier coefficients of
modular forms. We avoid computing the exponentially large integral
coefficients by working directly modulo a prime and computing isogenies
between elliptic curves via Velu's formulas.
Summer 2004
This term, the seminar was held at Imperial College and was
organised by Kevin Buzzard.
The seminar programme for Summer 2004
28 April
Dan Snaith (Imperial)
"Overconvergent Siegel modular forms"
5 May
Luis
Dieulefait (Barcelona)
"Existence of compatible families of Galois
representations and the Fontaine-Mazur conjecture for elliptic curves"
12 May
Bruno Kahn (Paris
7)
"Birational motives"
19 May
Fre
Vercauteren (Bristol)
"Zeta functions: the p-adic approach"
26 May
Sarah
Zerbes (Cambridge)
"Selmer groups over p-adic Lie extensions"
2 June
Adam Joyce
(Imperial)
"The Manin constant of modular abelian varieties"
The Manin constant of a (modular) elliptic curve was
introduced by Manin in a paper in the mid-70s, in which he also
conjectured that it is always 1. I shall give this definition and discuss
the results in the direction of proving the conjecture. I'll then
generalise the definition to abelian varieties of arbitrary dimension and
discuss the extensions of the above results to the general setting.
9 June
Brian Conrey
(AIM)
"Random matrix theory and ranks of elliptic curves"
16 June
Detlev
Hoffman (Nottingham)
"Isotropy of quadratic forms in finite and
infinite dimension"
23 June
Teruyoshi
Yoshida (Harvard, visiting Imperial)
"Non-abelian Lubin-Tate theory
and Deligne-Lusztig theory"
4 August
Chandrashekhar Khare
(Utah/Tata)
"Transcendental Galois representations"
| 14 January | Kevin Buzzard
(Imperial) "The 2-adic eigencurve at the boundary of weight space" Eigencurves are geometric objects parameterising certain modular forms. These objects were introduced by Coleman and Mazur in the mid-1990s and at the time very little was known about what they "looked like". Lloyd Kilford and I can write down equations for one of these eigencurves, and these equations are sufficiently explicit to enable us to pin down exactly what this eigencurve looks like near its boundary. In my talk I will give an introduction to the theory of eigencurves and then will explain a sketch of our result. | |
| 21 January | Alexei
Skorobogatov (Imperial) "Rational points on Enriques surfaces" The Enriques surfaces are cohomologically indistinguishable from rational surfaces, however unlike rational surfaces they have a non-trivial though very small fundamental group. For rational surfaces it is conjectured by Colliot-Th e and Sansuc, and is proved in many cases that the failure of the Hasse principle and weak approximation is controlled by the obstruction based on the Brauer group of the surface. In a joint work with David Harari we construct an Enriques surface over Q with an adelic point satisfying all the conditions provided by the Brauer group, but not in the closure of the set of Q-rational points. The proof uses descent to a torsor on this surface; the structure group of this torsor is a form of a (non-abelian) 1-dimensional orthogonal group. | |
| 28 January | Roger
Heath-Brown (Oxford) "Cayley's cubic surface" Roughly how many non-trivial primitive integer solutions does the equation 1/X0+1/X1+1/X2+1/X3=0have, in a large cube max |Xi| < B ? On clearing the denominators this gives us a Cayley's cubic surface. Manin's conjecture predicts a growth rate of order B(log B)6. This has now been proved, by using the Universal Torsor for the surface. The latter is an affine variety in 16 dimensional space, which encodes all the relevant divisibility information. The proof entails counting points on this affine variety, for which the key tool comes from the geometry of numbers. | |
| 4 February | Anthony Hayward
(King's) "Congruences satisfied by Stark units" "Refined abelian Stark conjectures" are special value conjectures for equivariant L-functions associated to Galois extensions of global fields at "s=0". There has been a proliferation of such conjectures since Stark's original work in the 1970s, notably those of Rubin and Popescu, who generalised Stark's abelian conjecture to account for differing orders of vanishing of the L-functions, and Gross, who gave congruences for the value of the L-function. These conjectures admit a natural refinement, due to Burns, which gives Gross-style congruences in the situation of Rubin's conjecture. The formulation of the refinement is inspired by the Equivariant Tamagawa Conjecture, and this over-riding conjecture provides a unifying overview of the field which was previously sorely lacking. After describing this general situation in the first half of the talk, I move on to study those situations in which systems of explicit units provide proofs of the Burns refinement, hopefully giving more of a sense of what these conjectures are "about". The proof involves details on the explicit units, a study of ray-class fields, and some unusual combinatorics and integer identities. | |
