The Langlands programme is a “grand unified theory” of mathematics: a vast network of conjectures that connect number theory to other areas of pure mathematics, such as representation theory, algebraic geometry, and harmonic analysis.

One of the fundamental principles underlying the Langlands conjectures is
*reciprocity*, which can be thought of as a magical bridge that connects
different mathematical worlds. This principle goes back centuries to the
foundational work of Euler, Legendre and Gauss on the law of quadratic
reciprocity. A celebrated modern instance of reciprocity is the correspondence
between modular forms and rational elliptic curves, which played a key role in
Wiles’s proof of Fermat’s Last Theorem [46
A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of
Math. (2)141, 443–551 (1995)
] and which relied on the
famous Taylor–Wiles method for proving modularity [43
R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras.
Ann. of Math. (2)141, 553–572 (1995)
].
Recently, the search for new reciprocity laws has begun to expand the scope of
the Langlands programme.

The *Ramanujan–Petersson conjecture* is an important consequence of the
Langlands programme, which goes back to a prediction Ramanujan made a century
ago about the size of the Fourier coefficients of a certain modular form
$Δ$, a highly symmetric function on the upper half plane. The
*Sato–Tate conjecture* is an equidistribution result about the number of
points of a given elliptic curve modulo varying primes, formulated half a
century ago. It is also a consequence of the Langlands programme.
In Section 1, I survey progress on these conjectures in two
fundamentally different settings: one setting in which there is a direct
connection to algebraic geometry (*modular curves*) and one setting in
which such a connection is missing (arithmetic hyperbolic $3$-manifolds, or
*Bianchi manifolds*).

*Shimura varieties* are certain highly symmetric algebraic varieties that
generalise modular curves and that provide, in many cases, a geometric
realisation of Langlands reciprocity. In Section 2, I explain a
new tool for understanding Shimura varieties called the *Hodge–Tate
period morphism*. This was introduced by Scholze in [35
P. Scholze, On torsion in the cohomology of locally symmetric varieties.
Ann. of Math. (2)182, 945–1066 (2015)
] and
refined in my joint work with Scholze [16
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties. Ann. of Math. (2)186, 649–766
(2017)
]. I then
discuss vanishing theorems for the cohomology of Shimura varieties proved using
the geometry of the Hodge–Tate period morphism [16
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties. Ann. of Math. (2)186, 649–766
(2017)
, 17
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties.
arXiv:1909.01898 (September 2019)
].

The *Calegari–Geraghty method* [11
F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles
method. Invent. Math.211, 297–433 (2018)
] vastly extends
the scope of the Taylor–Wiles method, though it is conjectural on an extension
of the Langlands programme to incorporate torsion in the cohomology of locally
symmetric spaces. In Section 3, I discuss joint
work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and
Thorne [1
P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm,
B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and
J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018)
], where we implement the Calegari–Geraghty method
unconditionally over *CM fields*, an important class of number fields that
contains imaginary quadratic fields as well as cyclotomic fields. This work
relies crucially on one of the vanishing theorems mentioned
above [17
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties.
arXiv:1909.01898 (September 2019)
], and has applications to both the
Ramanujan–Petersson and the Sato–Tate conjectures over CM fields.

**Remark 1****.** The Langlands programme is a beautiful but technical subject, with roots in
many different areas of mathematics. For a general mathematician,
Section 1 is the most accessible, as it highlights two concrete
consequences of the Langlands conjectures. The later
Sections 2 and 3 assume more
background in algebraic geometry and number theory.

I have prioritised references to well-written surveys above references to the original papers. I particularly recommend [21 M. Emerton. Langlands reciprocity: L-functions, automorphic forms, and Diophantine equations. To appear in The Genesis of the Langlands program (2020) ] for a historical account of Langlands reciprocity, [41 R. Taylor, Galois representations. Ann. Fac. Sci. Toulouse Math. (6)13, 73–119 (2004) ] for more background on the Langlands correspondence, and [36 P. Scholze, p-adic geometry. In Proceedings of the International Congress of Mathematicians – Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 899–933 (2018) ] for a cutting-edge account of the deep connections between arithmetic geometry and the Langlands programme.

## 1 The Ramanujan and Sato–Tate conjectures

### 1.1 Modular curves and Bianchi manifolds

The goal of this section is to discuss two fundamental examples of locally
symmetric spaces: *modular curves*, which have an algebraic structure, and
*Bianchi manifolds*, which do not. This dichotomy underlies the
fundamental difference between reciprocity laws over the field of rational
numbers $Q$ (and over real quadratic fields such as $Q(5 )$), and
reciprocity laws over imaginary quadratic fields such as $Q(i)$.

Let $G$ be a connected reductive group defined over $Q$, for example
$SL_{n}$, $GL_{n}$ or $Sp_{2n}$. We can then consider an
associated *symmetric space*$X$, endowed with an action of the real
points $G(R)$. This is roughly identified with $G(R)/K_{∞}$,
where $K_{∞}⊂G(R)$ is a maximal compact subgroup. We then want to
consider the action of certain arithmetic groups on $X$: more precisely we want
to restrict to finite index subgroups $Γ⊂G(Z)$ cut out by
congruence conditions. If $Γ$ is sufficiently small, we can form the
quotient $Γ\X$ and obtain a smooth orientable Riemannian
manifold, which is a *locally symmetric space* for $G$.

