EMS Magazine2021 / No. 119
MAG119DOI 10.4171/MAG-3

New frontiers in Langlands reciprocity

Ana Caraiani
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.

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\Delta, 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 33-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\mathbb{Q} (and over real quadratic fields such as Q(5)\mathbb{Q}(\sqrt{5})), and reciprocity laws over imaginary quadratic fields such as Q(i)\mathbb{Q}(i).

Let GG be a connected reductive group defined over Q\mathbb{Q}, for example SLn\mathrm{SL}_{n}, GLn\mathrm{GL}_{n} or Sp2n\mathrm{Sp}_{2n}. We can then consider an associated symmetric spaceXX, endowed with an action of the real points G(R)G(\mathbb{R}). This is roughly identified with G(R)/KG(\mathbb{R})/K_{\infty}, where KG(R)K_{\infty}\subset G(\mathbb{R}) is a maximal compact subgroup. We then want to consider the action of certain arithmetic groups on XX: more precisely we want to restrict to finite index subgroups ΓG(Z)\Gamma\subset G(\mathbb{Z}) cut out by congruence conditions. If Γ\Gamma is sufficiently small, we can form the quotient Γ\X\Gamma\backslash X and obtain a smooth orientable Riemannian manifold, which is a locally symmetric space for GG.

Example 2. If G=SL2/QG=\mathrm{SL}_{2}/\mathbb{Q}, the corresponding symmetric space is the upper-half plane

SL2(R)/SO2(R)H2:={zCIm z>0}\mathrm{SL}_{2}(\mathbb{R})/\mathrm{SO}_{2}(\mathbb{R})\simeq\mathbb{H}^{2}:=\{z\in\mathbb{C}\mid\mathrm{Im}\ z>0\}

endowed with the hyperbolic metric. The action of SL2(R)\mathrm{SL}_{2}(\mathbb{R}) on H2\mathbb{H}^{2} is by Möbius transformations:

zaz+bcz+d  for (abcd)SL2(R).z\mapsto\frac{az+b}{cz+d}\ \ \mathrm{for}\ \begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{R}).

For ΓSL2(Z)\Gamma\subset\mathrm{SL}_{2}(\mathbb{Z}) a finite index congruence subgroup (that is assumed sufficiently small), the quotients Γ\H2\Gamma\backslash\mathbb{H}^{2} are Riemann surfaces. These Riemann surfaces come from algebraic curves XΓX_{\Gamma} defined over Q\mathbb{Q} (or over finite extensions of Q\mathbb{Q}) called modular curves. A fundamental domain for a proper subgroup ΓSL2(Z)\Gamma\subset\mathrm{SL}_{2}(\mathbb{Z}) acting on H2\mathbb{H}^{2} is a finite union of translates of the fundamental domain in Figure 1.

Figure 1. A fundamental domain for SL2(Z)\mathrm{SL}_{2}(\mathbb{Z}) acting on H2\mathbb{H}^{2}

Example 3. If G=SL2/FG=\mathrm{SL}_{2}/F, where FF is an imaginary quadratic field1This can be viewed as a connected reductive group over Q\mathbb{Q} using a technical notion called the Weil restriction of scalars., the corresponding symmetric space is 33-dimensional hyperbolic space

SL2(C)/SU2(R)H3\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}_{2}(\mathbb{R})\simeq\mathbb{H}^{3}

and the locally symmetric spaces are called Bianchi manifolds. These are arithmetic hyperbolic 33-manifolds and, since their real dimension is odd, they do not admit a complex or algebraic structure.

The locally symmetric spaces for a group GG are important in what follows because they give a way to access automorphic representations of GG, the central objects of study in the Langlands programme. This is explained more in Section 2. For example, modular forms2These give rise to automorphic representations for the group SL2/Q\mathrm{SL}_{2}/\mathbb{Q}., which are holomorphic functions on H2\mathbb{H}^{2} that satisfy a transformation relation under some Γ\Gamma, 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 XΓX_{\Gamma} 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\Delta function. If zz is the variable on the upper-half plane H2\mathbb{H}^{2} and q=e2πizq=e^{2\pi iz}, Δ\Delta is given by the Fourier series expansion

Δ(z)=qn=1(1qn)24=n>0τ(n)qn.\Delta(z)=q\prod_{n=1}^{\infty}(1-q^{n})^{24}=\sum_{n>0}\tau(n)q^{n}.

In 1916, Ramanujan made three predictions about the behaviour of the Fourier coefficients τ(n)\tau(n). The first two of these were immediately proved by Mordell by studying the action on Δ\Delta 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)2p11/2|\tau(p)|\leq 2p^{{11}/{2}} for all primes pp.

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. A 378, 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 GLn\mathrm{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 GLn\mathrm{GL}_{n} are tempered.

Temperedness means roughly that the matrix coefficients of the representation are in L2+ϵL^{2+\epsilon} for all ϵ>0\epsilon>0. This singles out the building blocks of the category of irreducible admissible representations of pp-adic groups, such as GLn(Qp)\mathrm{GL}_{n}(\mathbb{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 GLn\mathrm{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 GLn\mathrm{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 pp-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 FF be a CM field and let π\pi be a regular algebraic, self-dual cuspidal automorphic representation of GLn/F\mathrm{GL}_{n}/F. Then π\pi 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 Theory 8, 1597–1646 (2014) ], I also complete the proof that the associated Galois representations are compatible with local Langlands3Local-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 FF be a CM field and π\pi be a cuspidal automorphic representation of GL2/F\mathrm{GL}_{2}/F of parallel weight 22. Then π\pi satisfies the generalised Ramanujan–Petersson conjecture.