| 11 February | Victor
Abrashkin (Durham) "Galois modules arising from Faltings's strict modules" A classical analogue of the concept of a finite flat group scheme over a complete discrete valuation ring loses all interesting properties in the equal characteristic case. In the past year Faltings proposed a modified definition of a group scheme with strict action and showed that it works perfectly in some situations. It will be explained in the talk that many known results about classification and arising Galois modules still hold for Faltings's modules. | |
| 18 February | Kanetomo
Sato (Nagoya) "p-adic ale Tate twists and arithmetic duality" For an algebraic variety X over a field k and a positive integer n invertible in k, the ale sheaf of n-th roots of unity and its tensor powers are called ( ale) Tate twists, and play fundamental roles both in number theory and in arithmetic geometry. In this talk, I will talk about a construction of p-adic Tate twists on regular arithmetic schemes (=schemes which are regular flat of finite type over Spec Z), and arithmetic duality theorems for p-adic Tate twists, which generalizes the classical Artin-Verdier duality theorem. | |
| 25 February | Toby Gee
(Imperial) "Companion forms over totally real fields" The companion forms conjecture was part of Serre's conjecture on the modularity of mod p Galois representations. This was proved in the early 90s by Gross and Coleman & Voloch. More recently Fred Diamond has conjectured extensions of these results to totally real fields; I will describe my recent progress on cases of these conjectures. | |
| 3 March | Aleksandra
Shlapentokh (East Carolina) "Hilbert's tenth problem and Mazur's conjectures" We discuss recent results concerning extensions of Hilbert's tenth problem to rings of integers of number fields and the field of rational numbers, and related conjectures of Mazur. | |
| 10 March | R is de la
Bret he (ENS Paris) "Counting points on varieties using universal torsors" We study the asymptotic order of the number of points of bounded height on certain varieties. Universal torsors turned out to be a useful tool to attack this kind of problem. We will give few examples to explain how to prove asymptotic estimations using universal torsors and tools of analytic number theory. | |
| 17 March | Nikolaos
Diamantis (Nottingham) "Second order cusp forms and L-functions" Second-order modular forms are functions that have recently appeared in several contexts: Eisenstein series formed with modular symbols, converse theorems of L-functions, percolation theory etc. They satisfy a functional equation that extends naturally that of the usual modular forms and their study is important for the topics that have motivated their introduction. We will discuss the ways they arise in various contexts, their classification and their L-functions. | |
| 24 March | Adam Logan
(Liverpool) "Heegner points on elliptic curves over real quadratic fields" Henri Darmon formulated a concrete version of a conjecture of Oda according to which it should be possible to construct rational points on elliptic curves over real quadratic fields defined over quadratic extensions of the field by integrating the Hilbert modular form associated to the curve. We present the conjecture together with some numerical evidence for it. This is joint work with Darmon. |
1/10/03 David Solomon, KCL
`Twisted Zeta-Functions, Stark Conjectures and Hilbert Symbols'
----------------------------------------------------------------
8/10/03 John Cremona, Nottingham:
`Explicit Higher Descents on Elliptic Curves'
----------------------------------------------------------------
15/10/03 Richard Hill, UCL:
`Fractional weights, Borcherds products and
the Congruence Subgroup Problem'
----------------------------------------------------------------
22/10/03 Tim Browning, Oxford:
`Counting rational points on singular cubic surfaces'
----------------------------------------------------------------
29/10/03 Christian Elsholtz, Royal Holloway College:
`Additive decompositions of the set of primes'
----------------------------------------------------------------
5/11/03 Daniel Delbourgo, Nottingham
"Euler characteristics of elliptic curves via p-adic modular forms"
----------------------------------------------------------------
12/11/03 Neil Dummigan, Sheffield,
`Critical values of tensor-product L-functions'
----------------------------------------------------------------
19/11/03 Dan Evans, Nottingham:
`Harmonic analysis on higher dimensional local fields'
----------------------------------------------------------------
26/11/03 Daniel Barsky, U Paris 13 [NOTE CHANGE OF DATE]
`Norms of Iwasawa series attached to totally real fields'
----------------------------------------------------------------
3/12/03 Nigel Byott, Exeter:
`Hopf-Galois strucutres of field extensions'
----------------------------------------------------------------
10/12/03 Prof. Igor Shparlinski, Macquarrie U.