**Example 2****.** If $G=SL_{2}/Q$, the corresponding symmetric space is the upper-half
plane

endowed with the hyperbolic metric. The action of $SL_{2}(R)$ on $H_{2}$ is by Möbius transformations:

For $Γ⊂SL_{2}(Z)$ a finite index congruence subgroup (that
is assumed sufficiently small), the quotients $Γ\H_{2}$
are Riemann surfaces. These Riemann surfaces come from algebraic curves
$X_{Γ}$ defined over $Q$ (or over finite extensions of
$Q$) called *modular curves*. A fundamental domain for a proper
subgroup $Γ⊂SL_{2}(Z)$ acting on $H_{2}$ is a finite
union of translates of the fundamental domain in
Figure 1.

**Example 3****.** If $G=SL_{2}/F$, where $F$ is an imaginary quadratic field^{1}This
can be viewed as a connected reductive group over $Q$ using a
technical notion called the Weil restriction of scalars., the corresponding
symmetric space is $3$-dimensional hyperbolic space

and the locally symmetric spaces are called *Bianchi manifolds*.
These are arithmetic hyperbolic $3$-manifolds and, since their real dimension
is odd, they do not admit a complex or algebraic structure.

The locally symmetric spaces for a group $G$ are important in what follows
because they give a way to access *automorphic representations* of $G$,
the central objects of study in the Langlands programme. This is explained more
in Section 2. For example, *modular forms*^{2}These
give rise to automorphic representations for the group $SL_{2}/Q$., which are
holomorphic functions on $H_{2}$ that satisfy a transformation relation
under some $Γ$, contribute to the first Betti cohomology of modular curves
(with possibly twisted coefficients).

Some locally symmetric spaces have an algebraic structure. If this happens,
they in fact come from smooth, quasi-projective varieties $X_{Γ}$ defined
over number fields, which are called *Shimura varieties*. The geometry of
Shimura varieties is a rich and fascinating subject in itself, that we discuss
more in Section 2. On the other hand, the Langlands programme is
much more mysterious beyond the setting of Shimura varieties, because there is
no obvious connection to algebraic geometry or arithmetic. We discuss this more
in Section 3.

### 1.2 The Ramanujan conjecture

A famous example of a modular form is Ramanujan’s $Δ$ function. If $z$ is the variable on the upper-half plane $H_{2}$ and $q=e_{2πiz}$, $Δ$ is given by the Fourier series expansion

In 1916, Ramanujan made three predictions about the behaviour of the Fourier
coefficients $τ(n)$. The first two of these were immediately proved by
Mordell by studying the action on $Δ$ of certain *Hecke operators*,
that we return to in Section 2. The *Ramanujan conjecture*,
which resisted attempts at proof for much longer, bounds the absolute value of
the Fourier coefficients: it states that $∣τ(p)∣≤2p_{11/2}$ for
all primes $p$.

Deligne finally established this bound in the early 1970’s, and this was one of the reasons for which he was awarded a Fields Medal in 1978. While the bound on the Fourier coefficients is purely a statement within harmonic analysis, the proof used the bridge of Langlands reciprocity and was ultimately obtained from a statement in arithmetic geometry. More precisely, Deligne’s proof of the Ramanujan conjecture went via the étale cohomology of modular curves, obtaining the desired bound as a consequence of his proof of the Weil conjectures for smooth projective varieties over finite fields.

The generalised *Ramanujan–Petersson conjecture* is a vast extension of
the above statement, with numerous applications across mathematics and computer
science. See, for example, the survey [31
W.-C. W. Li, The Ramanujan conjecture and its applications. Philos.
Trans. Roy. Soc. A378, 20180441, 14 (2020)
] for its applications
to extremal combinatorial objects called Ramanujan graphs. This more general
conjecture, which is part of Arthur’s conjectures on the automorphic spectrum
of $GL_{n}$ (see also the survey [34
P. Sarnak, Notes on the generalized Ramanujan conjectures. In Harmonic
analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4,
Amer. Math. Soc., Providence, RI, 659–685 (2005)
]), predicts that the
local components at finite places of cuspidal automorphic representations of
$GL_{n}$ are *tempered*.

Temperedness means roughly that the matrix coefficients of the
representation are in $L_{2+ϵ}$ for all $ϵ>0$. This singles
out the building blocks of the category of irreducible admissible
representations of $p$-adic groups, such as $GL_{n}(Q_{p})$, in the sense that
everything else can be constructed from tempered representations of smaller
groups. Tempered representations also play an important role in the *local
Langlands conjecture*, which relates them to arithmetic objects, essentially
representations of local Galois groups. For the group $GL_{n}$, local Langlands
is a theorem, proved by Harris–Taylor and Henniart in the early 2000’s, and
later reproved by Scholze.

For certain cuspidal automorphic representations of $GL_{n}$, which are global
objects built from the irreducible admissible representations mentioned above,
one can try to follow Deligne’s approach to the Ramanujan conjecture using the
étale cohomology of higher-dimensional Shimura varieties. When these
varieties have singular reduction, the arithmetic counterpart of the
Ramanujan–Petersson conjecture is Deligne’s *weight-monodromy
conjecture*. This goes beyond the Weil conjectures to predict that the étale
cohomology of smooth projective varieties over $p$-adic fields has a remarkably
elegant shape, even in the case of singular reduction.

In [12 A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles. Duke Math. J.161, 2311–2413 (2012) ], building on [18 L. Clozel, Purity reigns supreme. Int. Math. Res. Not. IMRN 328–346 (2013) , 39 S. W. Shin, Galois representations arising from some compact Shimura varieties. Ann. of Math. (2)173, 1645–1741 (2011) , 44 R. Taylor and T. Yoshida, Compatibility of local and global Langlands correspondences. J. Amer. Math. Soc.20, 467–493 (2007) ] and [25 M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies 151, Princeton University Press, Princeton, NJ (2001) ], I follow Deligne’s approach and complete the proof of the following result.

