On the modularity of elliptic curves over imaginary quadratic fields
(with James Newton)
|
PDF |
arXiv |
|
|
In this paper, we establish the modularity of every elliptic curve E/F,
where F runs over infinitely many imaginary quadratic fields, including
ℚ(√(-d)) for d=1,2,3,5. More precisely, let F be imaginary
quadratic and assume that the modular curve X0(15), which is
an elliptic curve of rank 0 over ℚ, also has rank 0 over F. Then we
prove that all elliptic curves over F are modular. More generally, when
F/ℚ is an imaginary CM field that does not contain a primitive fifth
root of unity, we prove the modularity of elliptic curves E/F under a
technical assumption on the image of the representation of Gal(F̅/F)
on E[3] or E[5].
The key new technical ingredient we use is a local-global compatibility
theorem for the p-adic Galois representations associated to torsion in
the cohomology of the relevant locally symmetric spaces. We establish
this result in the crystalline case, under some technical assumptions,
but allowing arbitrary dimension, arbitrarily large regular Hodge–Tate
weights, and allowing p to be small and highly ramified in the imaginary
CM field F.
|
The cohomology of Shimura varieties with torsion coefficients,
to appear in Proceedings of the ICM 2022.
|
PDF |
|
|
Video |
In this article, we survey recent work on some vanishing conjectures
for the cohomology of Shimura varieties with torsion coefficients, under
both local and global conditions. We discuss the p-adic geometry of
Shimura varieties and of the associated Hodge–Tate period morphism, and
explain how this can be used to make progress on these conjectures.
Finally, we describe some applications of these results, in particular
to the proof of the Sato–Tate conjecture for elliptic curves over CM
fields.
|
The geometric Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate Galois representations
(with Matthew Emerton, Toby Gee, and David Savitt)
|
PDF |
arXiv |
|
|
We establish a geometrisation of the Breuil–Mézard
conjecture for potentially Barsotti–Tate representations, as well as of
the weight part of Serre's conjecture, for moduli stacks of
two-dimensional mod p representations of the absolute Galois group of a
p-adic local field.
|
Components of moduli stacks of two-dimensional Galois representations
(with Matthew Emerton, Toby Gee, and David Savitt)
|
PDF |
arXiv |
|
|
In a previous article we introduced various moduli stacks of
two-dimensional tamely potentially Barsotti–Tate representations of the
absolute Galois group of a p-adic local field, as well as related moduli
stacks of Breuil–Kisin modules with descent data. We study the
irreducible components of these stacks, establishing in particular that
the components of the former are naturally indexed by certain Serre
weights.
|
Local geometry of moduli stacks of two-dimensional Galois representations
(with Matthew Emerton, Toby Gee, and David Savitt),
to appear in Proceedings of the International Colloquium on 'Arithmetic
Geometry', TIFR Mumbai, Jan. 6–10, 2020.
|
PDF |
arXiv |
|
|
We construct moduli stacks of two-dimensional mod p representations of
the absolute Galois group of a p-adic local field, as well as their
resolutions by moduli stacks of two-dimensional Breuil–Kisin modules
with tame descent data. We study the local geometry of these moduli
stacks by comparing them with local models of Shimura varieties at
hyperspecial and Iwahori level.
|
On the étale cohomology of Hilbert modular varieties with torsion coefficients
(with Matteo Tamiozzo), to appear in Compositio Math.
|
PDF |
arXiv |
|
|
We study the étale cohomology of Hilbert modular varieties, building
on the methods introduced for unitary Shimura varieties in [CS17, CS19]. We
obtain the analogous vanishing theorem: in the "generic" case, the
cohomology with torsion coefficients is concentrated in the middle
degree. We also probe the structure of the cohomology beyond the generic
case, obtaining bounds on the range of degrees where cohomology with
torsion coefficients can be non-zero. The proof is based on the
geometric Jacquet--Langlands functoriality established by Tian--Xiao and
avoids trace formula computations for the cohomology of Igusa varieties.
As an application, we show that, when p splits completely in the totally
real field and under certain technical assumptions, the p-adic local
Langlands correspondence for GL2(ℚp) occurs in the
completed homology of Hilbert modular varieties.
|
New frontiers in Langlands reciprocity, EMS Magazine (2021), no. 119.
|
PDF |
|
Journal |
|
In this survey, I discuss some recent developments at the crossroads
of arithmetic geometry and the Langlands programme. The emphasis is on
recent progress on the Ramanujan–Petersson and Sato–Tate
conjectures. This relies on new results about Shimura varieties and
torsion in the cohomology of locally symmetric spaces.
|
Vanishing theorems for Shimura varieties at unipotent level
(with Daniel R. Gulotta and Christian Johansson),
to appear in Journal of the EMS.
