The London-Paris Number Theory Seminar

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The London-Paris Number Theory Seminar meets twice per year, once in London and once in Paris. It is supported by grants from ANR Projet ArShiFo ANR-BLAN-0114, EPSRC Platform Grant EP/I019111/1, PerCoLaTor (Grant ANR-14-CE25), the Heilbronn Institute for Mathematical Research, and ERC Advanced Grant AAMOT.

London organizers: David Burns, Kevin Buzzard Fred Diamond, Yiannis Petridis, Alexei Skorobogatov, Andrei Yafaev, Sarah Zerbes.

Paris organizers: Matthew Morrow, Olivier Fouquet, Michael Harris, Marc Hindry, Benjamin Schraen, Jacques Tilouine.

24th meeting, London.

The 24nd meeting of the LPNTS took place in London at UCL, on 29th and 30th of May, 2018. The theme was "mod p automorphic forms".

The schedule was

Tuesday 29th:
1000--1100: Coffee
1100--1200: Payman Kassaei
1200--1400: lunch
1400-1500: Marie-France Vigneras
1500-1530: coffee
1530-1600: David Helm
1630-1730: Stefano Morra

Wednesday 30th:
0900-1000: Julien Hauseux
1000-1030: coffee
1030-1130: Pascal Boyer
1130-1230: Jack Thorne
Titles and abstracts:

Payman Kassaei (King's College London)
Title: Stratifications on mod p Shimura varieties and applications to automorphic forms
Abstract: In this talk, I will describe in an introductory way stratifications on certain mod p Shimura varieties and present some past and recent applications to arithmetic of automorphic forms.

Marie-France Vigneras (Jussieu)
Title: On supersingularity
Abstract: There are explicit relations between: Elliptic curves over $Q_p$ with supersingular reduction,
Irreducible admissible representations of $GL_2(Q_p)$ with supersingular reduction modulo $p$,
Supersingular simple modules modulo $p$ of the pro-$p$ Iwahori Hecke algebra of $GL_2(Q_p)$,
Irreducible 2-dimensional representations modulo $p$ of the Galois group of $Q_p$.
How does this generalize to a finite extension $F/Q_p$, a positive integer $n$, a reductive $p$-adic group?

Pascal Boyer (Paris 13)
Title: Torsion or not Torsion
Abstract: About the $\mathbb Z_l$-cohomology of Shimura varieties, on can be interested by killing the torsion using for example localization at some well chosen maximal ideal of some Hecke algebra. On the opposite point of view, we can ask for the arithmetic meaning of torsion classes so that we are led to the problem of the construction of such classes. In this talk we will try to tackle these two aspects in the particular case of Shimura varieties of Kottwitz-Harris- Taylor type.

Julien Hauseux (Lille)
Title: Extensions between generalised Steinberg representations
Let G be a p-adic reductive group. We compute the extensions between mod p smooth generalised Steinberg representations of G. This is part of a work in progress with Colmez, Dospinescu, and Nizioł.

David Helm (Imperial College)
Title: Towards a local Langlands correspondence in families for split groups in depth zero
Abstract: The local Langlands correspondence in families identifies the integral Bernstein centre for the group GL_n(F) with a certain ring of functions on the moduli space of Langlands parameters for GL_n. I will describe a conjectural generalization of this result to depth zero representations of a split reductive group G over F, and explain how a key part of the proof of local Langlands in families for GL_n generalizes to this case. This is joint work with Jean-Francois Dat, Rob Kurinczuk, and Gil Moss.

Stefano Morra (Montpellier)
Title: Local models for Galois representations, and applications to automorphic forms
Deformation spaces of finite at group schemes are a central subject in p-adic Hodge theory, and in arithmetic questions of a global nature. The description of their singularities in specific situations led to a wide horizon of achievements, from the proof of the conjectures of Serre (as generalized by Buzzard-Diamond- Jarvis, Schein, Gee and others) to the establishment of the Shimura-Taniyama- Weil conjecture and many more cases of modularity lifting theorems in dimension 2. Nevertheless outside a limited number of situations (Barsotti-Tate, ordinary, Fontaine- Laffaille) the geometry of more general Galois deformation spaces remains mysterious. In this talk we introduce an algebraic variety, refinement with monodromy of the local model of Pappas-Rapoport, which controls the structure of generic potentially crystalline deformation spaces of Galois representations in arbitrary dimension. In particular we illustrate how its geometry is predicted by and predicts generalized geometric interpretations of the weight part of Serre and Breuil-Mézard conjectures as proposed by Emerton and Gee. This is ongoing joint work with Daniel Le, Bao Viet Le Hung and Brandon Levin.

Jack Thorne (Cambridge)
Title : Odd Galois Representations
Abstract : A popular slogan is that "all Galois representations which appear in the cohomology of Shimura varieties are conjugate self-dual up to twist". However, fine words butter no parsnips, and this is on the face of it not quite true. I will discuss what one can say in this direction. This is joint work with Christian Johansson.

Previous meetings:

18th meeting (Imperial, 4--5/6/15)
19th meeting (Paris 13, 9/11/15)
20th meeting (UCL, 6--7/6/16)
21st meeting (Jussieu, 14--15/11/16)
22nd meeting (UCL, 5--6/6/17)
23rd meeting (Jussieu, 27--28/11/17)

This page is maintained by Kevin Buzzard.