`Euler Function: Smooth and Square'
-------------------------------------------------
Further details will appear when available.
This term it was held in the maths department of Imperial College and was organised by Kevin Buzzard.
----------------------------------------------------------------------
30 April --- TWO TALKS
14:15 Elmar Grosse-Kloenne (Muenster) "On twisted unit root L-functions
^^^^^ of families of varieties over
finite fields"
15:45 Steve Gelbart (Weizmann) "On lower bounds for automorphic
^^^^^ L-functions".
----------------------------------------------------------------------
7 May Allan Lauder (Oxford) "Deformation theory and the computation
of zeta functions"
----------------------------------------------------------------------
14 May Vic Snaith (Southampton) "On the Kummer--Vandiver conjecture"
----------------------------------------------------------------------
***Tuesday 20th May*** at 1600
Victor Rotger (Barcelona)---"Diophantine properties of fake
elliptic curves and their moduli spaces"
----------------------------------------------------------------------
21 May---Ed Nevens (Imperial)---TBA (something about moduli space
of abelian varieties and/or canonical
subgroups, perhaps)
----------------------------------------------------------------------
28 May---Neil Strickland (Sheffield)---"Elliptic cohomology"
----------------------------------------------------------------------
This term the seminar was held in the maths department of Imperial College on Wednesdays at 4.15 pm and was organised by Kevin Buzzard.
----------------------------------------------------------------------
15/1 Andrei Yafaev, Imperial
Title: `Descent on certain Shimura curves'
Abstract:
This is a joint work with Alexei Skorobogatov. Applying descent to
certain unramified coverings of Shimura curves we offer an explicit
method of constructing Shimura curves that do not satisfy Hasse
principle; the failure of the Hasse principle is being explained by
the Manin obstruction.
----------------------------------------------------------------------
22/1 Kevin Buzzard, Imperial
Title: `Overconvergent 2-adic modular forms'
Abstract:
This is joint work with Frank Calegari. Some computations I did (and
some known conjectures and theorems) led me to believe that in some
cases there are very precise formulae for the p-adic valuations of
the eigenvalues of T_p on various spaces of modular forms. Calegari
and I have made these conjectures completely explicit and precise in
the case p=2 and N=1 (for any weight k) and can prove them in some
cases using a combination of deep theorems of Coleman and elementary
combinatorial results involving hypergeometric function identities.
----------------------------------------------------------------------
29/1 I. Tomasic (Leeds)
"Weil conjectures--with a DIFFERENCE"
----------------------------------------------------------------------
5/2 A. Hayward (Kings)
"A conjectural class-number formula for
higher derivatives of abelian L-functions"
----------------------------------------------------------------------
12/2 R. Kucera (Brno)
"Cyclotomic units"
----------------------------------------------------------------------
19/2 (1430) J. Nekovar (Jussieu)
^^^^^^^^^^^
"On the parity of ranks of Selmer
groups associated to Hilbert modular forms"
19/2 (1615) T. Ochiai (Tokyo)
^^^^^^^^^^^
"Results and examples for Iwasawa
theory on Hida deformations."