|
PDF |
arXiv |
Journal |
|
We show that the compactly supported cohomology of Shimura varieties of
Hodge type of infinite Γ1(p∞)-level
(defined with respect to a Borel subgroup) vanishes above the middle degree,
under the assumption that the group of the Shimura datum splits at p. This
generalizes and strengthens the vanishing result proved in
"Shimura varieties at level Γ1(p∞) and
Galois representations". As an application of this vanishing theorem, we
prove a result on the codimensions of ordinary completed homology for the same
groups, analogous to conjectures of Calegari–Emerton for completed
(Borel–Moore) homology.
|
On the generic part of the cohomology of non-compact unitary Shimura varieties
(with Peter Scholze)
|
PDF |
arXiv |
|
|
We prove that the generic part of the mod l cohomology of Shimura varieties
associated to quasi-split unitary groups of even dimension is concentrated
above the middle degree, extending previous work to a non-compact case. The
result applies even to Eisenstein cohomology classes coming from the locally
symmetric space of the general linear group, and has been used in [ACC+18]
to get good control on these classes and deduce potential automorphy theorems
without any self-duality hypothesis.
Our main geometric result is a computation of the fibers of the
Hodge–Tate period map on compactified Shimura varieties, in terms of
similarly compactified Igusa varieties.
|
Potential automorphy over CM fields (with Patrick B. Allen, Frank Calegari,
Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard
Taylor, and Jack A. Thorne), Annals of Math. 197 (2023), no. 3, 897–1113.
|
PDF |
arXiv |
Journal |
|
Let F be a CM number field. We prove modularity lifting theorems for regular
n-dimensional Galois representations over F without any self-duality condition.
We deduce that all elliptic curves E over F are potentially modular, and
furthermore satisfy the Sato–Tate conjecture. As an application of a
different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal
automorphic representations for GL2(𝔸F).
|
Perfectoid Shimura varieties, in Perfectoid spaces: Lectures from
the 2017 Arizona Winter School.
|
PDF |
|
Book |
Video |
This is an expanded version of the lecture notes for the minicourse I gave at
the 2017 Arizona Winter School. In these notes, I discuss Scholze's
construction of Galois representations for torsion classes in the cohomology
of locally symmetric spaces for GLn, with a focus on his proof that
Shimura varieties of Hodge type with infinite level at p acquire the structure
of perfectoid spaces. I also briefly discuss some recent vanishing results
for the cohomology of Shimura varieties with infinite level at p.
|
Shimura varieties at level Γ1(p∞) and Galois representations
(with Daniel R. Gulotta, Chi-Yun Hsu, Christian Johansson, Lucia Mocz,
Emanuel Reinecke, and Sheng-Chi Shih), Compositio Math. 156 (2020), no. 6, 1152–1230.
|
PDF |
arXiv |
Journal |
Video |
We show that the compactly supported cohomology of certain U(n,n) or
Sp(2n)-Shimura varieties with
Γ1(p∞)-level vanishes above the middle
degree. The only assumption is that we work over a CM field F in which
the prime p splits completely. We also give an application to Galois
representations for torsion in the cohomology of the locally symmetric
spaces for GLn/F. More precisely, we use the vanishing result
for Shimura varieties to eliminate the nilpotent ideal in the
construction of these Galois representations. This strengthens recent
results of Scholze and Newton-Thorne.
|
Patching and the p-adic Langlands program for GL(2,ℚp)
(with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas,
and Sug Woo Shin), Compositio Math. 154 (2018), no. 3, 503–548.
|
PDF |
arXiv |
Journal |
|
We present a new construction of the p-adic local Langlands correspondence for
GL(2, ℚp) via the patching method of Taylor–Wiles and
Kisin. This construction sheds light on the relationship between the various
other approaches to both the local and global aspects of the p-adic Langlands
program; in particular, it gives a new proof of many cases of the second
author's local-global compatibility theorem, and relaxes a hypothesis on
the local mod p representation in that theorem.
|
Kisin modules with descent data and parahoric local models
(with Brandon Levin),
Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 1, 181–213.
|
PDF |
arXiv |
Journal |
|
We construct a moduli space Yμ,τ of Kisin modules with
tame descent datum τ and with fixed p-adic Hodge type μ, for some
finite extension K/ℚp. We show that this space is
smoothly equivalent to the local model for
ResK/ℚpGLn, cocharacter {μ}, and
parahoric level structure. We use this to construct the analogue of
Kottwitz–Rapoport strata on the special fiber Yμ,τ
indexed by the μ-admissible set. We also relate Yμ,τ
to potentially crystalline Galois deformation rings.
|
On the generic part of the cohomology of compact unitary Shimura varieties
(with Peter Scholze), Annals of Math. 186 (2017), no. 3, 649–766.
|
PDF |
arXiv |
Journal |
|
|
The goal of this paper is to show that the cohomology of compact unitary
Shimura varieties is concentrated in the middle degree and torsion-free,
after localizing at a maximal ideal of the Hecke algebra satisfying a
suitable genericity assumption. Along the way, we establish various
foundational results on the geometry of the Hodge-Tate period map. In
particular, we compare the fibres of the Hodge-Tate period map with Igusa
varieties.
|
p-adic q-expansion principles on unitary Shimura varieties (with Ellen
Eischen, Jessica Fintzen, Elena Mantovan, and Ila Varma),
Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop.