**Theorem 4**

**.**

*Let $F$ be a CM field and let $π$ be a regular algebraic, self-dual cuspidal
automorphic representation of $GL_{n}/F$. Then $π$ satisfies the generalised
Ramanujan–Petersson conjecture.*

The global Langlands correspondence relates automorphic representations to
global Galois representations. The direction from automorphic to Galois is best
understood in the setting of Theorem 4, which is the
so-called “self-dual case”. This has been a milestone achievement in the
field: it required the combined effort of many people over several decades,
including Kottwitz, Clozel, Harris, Taylor, Shin, and Chenevier, and was built
on fundamental contributions by Arthur, Laumon, Ngô and Waldspurger.
In [12
A. Caraiani, Local-global compatibility and the action of monodromy on nearby
cycles. Duke Math. J.161, 2311–2413 (2012)
, 13
A. Caraiani, Monodromy and local-global compatibility for l=p.
Algebra Number Theory8, 1597–1646 (2014)
], I also complete the proof that the
associated Galois representations are compatible with local
Langlands^{3}Local-global compatibility is a crucial property one expects
from the Langlands correspondence, which generalises the compatibility between
local and global class field theory., by establishing new instances of the
weight-monodromy conjecture for Shimura varieties.

More recently, in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, I obtained an application to the Ramanujan–Petersson conjecture beyond the self-dual case. This is the first instance where this conjecture is not deduced from the Weil conjectures, but rather by an approximation of the very different strategy outlined by Langlands in [30 R. P. Langlands, Problems in the theory of automorphic forms. In Lectures in modern analysis and applications, III, 18–61. Lecture Notes in Math., Vol. 170 (1970) ].

**Theorem 5**([1 P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018) ])

**.**

*Let $F$ be a CM field and $π$ be a cuspidal automorphic representation of
$GL_{2}/F$ of parallel weight $2$. Then $π$ satisfies the
generalised Ramanujan–Petersson conjecture.*

The condition on the weight means that $π$ contributes to the Betti cohomology with constant coefficients of the relevant locally symmetric space, which is for example a Bianchi manifold. These locally symmetric spaces do not have an algebraic structure, so one cannot appeal directly to arithmetic geometry. We come back to discuss the strategy for the proof of Theorem 5 in Section 3.

### 1.3 The Sato–Tate conjecture

An *elliptic curve* is a smooth, projective curve of genus one together
with a specified point. If $F$ is a number field, an elliptic curve defined
over $F$ can be described as a plane curve, given by (the homogenisation of) a
cubic equation of the form $y_{2}=x_{3}+ax+b$ with $a,b∈F$.

Such an elliptic curve $E/F$, if it does not have complex multiplication, is
expected to satisfy the *Sato–Tate conjecture*. When $p$ is a
prime of $F$ over which $E$ has good reduction, the number

(where $k(p)$ denotes the residue field at $p$, of cardinality $q_{p}$) is contained in the interval $[−1,1]$ by a result of Hasse; this is also a special case of Deligne’s result on the Weil conjectures. The Sato–Tate conjecture, formulated in the 1960’s, states that, as $p$ runs over all the primes of $F$ over which $E$ has good reduction, these numbers become equidistributed in $[−1,1]$ with respect to the semicircle probability measure $π2 1−x_{2} dx$.

**Remark 6****.** The condition for an elliptic curve to have complex multiplication is very
special, and in that case the probability distribution is different and
well-understood. See [40
A. V. Sutherland, Sato–Tate distributions. In Analytic methods in
arithmetic geometry, Contemp. Math. 740, Amer. Math. Soc., Providence, RI,
197–248 (2019)
] for a survey on Sato–Tate-type
conjectures, which explains the expected distributions, and [26
N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and
monodromy. American Mathematical Society Colloquium Publications 45,
American Mathematical Society, Providence, RI (1999)
]
for the more general conceptual framework that underlies this conjecture.

According to the Langlands reciprocity conjecture, any elliptic curve $E/F$ is
also expected to come from an automorphic representation of $GL_{2}$
over $F$. If this is the case, we say that $E$ is *automorphic*. The
precise relationship between elliptic curves and automorphic representations
can be expressed as an equality of the two *$L$-functions* associated to
them. $L$-functions are complex analytic functions that generalise the Riemann
zeta function and that remember deep arithmetic information about the original
objects.

For example, the $L$-functions of all elliptic curves defined over $Q$ are known to come from modular forms, by work of Breuil–Conrad–Diamond–Taylor [9 C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises. J. Amer. Math. Soc.14, 843–939 (2001) ] building on [46 A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2)141, 443–551 (1995) ] and [43 R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2)141, 553–572 (1995) ]. The analogous result for elliptic curves defined over real quadratic fields was later proved by Freitas–Le Hung–Siksek [22 N. Freitas, B. V. Le Hung and S. Siksek, Elliptic curves over real quadratic fields are modular. Invent. Math.201, 159–206 (2015) ]. The $L$-functions of elliptic curves over imaginary quadratic fields are expected to come from classes in the cohomology of Bianchi manifolds, but this case has historically been much more mysterious.

Soon after the Sato–Tate conjecture was formulated, Serre and Tate discovered
that the correct distribution would follow from the expected analytic
properties of the symmetric power $L$-functions of $E$. In turn, these analytic
properties would follow if one knew the automorphy of $E$ and all its symmetric
powers. This argument is explained in [37
J.-P. Serre, Abelian l-adic representations and elliptic curves.
McGill University lecture notes written with the collaboration of Willem Kuyk
and John Labute, W. A. Benjamin, Inc., New York-Amsterdam (1968)
] and uses Tauberian theorems
in analytic number theory: the techniques are essentially those that led to the
proof of the prime number theorem. In fact, to establish the correct
distribution, it suffices to know that $E$ and its symmetric powers are
*potentially automorphic*: this means they become automorphic after base
change to some Galois field extension $F_{′}$ of $F$.