----------------------------------------------------------------------
26/2 H. Narita (Tokyo)
"Fourier-Jacobi expansion of certain
automorphic forms on Sp(1,q)"
----------------------------------------------------------------------
5/3 M. Breuning (Kings)
TBA (something about local epsilon constants)
----------------------------------------------------------------------
12/3 D. Harari (Strasbourg)
"Arithmetic duality theorems for 1-motives"
----------------------------------------------------------------------
19/3 N. Broberg (Durham)
"Counting rational points on finite
covers of the projective plane"
----------------------------------------------------------------------
9/4
1400: A. Yakovlev
^^^^
"Multiplicative Galois modules in local fields"
1530: G. Henniart
^^^^
"Expliciting the Langlands conjecture:the tame case"
-------------------------------------------------------------------------
This term, the seminar was held on Wednesdays, in room 423 of KCL, and was organised by Dr David Solomon. Also on Wednesdays in King's this term was the London Number Theory Study Group, which met from 2:45 to 3:45 pm.
The seminar programme
9/10 FIRST MEETING: EXCEPTIONALLY A DOUBLE-HEADER STARTING AT 2:45 in room 436:
2:45 - 3:45 (room 436) Amnon Besser, Ben Gurion University,
4:15 - 5:15 (room 423) Takao Yamazaki, Tsukuba University,
Title: `On the structure of Chow groups of
surfaces over local fields'
Abstract: Let X be a surface over a p-adic field with good reduction
and let Y be its special fiber. We consider the structure of
the Chow group CH0(X) of zero-cycles on X. If we write T(X)
for the kernel of the Albanese map of X, then the structure of
the quotient group CH0(X)/T(X) is well understood. Hence we only
have to study T(X). Let D(X) be the maximal divisible subgroup
of T(X). Then, it is conjectured that F(X) = T(X)/D(X) is finite
and that F(X) is isomorphic to the Albanese kernel T(Y) of Y
modulo p-primary torsion. On the contrary, we shall show that the
p-primary torsion subgroup of F(X) can be arbitrary large
even though we fix the special fiber Y.
----------------------------------------------------------------------
16/10 Mohammed Saidi, University of Durham
Title: `On the fundamental group of complete curves in positive
characteristics'
----------------------------------------------------------------------
30/10 Richard Hill, UCL
Title: `Shintani Cocyles on GL_n'
------------------------------------------------------------------------
6/11 No Seminar (Reading Week)
------------------------------------------------------------------------
13/11 Sey Yoon Kim, KCL
Title: `On the Equivariant Tamagawa Number Conjecture
for certain Quaternion Fields '
Abstract:
Let L/K be a finite extension of number fields. Then the equivariant
Tamagawa number conjecture relates the values of the Artin
L-functions of L/K at integers to various algebraic data of L/K, and
in particular, the conjecture at s=0,1 implies Chinburg's root number
conjecture for L/K. We explain the conjecture at s=0 for abelian
extensions over Q; then prove it for a family of biquadratic abelian
extensions over Q to lift a 1989 result of Chinburg on his conjecture
for the case of quaternion extensions over Q.
-------------------------------------------------------------------------
20/11 Martin Taylor, U.M.I.S.T.
Title: `Arithmetic Euler Characteristics'
Abstract:
I shall start by recalling the basic theory and constructions for
Euler characteristics of varieties which support an action by a
finite group. These ideas then extend firstly to the construction of
equivariant Euler characteristics of arithmetic varieties, and then
more generally to Euler characteristics which take into account
metrics and signatures.
-------------------------------------------------------------------------
27/11 Robin Chapman, Exeter
Title: `Hermitian structures on lattices'
Abstract:
We consider lattices in Euclidean space Rn. If n is even, Rn can be
given a structure of a complex vector space in many ways. Given a
lattice L we investigate which C-structures on Rn have a Hermitian
form compatible with the Euclidean structure on Rn and for which L
becomes an O-module for some quadratic order O. In some cases we
determine explicitly the O-module structure of L.