Springer International Publishing (2016), 197–243.
|
PDF |
arXiv |
Book |
|
We formulate and prove certain vanishing theorems for p-adic automorphic
forms on unitary groups of arbitrary signature. The p-adic q-expansion
principle for p-adic modular forms on the Igusa tower says that if the
coefficients of (sufficiently many of) the q-expansions of a p-adic
modular form f are zero, then f vanishes everywhere on the Igusa tower.
There is no p-adic q-expansion principle for unitary groups of arbitrary
signature in the literature. By replacing q-expansions with Serre-Tate
expansions (expansions in terms of Serre-Tate deformation coordinates) and
replacing modular forms with automorphic forms on unitary groups of
arbitrary signature, we prove an analogue of the p-adic q-expansion
principle. More precisely, we show that if the coefficients of
(sufficiently many of) the Serre-Tate expansions of a p-adic automorphic
form f on the Igusa tower (over a unitary Shimura variety) are zero, then
f vanishes identically on the Igusa tower.
This paper also contains a substantial expository component. In
particular, the expository component serves as a complement to Hida's
extensive work on p-adic automorphic forms.
|
On the image of complex conjugation in certain Galois representations
(with Bao V. Le Hung), Compositio Math. 152 (2016), no. 7,
1476–1488.
|
PDF |
arXiv |
Journal |
|
We compute the image of any choice of complex conjugation on the Galois
representations associated to regular algebraic cuspidal automorphic
representations and to torsion classes in the cohomology of locally
symmetric spaces for GLn over a totally real field F.
|
Patching and the p-adic local Langlands correspondence (with Matthew
Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas, and Sug
Woo Shin), Cambridge Journal of Math. 4 (2016), no. 2, 197–287.
|
PDF |
arXiv |
Journal |
Video |
We use the patching method of Taylor–Wiles and Kisin to construct a
candidate for the p-adic local Langlands correspondence for
GLn(F), F a finite extension of ℚp. We use our
construction to prove many new cases of the Breuil–Schneider
conjecture.
|
Monodromy and local-global compatibility for l=p, Algebra Number Theory 8
(2014), no. 7, 1597–1646.
|
PDF |
arXiv |
Journal |
|
We strengthen the compatibility between local and global Langlands
correspondences for GLn when n is even and l=p. Let L be a CM
field and Π a cuspidal automorphic representation of
GLn(AL) which
is conjugate self-dual and regular algebraic. In this case, there is an
l-adic Galois representation associated to Π, which is known to be
compatible with local Langlands in almost all cases when l=p by recent
work of Barnet-Lamb, Gee, Geraghty and Taylor. The compatibility was
proved only up to semisimplification unless Π has Shin-regular weight.
We extend the compatibility to Frobenius semisimplification in all cases
by identifying the monodromy operator on the global side. To achieve this,
we derive a generalization of Mokrane's weight spectral sequence for log
crystalline cohomology.
|
Local-global compatibility and the action of monodromy on nearby cycles,
Duke Math. J. 161 (2012), no. 12, 2311–2413.
|
PDF |
arXiv |
Journal |
Video |
We strengthen the local-global compatibility of Langlands correspondences
for GLn in the case when n is even and l≠p. Let L be
a CM field and Π be a cuspidal automorphic representation of
GLn(AL) which is conjugate self-dual. Assume that
Π∞ is cohomological and not "slightly regular", as
defined by Shin. In this case, Chenevier and Harris constructed an
l-adic Galois representation Rl(Π) and proved the
local-global compatibility up to semisimplification at primes v not
dividing l. We extend this compatibility by showing that the Frobenius
semisimplification of the restriction of Rl(Π) to the
decomposition group at v corresponds to the image of Πv via
the local Langlands correspondence. We follow the strategy of
Taylor-Yoshida, where it was assumed that Π is square-integrable at a
finite place. To make the argument work, we study the action of the
monodromy operator N on the complex of nearby cycles on a scheme which is
locally etale over a product of semistable schemes and derive a
generalization of the weight-spectral sequence in this case. We also prove
the Ramanujan–Petersson conjecture for Π as above.
|
|