The Sato–Tate conjecture for elliptic curves defined over totally real fields was proved in most cases by Clozel, Harris, Shepherd-Barron, and Taylor [19 L. Clozel, M. Harris and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. Publ. Math. Inst. Hautes Études Sci. 1–181 (2008) , 24 M. Harris, N. Shepherd-Barron and R. Taylor, A family of Calabi-Yau varieties and potential automorphy. Ann. of Math. (2)171, 779–813 (2010) , 42 R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 183–239 (2008) ], and completed in work of Barnet-Lamb–Geraghty–Harris–Taylor around 2010 [4 T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor, A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci.47, 29–98 (2011) ]. This relied on the potential automorphy of symmetric powers, which could be established in the self-dual setting using a generalisation of the Taylor–Wiles method. However, the method broke down for elliptic curves defined over imaginary quadratic fields or more general CM fields. In Section 3, we explain how to overcome the barrier to treating elliptic curves defined over CM fields and obtain the following result.

**Theorem 7**([1 P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018) ])

**.**

*Let $F$ be a CM field and $E/F$ be an elliptic curve that does not have complex
multiplication. Then $E$ is potentially automorphic and satisfies the
Sato–Tate conjecture.*

**Remark 8****.** Both Theorems 5 and 7 rely crucially on
the vanishing theorem for Shimura varieties proved
in [17
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties.
arXiv:1909.01898 (September 2019)
], which is discussed
in Section 2.

**Remark 9****.** The beautiful work of Boxer–Calegari–Gee–Pilloni [7
G. Boxer, F. Calegari, T. Gee, and V. Pilloni. Abelian Surfaces over totally real fields are Potentially Modular.
arXiv:1812.09269 (December 2018)
], completed at
the same time as [1
P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm,
B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and
J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018)
], proves the potential automorphy of elliptic
curves in Theorem 7 independently, and they are even able to
show the potential automorphy of abelian surfaces over totally real fields.
Moreover, in the recent paper [2
P. B. B. Allen, C. Khare, and J. A. Thorne. Modularity of GL2(𝔽p)-representations over CM
fields. arXiv:1910.12986 (October 2019)
], Allen–Khare–Thorne
establish actual automorphy of elliptic curves in certain cases (rather than
potential automorphy). All of this is hopefully only the beginning of a
fascinating story over CM fields!

## 2 Vanishing theorems for Shimura varieties with torsion coefficients

### 2.1 Shimura varieties

Recall that, if the locally symmetric spaces for a group $G/Q$ have an algebraic structure, they in fact come from smooth, quasi-projective varieties $X_{Γ}$ defined over number fields, which are called Shimura varieties.

The pair $(G,X)$ must satisfy certain axioms in order for the corresponding locally symmetric spaces to come from Shimura varieties. The key point is for the symmetric space $X$ to be a Hermitian symmetric domain (or a finite disjoint union thereof). There is a complete classification of groups $G$ for which this holds. For example, the symplectic group $Sp_{2n}$ and the unitary group $U(n,n)$ give rise to Shimura varieties, which can be described in terms of moduli spaces of abelian varieties equipped with additional structures.

**Remark 10****.** Some locally symmetric spaces that are not Shimura varieties can still
be studied by relating them to Shimura varieties. For example, Bianchi manifolds
can be realised in the boundary of certain compactifications of Shimura
varieties attached to the unitary group $U(2,2)$. We come back to
this in Section 3.

Recall also that the locally symmetric spaces for a group $G$ give a way to
access automorphic representations of $G$. More precisely, as the congruence
subgroup $Γ⊂G(Z)$ varies, we have a tower of locally symmetric
spaces. The symmetries of this tower induce correspondences on each individual
space $Γ\X$ called Hecke operators^{4}To discuss Hecke
operators rigorously, we should use the adelic perspective on locally
symmetric spaces and Shimura varieties. The resulting spaces would be
disjoint unions of finitely many copies of $Γ\X$. We ignore
this subtlety here and later on in the text.. Keeping track of the various
Hecke operators, we obtain an action of a commutative Hecke algebra
$T$ on the Betti cohomology $H_{i}(Γ\X,C)$. The work
of Matsushima, Franke and others shows that the systems of eigenvalues of
$T$ that occur in $H_{i}(Γ\X,C)$ come from certain
automorphic representations of $G$.

In addition to the Hecke symmetry, the cohomology of Shimura varieties also has a Galois symmetry, because Shimura varieties are defined over number fields. Because of these two kinds of symmetries, Shimura varieties give, in many cases, a geometric realisation of the global Langlands correspondence between automorphic and Galois representations.

One can ask a more precise question, about the range of degrees of cohomology to which any particular automorphic representation can contribute. Assume, for simplicity, that $X_{Γ}(C)$ is a compact Shimura variety. Then Borel–Wallach [6 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Second ed., Mathematical Surveys and Monographs 67, American Mathematical Society, Providence, RI (2000) ] show that, if $π$ is an automorphic representation whose component at $∞$ is a tempered representation of $G(R)$, then $π$ can only contribute to $H_{i}(X_{Γ}(C),C)$ in the middle degree $i=dim_{C}X_{Γ}$. This result, like the Ramanujan–Petersson conjecture, also fits within the framework of Arthur’s conjectures [3 J. Arthur, Unipotent automorphic representations: Conjectures. 171–172, 13–71 (1989) ].