-------------------------------------------------------------------------
4/12 David Burns, King's College London
Title: `Nearly perfect complexes and Weil-etale cohomology'
Abstract:
We describe a more conceptual approach to the construction of Euler
characteristics of nearly perfect complexes which was recently
introduced by Chinburg, Kolster, Pappas and Snaith. We then discuss
certain applications of our approach in the context of Lichtenbaum's
theory of Weil- ale cohomology.
-------------------------------------------------------------------------
11/12 Rob de Jeu, Durham
Title: `Zagier's conjecture and (p-adic) regulators'
Abstract:
Let k be a number field. There is a classical relation between the
residue of the zeta function of k, zetak(s), at s=1, and the
regulator of the group of units of its ring of integers. Borel proved
a similar relation between zetak(n) and K2n-1(k) for n>=2.
The K-groups are difficult to describe explicitly. We
discuss a conjecture of Zagier on how this could be done, and
describe Borel's regulator (as well as a p-adic regulator) in this
context, involving polylogarithms.
This term it was held in the Maths department of Imperial College and organised by Kevin Buzzard.
24 April Frazer Jarvis (Sheffield)
"Points on Fermat curves over real quadratic fields"
1 May Jayanta Manoharmayum (Sheffield)
"modularity of GL2(F7) Galois representations"
*2 May* Helena Verrill (Hannover)
"Transportable modular symbols"
8 May Alexei Skorobogatov (Imperial)
"Some new cases of the Hasse principle and weak approximation"
15 May Dan Jacobs (Imperial)
Slopes of Compact Operators
22 May Ben Green (Cambridge)
"Counting sumfree sets in abelian groups"
29 May Lloyd Kilford (Imperial)
"Slopes of overconvergent 2-adic modular forms"
5 June Denis Petrequin (Cambridge)
"Chern classes and cycle classes in rigid cohomology"
12 June Oliver Bltel (Heidelberg)
TBA
*13 June* Oliver Bueltel (Heidelberg)
TBA (continued).
19 June Chad Schoen (Duke)
"Torsion in the Chow group"
This term it was held in the Maths department of Imperial College and organised by Kevin Buzzard.
Jan 16 Andrei Yafaev (Imperial)
"Galois orbits of abelian varieties with complex multiplication
Jan 23 Denis Benois (Bordeaux)
"On Tamagawa numbers of crystalline representations"
Jan 30 Tony Scholl (Cambridge)
"Local epsilon-factors and tensor products of
representations of GL(2)"
Feb 6 Richard Hill (UCL)
"something to do with metaplectic groups"
Feb 13 John Coates (Cambridge)
"Iwasawa algebras and arithmetic"
Feb 20 Tim Dokchitser (Durham)
"TBA"
Feb 27 Susan Howson (Nottingham)
"Applications of Euler Characteristics to
non-Abelian Iwasawa Theory
Mar 6 Shaun Stevens (Oxford)
"TBA"
Mar 13 John Wilson (Oxford)
"Abelian surfaces with real multiplication"
Mar 29 David Solomon (Kings)
"Abelian Stark Conjectures in Z_p-extensions"
Oct 10 Andrei Yafaev (Imperial)
"Special points on Shimura varieties"
Abstract
A conjecture of Andre and Oort predicts that irreducible
components of a Zariski closure of a set of special points in
a Shimura variety are subvarieties of Hodge type. This talk
is devoted to a recent result towards this conjecture
obtained in a joint work with Bas Edixhoven.
