Recent progress on Langlands reciprocity for GL_{n}:
Shimura varieties and beyond
(with Sug Woo Shin)

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The goal of these lecture notes is to survey progress on the global
Langlands reciprocity conjecture for GL_{n} over number fields
from the last decade and a half. We highlight results and conjectures on
Shimura varieties and more general locally symmetric spaces, with a view
towards the Calegari–Geraghty method to prove modularity lifting
theorems beyond the classical setting of Taylor–Wiles.

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 X_{0}(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 localglobal compatibility
theorem for the padic 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.

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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 padic 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 twodimensional 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
twodimensional mod p representations of the absolute Galois group of a
padic local field.

Components of moduli stacks of twodimensional Galois representations
(with Matthew Emerton, Toby Gee, and David Savitt)

PDF 
arXiv 


In a previous article we introduced various moduli stacks of
twodimensional tamely potentially Barsotti–Tate representations of the
absolute Galois group of a padic 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 twodimensional 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 twodimensional mod p representations of
the absolute Galois group of a padic local field, as well as their
resolutions by moduli stacks of twodimensional 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 nonzero. The proof is based on the
geometric JacquetLanglands functoriality established by TianXiao 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 padic local
Langlands correspondence for GL_{2}(ℚ_{p}) occurs in the
completed homology of Hilbert modular varieties.

New frontiers in Langlands reciprocity, EMS Magazine (2021), no. 119.

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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),
J. Eur. Math. Soc. 25 (2023), no. 3, 869–911.

PDF 
arXiv 
Journal 

We show that the compactly supported cohomology of Shimura varieties of
Hodge type of infinite Γ_{1}(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 Γ_{1}(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 noncompact unitary Shimura varieties
(with Peter Scholze), to appear in Annals of Math.

PDF 
arXiv 


We prove that the generic part of the mod l cohomology of Shimura varieties
associated to quasisplit unitary groups of even dimension is concentrated
above the middle degree, extending previous work to a noncompact 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 selfduality 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
ndimensional Galois representations over F without any selfduality 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 GL_{2}(𝔸_{F}).

Perfectoid Shimura varieties, in Perfectoid spaces: Lectures from
the 2017 Arizona Winter School.

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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 GL_{n}, 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 Γ_{1}(p^{∞}) and Galois representations
(with Daniel R. Gulotta, ChiYun Hsu, Christian Johansson, Lucia Mocz,
Emanuel Reinecke, and ShengChi 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
Γ_{1}(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 GL_{n}/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 NewtonThorne.

Patching and the padic Langlands program for GL(2,ℚ_{p})
(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 padic local Langlands correspondence for
GL(2, ℚ_{p}) 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 padic Langlands
program; in particular, it gives a new proof of many cases of the second
author's localglobal 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 Y^{μ,τ} of Kisin modules with
tame descent datum τ and with fixed padic Hodge type μ, for some
finite extension K/ℚ_{p}. We show that this space is
smoothly equivalent to the local model for
Res_{K/ℚp}GL_{n}, cocharacter {μ}, and
parahoric level structure. We use this to construct the analogue of
Kottwitz–Rapoport strata on the special fiber Y^{μ,τ}
indexed by the μadmissible set. We also relate Y^{μ,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 torsionfree,
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 HodgeTate period map. In
particular, we compare the fibres of the HodgeTate period map with Igusa
varieties.

padic qexpansion 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 padic automorphic
forms on unitary groups of arbitrary signature. The padic qexpansion
principle for padic modular forms on the Igusa tower says that if the
coefficients of (sufficiently many of) the qexpansions of a padic
modular form f are zero, then f vanishes everywhere on the Igusa tower.
There is no padic qexpansion principle for unitary groups of arbitrary
signature in the literature. By replacing qexpansions with SerreTate
expansions (expansions in terms of SerreTate deformation coordinates) and
replacing modular forms with automorphic forms on unitary groups of
arbitrary signature, we prove an analogue of the padic qexpansion
principle. More precisely, we show that if the coefficients of
(sufficiently many of) the SerreTate expansions of a padic 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 padic 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 GL_{n} over a totally real field F.

Patching and the padic 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 padic local Langlands correspondence for
GL_{n}(F), F a finite extension of ℚ_{p}. We use our
construction to prove many new cases of the Breuil–Schneider
conjecture.

Monodromy and localglobal 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 GL_{n} when n is even and l=p. Let L be a CM
field and Π a cuspidal automorphic representation of
GL_{n}(A_{L}) which
is conjugate selfdual and regular algebraic. In this case, there is an
ladic 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 BarnetLamb, Gee, Geraghty and Taylor. The compatibility was
proved only up to semisimplification unless Π has Shinregular 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.

Localglobal 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 localglobal compatibility of Langlands correspondences
for GL_{n} in the case when n is even and l≠p. Let L be
a CM field and Π be a cuspidal automorphic representation of
GL_{n}(A_{L}) which is conjugate selfdual. Assume that
Π_{∞} is cohomological and not "slightly regular", as
defined by Shin. In this case, Chenevier and Harris constructed an
ladic Galois representation R_{l}(Π) and proved the
localglobal compatibility up to semisimplification at primes v not
dividing l. We extend this compatibility by showing that the Frobenius
semisimplification of the restriction of R_{l}(Π) to the
decomposition group at v corresponds to the image of Π_{v} via
the local Langlands correspondence. We follow the strategy of
TaylorYoshida, where it was assumed that Π is squareintegrable 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 weightspectral sequence in this case. We also prove
the Ramanujan–Petersson conjecture for Π as above.