**Question 11****.** The upshot of the Borel–Wallach result is that the cohomology of a Shimura
variety $X_{Γ}$ with $C$-coefficients is somehow degenerate outside the
middle degree. Can we extend this to torsion coefficients, such as
$H_{i}(X_{Γ}(C),F_{ℓ})$?

More precise versions of this question are formulated as conjectures in [11 F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles method. Invent. Math.211, 297–433 (2018) ] and [20 M. Emerton, Completed cohomology and the p-adic Langlands program. In Proceedings of the International Congress of Mathematicians – Seoul 2014. Vol. II, Kyung Moon Sa, Seoul, 319–342 (2014) ]. These are motivated by the Calegari–Geraghty method, which is discussed in Section 3, and by the search for a mod $ℓ$ analogue of Arthur’s conjectures. In the next two subsections, we explain a new tool that can be used to compute $H_{i}(X_{Γ}(C),F_{ℓ})$ and discuss our results towards Question 11.

### 2.2 The Hodge–Tate period morphism

This morphism was introduced by Scholze in his breakthrough paper [35 P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2)182, 945–1066 (2015) ] and gives a completely new way to access the geometry and cohomology of Shimura varieties.

In the case of the modular curve, the Hodge–Tate period morphism is a $p$-adic analogue of the following complex picture, where the map on the right is the standard holomorphic embedding of the upper-half plane $H_{2}$ into the Riemann sphere $P_{1}(C)$:

This picture has the following moduli interpretation. First, $X_{Γ}$ is a moduli space of elliptic curves equipped with some additional structures (determined by $Γ$). The upper-half plane $H_{2}$ is the universal cover of $X_{Γ}(C)=Γ\H_{2}$; it parametrises (positive) complex structures one can put on a two-dimensional real vector space. This amounts to parameterising Hodge structures of elliptic curves, i.e. direct sum decompositions:

with $H_{1}(E,O_{E})=H_{0}(E,Ω_{E}) $. The morphism $π_{dR}$ sends the Hodge decomposition to the associated Hodge filtration

This is an example of a *period morphism*. One can construct such a
diagram for higher-dimensional Shimura varieties as well, and this has played
an important role in studying automorphic forms on Shimura varieties from a
geometric point of view.

The Hodge–Tate period morphism is based on the Hodge–Tate filtration on étale cohomology, tracing back to foundational work in $p$-adic Hodge theory by Tate and Faltings. Let $p$ be a prime and let $C$ be the $p$-adic completion of an algebraic closure of $Q_{p}$, which will play a role analogous to that of $C$ in what follows. If $E/C$ is an elliptic curve, its étale cohomology admits a Hodge–Tate filtration:

See Bhatt’s article in [5 B. Bhatt, A. Caraiani, K. S. Kedlaya and J. Weinstein, Perfectoid spaces. Mathematical Surveys and Monographs 242, American Mathematical Society, Providence, RI (2019) ] for an excellent survey on $p$-adic Hodge theory and more details on the Hodge–Tate filtration. Instead of viewing the curve $X_{Γ}$ as a Riemann surface, we view it as an adic space $X_{Γ}$, a kind of $p$-adic analytic space introduced by Huber. Then there exists a diagram

where $X_{Γ}(p_{∞})$, which is roughly the inverse limit of modular curves $X_{Γ(p_{n})}$ with increasing level at $p$, is a perfectoid space. Over a point of $X_{Γ(p_{∞})}$ corresponding to an elliptic curve $E/C$, we have a trivialisation of $H_{eˊt}(E,Z_{p})≃Z_{p}$. This point gets sent under $π_{HT}$ to the line

For higher-dimensional Shimura varieties, the following result describes the geometry of the Hodge–Tate period morphism in detail. While the statement of Theorem 12 involves much non-trivial arithmetic geometry, it has applications to Theorems 16 and 17 below, whose statements are substantially more elementary.

**Theorem 12**([35 P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2)182, 945–1066 (2015) , 16 A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2)186, 649–766 (2017) ])

**.**

*Let $X_{Γ}$ be a Shimura variety of Hodge type associated to a connected
reductive group $G$. Let $μ$ denote the conjugacy class of Hodge cocharacters
and let $Fℓ_{G,μ}:=G/P_{μ}$ denote the corresponding flag
variety, considered as an adic space over a $p$-adic completion of the reflex
field.*

*There exists a unique perfectoid space*$X_{Γ(p_{∞})}$*which can be identified with the inverse limit of the adic spaces*$(X_{Γ(p_{n})})_{n}$*.**There exists a Hodge–Tate period morphism*$π_{HT}:X_{Γ(p_{∞})}→Fℓ_{G,μ},$*which is*$G(Q_{p})$*-equivariant.**There exists a Newton stratification*$Fℓ_{G,μ}=b∈B(G,μ)⨆ Fℓ_{G,μ}$*into locally closed strata.**If*$X_{Γ}$*is compact and of PEL type, and*$xˉ$*is a geometric point of the Newton stratum*$Fℓ_{G,μ}$*, we identify the fiber*$π_{HT}(xˉ)$*with a “perfectoid” version of an Igusa variety*$Ig_{b}$*.*

**Remark 13****.** The first two parts of Theorem 12 are due to Scholze^{5}Up to
the precise identification of the target of the Hodge–Tate period morphism as
the flag variety $Fℓ_{G,μ}$ in all cases, which is done
in [16
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties. Ann. of Math. (2)186, 649–766
(2017)
]. and play the lead role in his breakthrough
construction of Galois representations for torsion in the cohomology of locally
symmetric spaces. There are many surveys of this result; see for
example [33
S. Morel, Construction de représentations galoisiennes de torsion
[d’après Peter Scholze]. Astérisque Exp. No. 1102, 449–473
(2016)
] or [45
J. Weinstein, Reciprocity laws and Galois representations: Recent
breakthroughs. Bull. Amer. Math. Soc. (N.S.)53, 1–39 (2016)
]. For more details on the Hodge–Tate
period morphism, see also the last article in [5
B. Bhatt, A. Caraiani, K. S. Kedlaya and J. Weinstein, Perfectoid
spaces. Mathematical Surveys and Monographs 242, American Mathematical
Society, Providence, RI (2019)
].