Oct 17 Sir Peter Swinnerton-Dyer (Cambridge)
"Rational points on certain Kummer surfaces"
Abstract
Most of this seminar represents joint work with Alexei
Skorobogatov. Let E_1,E_2 be elliptic curves defined over
an algebraic number field k, and let F_i:y_i^2=f_i(x_i)
with f_i quartic be a 2-covering of E_i. Then
V:y^2=f_1(x_1)f_2(x_2) is a Kummer surface associated with
the Abelian surface E_1\times E_2. In the special case when
E_1 and E_2 have all their 2-division points defined over
k, I shall show that the Hasse Principle holds for V
provided that (i) the Tate-Safarevi\v{c} groups of all the
twists of E_1 and E_2 are finite, (ii) a certain rather
weak technical condition holds. Here (i) is necessitated by
the method of proof, but it is generally believed to be true;
(ii) can be shown to be strictly stronger than the true
necessary and sufficient condition, which is conjectured zto
be the absence of a Brauer-Manin obstruction. Note that
Schinzel's Hypothesis does not appear. The methods used have
much in common with those used for diagonal cubic surfaces
a_0X_0^3+a_1X_1^3+a_2X_2^3+a_3X_3^3=0 but some stages of the
argument for the latter are much more complicated.
Comparisons will be made between the two.
Oct 24 Michael Spiess (Nottingham)
"Monodromy modules and derivatives of p-adic L-functions"
Oct 31 Jonathan Dee (Imperial)
"Phi-Gamma-modules and families of p-adic Galois representations"
Nov 7 Jean-Louis Colliot-Th e (Orsay)
"Linear algebraic groups over two-dimensional fields"
Abstract
Let $k$ be an algebraically closed field of characteristic
zero. Let $K$ be either a function field in two variables
over $k$ or the fraction field of a $2$-dimensional,
excellent, strictly henselian local domain with residue field
$k$. We show that linear algebraic groups over such a field
$K$ satisfy properties which are familar in the context of
number fields: finiteness of $R$-equivalence, Hasse principle
forprincipal homogeneous spaces of simply connected
groups,Hasse principle for complete homogeneous spaces.
This is joint work with P. Gille (Orsay) and R. Parimala
(Mumbai).
Nov 14 Kevin Buzzard (Imperial)
"The eigenvariety"
Abstract:
For a general reductive group, people are beginning to
believe that certain classes of automorphic forms on this
group lie naturally in p-adic analytic families, as the
weight varies. For GL_1 one can formulate a precise statement
and its proof is an easy consequence of global class field
theory. For GL_2 over Q, Coleman and Mazur have constructed
families interpolating classical holomorphic modular forms,
and have gone onto construct a geometric object, the
eigencurve, parameterising the forms. I will explain that if
one is willing to do a little rigid geometry then one can
generalise much of the Coleman-Mazur construction to a much
wider setting, and hence construct "the eigenvariety" in much
greater generality.
Nov 21 Vic Snaith (Southampton)
"Relative K_0, Fitting ideals and the Stickelberger phenomenon"
Nov 28 Burt Totaro (Cambridge)
"Rational points on homogeneous spaces, and the group E_8"
Abstract: A homogeneous variety over a field need not have a
rational point over the same field. The simplest example is a
conic curve, which in general has a rational point only over
a quadratic extension field. More generally, given a
semisimple group G, and a homogeneous G-variety over an
arbitrary field, we can ask what degree of field extension we
need in order to find a rational point. I will explain what
is known about this problem, both for the classical groups
and the exceptional groups such as E_8.
Dec 5 Otmar Venjakob (Cambridge)
"Iwasawa theory of p-adic Lie extensions"
Abstract:
The most prominent example of a (non-abelian) p-adic Lie
extension K of a number field k arises maybe by adjoining to
k the p-power division points of an elliptic curve E (over k)
without complex conjugation. If G denotes the Galois group of
K/k one can study the (Pontryagin dual of the) Selmer group
of E over K as a module over the completed group algebra R(G)
of G. In this situation it is also reasonable - though not at
all obvious - to speak about pseudonull modules. One basic
result is that the Selmer group does not contain any nonzero
pseudo-null submodule (under certain conditions). If there is
enough time we are going to discuss also some features of the
general structure theory of torsion R(G)-modules up to
pseudo-isomorphism, which was proven by Coates, Schneider and
Sujatha recently.