**Remark 14****.** Igusa varieties were introduced by Harris–Taylor as part of their proof of
local Langlands for $GL_{n}$, and generalised by Mantovan.
Rapoport–Zink spaces are local analogues of Shimura varieties, which provide a
geometric realisation of the local Langlands correspondence. The computation
of the fibers of $π_{HT}$ suffices for applications to
Theorems 16 and 17 below, but
in [16
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties. Ann. of Math. (2)186, 649–766
(2017)
], we go further and prove a conceptually
cleaner version of Mantovan’s product formula [32
E. Mantovan, On the cohomology of certain PEL-type Shimura varieties.
Duke Math. J.129, 573–610 (2005)
], which relates
Shimura varieties, Igusa varieties and Rapoport–Zink spaces.

**Remark 15****.** In [17
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties.
arXiv:1909.01898 (September 2019)
] we extend part (4) of
Theorem 12 to $U(n,n)$-Shimura varieties, which are
non-compact. We compute the fibers of $π_{HT}$ for both the minimal
and toroidal compactifications of these Shimura varieties, and relate them to
partial minimal and toroidal compactifications of Igusa varieties.

### 2.3 Vanishing theorems

In order to address Question 11, we would like to compute the
localisation $H_{∗}(X_{Γ},F_{ℓ})_{m}$, where the maximal ideal
$m⊂T$ is equivalent to a mod $ℓ$ system of Hecke
eigenvalues. Using the Hodge–Tate period morphism at an auxiliary prime $p=ℓ$^{6}Here, we assume that the Hecke operators in $T$ are
all supported at primes different from $p$., we obtain an action of
$T$ on the complex of sheaves $Rπ_{HT∗}F_{ℓ}$ living over
$Fℓ_{G,μ}$, and we are reduced to understanding the localisation
$(Rπ_{HT∗}F_{ℓ})_{m}$. By the properties of $π_{HT}$, this behaves
similarly to a perverse sheaf, which is the key to controlling the degrees in
which $(Rπ_{HT∗}F_{ℓ})_{m}$ can have non-zero cohomology. We make
these ideas rigorous in [16
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties. Ann. of Math. (2)186, 649–766
(2017)
, 17
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties.
arXiv:1909.01898 (September 2019)
] for unitary Shimura varieties, under some mild
technical assumptions.

Let $F=F_{+}⋅E$ be a CM field, with maximal totally real field $F_{+}=Q$ and $E$ imaginary quadratic. Let $G$ be a unitary group preserving a
skew-Hermitian form on $F_{m}$. Assume that $G$ is quasi-split at all finite
places. Let $m⊂T$ be a system of Hecke eigenvalues that occurs
in $H_{i}(X_{Γ},F_{ℓ})$. Assume $m$ is *generic* at an auxiliary
prime $p=ℓ$^{7}See [17
A. Caraiani and P. Scholze, On the generic part of the cohomology of compact
unitary Shimura varieties.
arXiv:1909.01898 (September 2019)
, Theorem 1.1] for the
precise condition, which is technical, but explicit. This condition should be
thought of as a mod $ℓ$ analogue of temperedness.. This condition
guarantees that all lifts of $m$ to characteristic $0$ are as simple as
possible at $p$, from a representation-theoretic point of view: they are
generic principal series representations of $G(Q_{p})$.

**Theorem 16**([16 A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2)186, 649–766 (2017) ])

**.**

*If $X_{Γ}$ is compact and $m$ is generic, then $H_{i}(X_{Γ}(C),F_{ℓ})_{m}$ is concentrated in the middle degree $i=dim_{C}X_{Γ}$.*

In the non-compact case, genericity, which is a local condition at an auxiliary prime $p=ℓ$, is not enough. We also need a global condition to control the boundary of the Shimura variety. To formulate the global condition, we consider the semi-simple Galois representation $ρˉ _{m}$ associated to the system of eigenvalues $m$ by [35 P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2)182, 945–1066 (2015) ]; the existence of $ρˉ _{m}$ is an instance of the global Langlands correspondence in the torsion setting. We want to assume that $ρˉ _{m}$ is not too degenerate; this amounts to bounding the number of its absolutely irreducible factors.

**Theorem 17**([17 A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. arXiv:1909.01898 (September 2019) ])

**.**

*If $X_{Γ}$ is a $U(n,n)$-Shimura variety (so $m$ is even and
$G$ is quasi-split at the infinite places as well), $m$ is generic,
and $ρˉ _{m}$ has at most two absolutely irreducible factors, then:*

$H_{c}(X_{Γ}(C),F_{ℓ})_{m}$

*is concentrated in degrees*$i≤dim_{C}X_{Γ}$*, and*$H_{i}(X_{Γ}(C),F_{ℓ})_{m}$

*is concentrated in degrees*$i≥dim_{C}X_{Γ}$*.*

**Remark 18****.** There are previous results in this direction, due to Dimitrov, Shin,
Emerton–Gee, and especially Lan–Suh [28
K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on
compact PEL-type Shimura varieties. Duke Math. J.161,
1113–1170 (2012)
, 29
K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on
general PEL-type Shimura varieties. Adv. Math.242,
228–286 (2013)
]. Compared to
previous work, our result is sharper and better adapted to applications. There
is also intriguing ongoing work of Boyer [8
P. Boyer, Sur la torsion dans la cohomologie des variétés de Shimura
de Kottwitz–Harris–Taylor. J. Inst. Math. Jussieu18,
499–517 (2019)
], which proves a stronger
result in the special case of Harris–Taylor Shimura varieties: he goes beyond
genericity and investigates the distribution of non-generic systems of Hecke
eigenvalues.

**Remark 19****.** The idea of the proof in the compact case is the following: start with a
top-dimensional Newton stratum $Fℓ_{G,μ}⊂Fℓ_{G,μ}$ in the support of $(Rπ_{HT,∗}F_{ℓ})_{m}$.
Since the complex $(Rπ_{HT,∗}F_{ℓ})_{m}$ behaves like a perverse sheaf,
its restriction to $Fℓ_{G,μ}$ is concentrated in one degree.
Therefore, $(Rπ_{HT,∗}Q_{ℓ})_{m}$ is also concentrated in one degree
over $Fℓ_{G,μ}$. On the other hand, we can compute the
alternating sum of cohomology groups of $Ig_{b}$ with
$Q_{ℓ}$-coefficients, using the trace formula and work of
Shin [38
S. W. Shin, A stable trace formula for Igusa varieties. J. Inst. Math.
Jussieu9, 847–895 (2010)
]. In the end, the genericity condition is contradicted
unless $b$ corresponds to the zero-dimensional ordinary stratum. The upshot is
that $(Rπ_{HT,∗}F_{ℓ})_{m}$ is concentrated in one degree over a
zero-dimensional stratum!

**Remark 20****.** In parallel to Question 11, one can also study the cohomology of
locally symmetric spaces with torsion coefficients and with increasing level at
$p$. The resulting structure is called *completed cohomology* and was
introduced by Emerton as a general framework for studying congruences modulo
$p_{k}$ between automorphic forms. Motivated by heuristics coming from the
$p$-adic Langlands programme, Calegari–Emerton [10
F. Calegari and M. Emerton, Completed cohomology – A survey. In
Non-abelian fundamental groups and Iwasawa theory, London Math. Soc.
Lecture Note Ser. 393, Cambridge Univ. Press, Cambridge, 239–257 (2012)
] formulated a
vanishing conjecture for completed cohomology. For most Shimura varieties, the
Calegari–Emerton conjecture is now a theorem due to Scholze and
Hansen–Johansson.

In [14
A. Caraiani, D. R. Gulotta, C.-Y. Hsu, C. Johansson, L. Mocz, E. Reinecke and
S.-C. Shih, Shimura varieties at level Γ1(p∞) and Galois
representations. Compos. Math.156, 1152–1230 (2020)
, 15
A. Caraiani, D. R. Gulotta, and C. Johansson. Vanishing theorems for Shimura varieties at unipotent level. arXiv:1910.09214 (October 2019)
], we prove a vanishing result for the compactly
supported cohomology of Shimura varieties of Hodge type with unipotent level at
$p$. The only assumption is that the group $G$ giving rise to the Shimura
variety is split over $Q_{p}$. This result is stronger than what
Calegari–Emerton conjectured, and it also points towards analogues of
Theorems 16 and 17 for $ℓ=p$, with
*generic* replaced by *ordinary* in the sense of Hida.

## 3 Potential automorphy over CM fields

Theorem 5 on the Ramanujan–Petersson conjecture and Theorem 7 on the Sato–Tate conjecture would follow if we knew that all the symmetric powers of the associated Galois representations were automorphic, or even just potentially automorphic. The original method developed by Taylor–Wiles is a powerful technique for proving automorphy, but it is restricted to settings where a certain numerical criterion holds: these are roughly the settings where the objects on the automorphic side arise from the middle degree cohomology of a Shimura variety.

When $F$ is a number field, the locally symmetric spaces for $GL_{n}/F$, such as the Bianchi manifolds discussed in Example 3, do not have an algebraic structure (outside very special cases). Calegari–Geraghty [11 F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles method. Invent. Math.211, 297–433 (2018) ] proposed an extension of the Taylor–Wiles method to general number fields $F$, conjectural on a precise understanding of the cohomology of locally symmetric spaces for $GL_{n}/F$. Part of their insight was to realise the central role played by torsion classes in the cohomology of these locally symmetric spaces, which should be thought of as modulo $p_{k}$ versions of automorphic forms and treated on equal footing with their characteristic $0$ counterparts. Another part of their insight was to reinterpret the failure of the Taylor–Wiles numerical criterion in terms of certain non-negative integers $q_{0}$, $l_{0}$ seen on the automorphic side.

The Calegari–Geraghty method gives an automorphy lifting result for $GL_{n}/F$ as long as the following prerequisites are in place:

The construction of Galois representations associated to classes in the cohomology with $Z_{p}$ coefficients of the locally symmetric spaces for $GL_{n}/F$.

Local-global compatibility for these Galois representations at all primes of $F$, including at primes above $p$.

A vanishing conjecture for the cohomology with $Z_{p}$ coefficients outside the range of degrees $[q_{0},q_{0}+l_{0}]$, under an appropriate non-degeneracy condition.

**Remark 21****.** For Shimura varieties, the third problem is closely related to
Theorems 16 and 17, since in that case
$q_{0}$ is the middle degree of cohomology and $l_{0}=0$. For $3$-dimensional
Bianchi manifolds, the third problem says that the non-degenerate part of
cohomology is concentrated in degrees $1$ and $2$; this can be checked by hand.
For general locally symmetric spaces that do not have an algebraic structure,
this problem most likely lies deeper than the first two.