Dec 12 No Talk
This term, the seminar was held on Wednesdays in room 423 of KCL, and was organised by Dr David Solomon.
| Jun 27 | Stephen Lichtenbaum (Brown University, USA) -
(Final Seminar) |
| Jun 26 | Frank Calegari
Fontaine proved in 1985 that there do not exist any Abelian varieties over Z. Jacobians of the modular curves X_0(p^n) (non-zero for sufficiently large n) provide examples of Abelian varieties over Z[1/p] for each p. If, however, we restrict our attention to semistable Abelian varieties,then combining Fontaine's theorem, recent papers of Brumer-Kramer and of Schoof, and some new ideas and results, we prove the following: There exists a semistable Abelian variety over Z[1/n] with n squarefree if and only if n is not in the set: {1,2,3,5,6,7,10,13}. |
| Jun 26 | William Stein
During the past few years, Barry Mazur, myself, and others have studied visible subgroups of Shafarevich-Tate groups of abelian varieties. Recently, I've been studying visibility of Mordell-Weil groups of abelian varieties. In this talk, I will very briefly review some results about visibility of Shafarevich-Tate groups, then discuss some of what I've been able to prove about their counterparts in the context of visibility of Mordell-Weil groups. In particular, I will show that Mordell-Weil groups of elliptic curves over Q are visible in modular abelian varieties. |
| Jun 15 | Helena Verrill
|
| Jun 13 | Romyar Sharifi -
Abstract: We give an explicit description of generators of the ith unit groups of K = Qpzetapn as Galois submodules of the multiplicative group of K. We can use this to determine the ramification groups of degree pn Kummer extensions of K which are Galois over Qp. |
| Jun 6 | Richard Hill (University College, London) -
|
| May 30 | Prof. V. Nikulin (Liverpool) -
|
| May 23 | Daniel Delbourgo (Nottingham) -
|
| My 16 | Rob de Jeu (Durham) -
|
This term it was held in the maths department of Imperial College, and was organised by Kevin Buzzard.
| Jan 17 | Kevin Buzzard (Imperial) -
|
| Jan 24 | Al Weiss (U Alberta)
|
| Jan 31 | Andreas Langer (Bielefeld)
|
| Feb 7 | Colin Bushnell (Kings)
|
| Feb 14 | Alexei Skorobogatov (Imperial)
|
| Feb 21 | Christophe Cornut (Strasbourg)
|
| Feb 28 | Keith Ball (UCL)
|
| Mar 7 | Werner Hoffman (Humboldt U)
|
| Mar 14 | Anupam Saikia (Cambridge)
|
| Mar 21 | Victor Flynn (Liverpool)
|
| Abstract: We shall discuss the idea of finding all rational points on a curve C by first finding an associated collection of curves whose rational points cover those of C. This classical technique has recently been given a new lease of life by being combined with descent techniques on Jacobians of curves and Chabauty techniques. We shall survey recent applications during the last 5 years which have used Chabauty techniques and covering collections of curves of genus 2 obtained from pullbacks along isogenies on their Jacobians. |
This term it was held in the maths department of Imperial College, and was organised by Alexei Skorobogatov.
| Oct 11 | Kevin Buzzard (IC)
|
| Oct 18 | Alexei Skorobogatov (Imperial)
|
| Oct 25 | David Solomon (KCL)
|
| Nov 1 | Kevin Buzzard (IC)
|
| Nov 8 | Jan Nekovar (Cambridge)
|
| Nov 15 | Tom Fisher (Cambridge)
|
| Nov 22 | Roger Heath-Brown (Oxford)
|
| Nov 29 | Jean-Robert Belliard (Nottingham)
|
| Dec 6 | Richard Hill (UCL)
|
| May 24 | Richard Hill (UCL)
|
| May 31 | Frazer Jarvis (University of Sheffield)
|
| Jun 7 | Alexei Skorobogatov (ICL)
|
| Jun 14 | Anton Deitmar (Exeter)
|
| Jun 21 | No seminar scheduled |
| Jun 27 | Cornelius Greither (U. der Bundeswehr, Munich)
|
| Jan 26 | Neil Dummigan (Oxford)
|
| Feb 2 | Susan Howson (Nottingham)
|
| Feb 9 | David Burns (KCL)
|
| Feb 16 | Victor Abrashkin
|
| Feb 23 | David Burns (KCL)
|
| Mar 1 | Richard Hill
|
| Mar 8 | Robert Vaughan
|
| Mar 15 | David Solomon
|
| Mar 22 | David Solomon
|
From the 4th of November, the seminars moved to Imperial College and were organised by Dr Kevin Buzzard.
| Cambridge-Oxford-Warwick (COW) algebraic geometry seminar, room 642 IC | ||
| 2.00 | Nick Shepherd-Barron (Cambridge CMS) -
| |
| 3.15 | Paul Seidel -
| |
Abstract:
In 1989 Glenn Stevens showed how the `periods' of
Eisenstein series could be used to define certain families of 1-cocycles
on GL2(Q) whose values can be expressed in terms of Dedekind
Sums (for example, those appearing in the transformation formula for
Dedekind's eta-function). In fact, the cocycle relation is equivalent to
(a generalisation of) the classical Dedekind Reciprocity Law for these
sums. Stevens also showed how these cocycles could be used to evaluate the
partial zeta-functions of real quadratic fields at non-positive integers.
In 1992, Robert Sczech constructed 1-cocycles on GL2(Q) by a
different method, using only real analysis. He too showed how they related
to Dedekind Sums and partial zeta-values. (Indeed they are very closely
related to Stevens'). In 1993 Sczech extended his construction to
GLn(Q), producing (n-1)-cocycles that are similarly `universal'
for the evaluation of partial zeta-values over totally real fields of
degree n.
In this talk, I shall present a third construction of cocycles on
GLn(Q) for n=2 and 3, which was inspired by Shintani's formulae
for partial zeta-values. They are again closely related to Stevens' and
Sczech's but the construction is algebraic and elementary. I shall also
explain the connections with Dedekind Sums and mention (time allowing)
partial zeta-values, p-adic interpolation, and the `challenge of higher
dimensions'.
Abstract:
Let S be an algebraic group over a field k, and X be a
k-variety. We denote by X' the same variety considered over the algebraic
closure of k. If Y'/X' is a torsor under S (equipped with a suitable
Galois action), then the obstruction for Y'/X' to come from some Y/X
(defined over k) lies in the second cohomology set of S. If X contains a
k-point, then this class is neutral. In some arithmetically meaningful
cases, e.g. X a principal homogeneous space of a semi-simple group over a
totally imaginary number field k, and Y'/X' is the universal covering, the
converse is also true. Using these ideas one can give a short proof of an
old theorem of Sansuc that the Manin obstruction is the only obstruction
to the Hasse principle for principal homogeneous spaces of semi- simple
groups over number fields (here S is Abelian).
Abstract:
The talk will be about a non-classical generalisation of
the Cassels-Tate pairing. It is defined, roughly speaking, on the
non-generic part of the dual of the Selmer group. The existence of the
pairing has non-trivial consequences; for example, one can deduce results
about the parity of the (co)-rank of the Selmer group.
Abstract:
Recent deep work of Coleman has shown what people have
suspected now for a long time - namely that many modular forms "come in
families". Coleman's work establishes (a slightly weak form of) a
conjecture of Gouvea and Mazur. I will explain the conjecture and show how
one can use a completely different (and much simpler method) to attack it.
The simpler method does not give results as strong as Coleman's (one only
gets "continuity" results rather than "analyticity" results), but has the
advantage that it generalises much more easily to other kinds of modular
forms.