When $F$ is a CM field, the first problem was solved by Scholze in [35 P. Scholze, On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2)182, 945–1066 (2015) ], strengthening previous results of Harris–Lan–Taylor–Thorne [23 M. Harris, K.-W. Lan, R. Taylor and J. Thorne, On the rigid cohomology of certain Shimura varieties. Res. Math. Sci.3, Paper No. 37, 308 (2016) ] for characteristic $0$ coefficients. After completing [16 A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2)186, 649–766 (2017) ], it became clear to Scholze and me that a non-compact version of Theorem 16 would give a strategy to attack the second (rather than the third!) problem over CM fields. In joint work with Scholze, I set out to prove Theorem 17 and, in November 2016, I co-organised with Taylor an “emerging topics” working group at the IAS, whose goal was to explore this strategy and its consequences. The working group was a resounding success and it led to the paper [1 P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018) ], where we implement the Calegari–Geraghty method in arbitrary dimension for the first time and obtain as consequences Theorems 5 and 7.

The solution to the first problem above, i.e., the construction of Galois
representations, is much more subtle than in the self-dual case, because one
cannot directly use the étale cohomology of Shimura varieties. Instead, the
starting point for both [23
M. Harris, K.-W. Lan, R. Taylor and J. Thorne, On the rigid cohomology of
certain Shimura varieties. Res. Math. Sci.3, Paper No. 37,
308 (2016)
] and [35
P. Scholze, On torsion in the cohomology of locally symmetric varieties.
Ann. of Math. (2)182, 945–1066 (2015)
] is to realise
the locally symmetric spaces for $GL_{n}/F$ in the boundary of the
*Borel–Serre compactification* of $U(n,n)$-Shimura varieties.
The Borel–Serre compactification is a real manifold with corners, which is
homotopy equivalent to the original $U(n,n)$-Shimura variety. In the
torsion setting, Scholze constructs the desired Galois representations by
congruences, using the Hodge–Tate period morphism for the
$U(n,n)$-Shimura variety. This increases the level at primes of $F$
dividing $p$, and makes the second problem, local-global compatibility,
particularly tricky at these primes.

In [1 P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018) ], we begin to solve the second problem, by establishing the first instances of local-global compatibility at primes of $F$ dividing $p$. We need a delicate argument to understand the boundary of the Borel–Serre compactification, which combines algebraic topology and modular representation theory. In addition, Theorem 17 is the crucial new ingredient: in the middle degree, it implies that classes from the boundary lift to the cohomology of a $U(n,n)$-Shimura variety with $Q_{p}$-coefficients, while remembering the level and weight at primes of $F$ dividing $p$.

The proofs of Theorems 5 and 7 use the Calegari–Geraghty method, together with solutions to the first two problems discussed above. The third problem was not solved with $Z_{p}$ coefficients. By an insight of Khare–Thorne [27 C. B. Khare and J. A. Thorne, Potential automorphy and the Leopoldt conjecture. Amer. J. Math.139, 1205–1273 (2017) ], this problem could be replaced by its $Q_{p}$ coefficient analogue in certain settings. One of the main challenges in [1 P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. A. Thorne. Potential automorphy over CM fields. arXiv:1812.09999 (December 2018) ] was to make this insight compatible with other techniques in automorphy lifting, which rely on reduction modulo $p$. We resolve this challenge by considering reduction modulo $p$ from a derived perspective. Outside low-dimensional cases, such as Bianchi manifolds, or Shimura varieties, the third problem remains open for $Z_{p}$ coefficients.

*Acknowledgements*

This article was written in relation to my being awarded one of the 2020 Prizes of the European Mathematical Society. I wish to dedicate this article to my father, Cornel Caraiani (1954–2020), who inspired my love of mathematics.

I have been lucky to have many wonderful mentors and collaborators and I am grateful to all of them for the mathematics they have taught me. In addition, I especially want to thank Matthew Emerton, Toby Gee, Sophie Morel, James Newton, Peter Scholze, and Richard Taylor for generously sharing their ideas with me over the years, and for their substantial moral and professional support.

I am also grateful to Toby Gee, James Newton, Steven Sivek, and Matteo Tamiozzo for comments on an earlier version of this article.

^{1}This can be viewed as a connected reductive group over $Q$ using a technical notion called the Weil restriction of scalars.

^{2}These give rise to automorphic representations for the group $SL_{2}/Q$.

^{3}Local-global compatibility is a crucial property one expects from the Langlands correspondence, which generalises the compatibility between local and global class field theory.

^{4}To discuss Hecke operators rigorously, we should use the adelic perspective on locally symmetric spaces and Shimura varieties. The resulting spaces would be disjoint unions of finitely many copies of $Γ\X$. We ignore this subtlety here and later on in the text.

^{5}Up to the precise identification of the target of the Hodge–Tate period morphism as the flag variety $Fℓ_{G,μ}$ in all cases, which is done in [16 A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. Ann. of Math. (2)186, 649–766 (2017) ].

^{6}Here, we assume that the Hecke operators in $T$ are all supported at primes different from $p$.

^{7}See [17 A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. arXiv:1909.01898 (September 2019) , Theorem 1.1] for the precise condition, which is technical, but explicit. This condition should be thought of as a mod $ℓ$ analogue of temperedness.

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## Cite this article

Ana Caraiani, New frontiers in Langlands reciprocity. Eur. Math. Soc. Mag. 119 (2021), pp. 8–16

DOI 10.4171/MAG/3