The condition on the weight means that π\pi 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 FF is a number field, an elliptic curve defined over FF can be described as a plane curve, given by (the homogenisation of) a cubic equation of the form y2=x3+ax+by^{2}=x^{3}+ax+b with a,bFa,b\in F.

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

1+qp#E(k(p))2qp\frac{1+q_{\mathfrak{p}}-\#E\bigl(k(\mathfrak{p})\bigr)}{2\sqrt{q_{\mathfrak{p}}}}

(where k(p)k(\mathfrak{p}) denotes the residue field at p\mathfrak{p}, of cardinality qpq_{\mathfrak{p}}) is contained in the interval [1,1][-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\mathfrak{p} runs over all the primes of FF over which EE has good reduction, these numbers become equidistributed in [1,1][-1,1] with respect to the semicircle probability measure 2π1x2dx\frac{2}{\pi}\sqrt{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/FE/F is also expected to come from an automorphic representation of GL2\mathrm{GL}_{2} over FF. If this is the case, we say that EE is automorphic. The precise relationship between elliptic curves and automorphic representations can be expressed as an equality of the two LL-functions associated to them. LL-functions are complex analytic functions that generalise the Riemann zeta function and that remember deep arithmetic information about the original objects.

For example, the LL-functions of all elliptic curves defined over Q\mathbb{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 LL-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 LL-functions of EE. In turn, these analytic properties would follow if one knew the automorphy of EE 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 EE and its symmetric powers are potentially automorphic: this means they become automorphic after base change to some Galois field extension FF^{\prime} of FF.

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 FF be a CM field and E/FE/F be an elliptic curve that does not have complex multiplication. Then EE 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/QG/\mathbb{Q} have an algebraic structure, they in fact come from smooth, quasi-projective varieties XΓX_{\Gamma} defined over number fields, which are called Shimura varieties.

The pair (G,X)(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 XX to be a Hermitian symmetric domain (or a finite disjoint union thereof). There is a complete classification of groups GG for which this holds. For example, the symplectic group Sp2n\mathrm{Sp}_{2n} and the unitary group U(n,n)\mathrm{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)\mathrm{U}(2,2). We come back to this in Section 3.

Recall also that the locally symmetric spaces for a group GG give a way to access automorphic representations of GG. More precisely, as the congruence subgroup ΓG(Z)\Gamma\subset G(\mathbb{Z}) varies, we have a tower of locally symmetric spaces. The symmetries of this tower induce correspondences on each individual space Γ\X\Gamma\backslash X called Hecke operators4To 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\Gamma\backslash 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\mathbb{T} on the Betti cohomology Hi(Γ\X,C)H^{i}(\Gamma\backslash X,\mathbb{C}). The work of Matsushima, Franke and others shows that the systems of eigenvalues of T\mathbb{T} that occur in Hi(Γ\X,C)H^{i}(\Gamma\backslash X,\mathbb{C}) come from certain automorphic representations of GG.

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 XΓ(C)X_{\Gamma}(\mathbb{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\pi is an automorphic representation whose component at \infty is a tempered representation of G(R)G(\mathbb{R}), then π\pi can only contribute to Hi(XΓ(C),C)H^{i}(X_{\Gamma}(\mathbb{C}),\mathbb{C}) in the middle degree i=dimCXΓi=\dim_{\mathbb{C}}X_{\Gamma}. 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 XΓX_{\Gamma} with C\mathbb{C}-coefficients is somehow degenerate outside the middle degree. Can we extend this to torsion coefficients, such as Hi(XΓ(C),F)H^{i}(X_{\Gamma}(\mathbb{C}),\mathbb{F}_{\ell})?

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 \ell analogue of Arthur’s conjectures. In the next two subsections, we explain a new tool that can be used to compute Hi(XΓ(C),F)H^{i}(X_{\Gamma}(\mathbb{C}),\mathbb{F}_{\ell}) 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 pp-adic analogue of the following complex picture, where the map on the right is the standard holomorphic embedding of the upper-half plane H2\mathbb{H}^{2} into the Riemann sphere P1(C)\mathbb{P}^{1}(\mathbb{C}):

This picture has the following moduli interpretation. First, XΓX_{\Gamma} is a moduli space of elliptic curves equipped with some additional structures (determined by Γ\Gamma). The upper-half plane H2\mathbb{H}^{2} is the universal cover of XΓ(C)=Γ\H2X_{\Gamma}(\mathbb{C})=\Gamma\backslash\mathbb{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:

C2=H1(E(C),C)H0(E,ΩE1)H1(E,OE)\mathbb{C}^{2}=H^{1}(E(\mathbb{C}),\mathbb{C})\simeq H^{0}(E,\Omega^{1}_{E})\oplus H^{1}(E,\mathcal{O}_{E})

with H1(E,OE)=H0(E,ΩE1)H^{1}(E,\mathcal{O}_{E})=\overline{H^{0}(E,\Omega^{1}_{E})}. The morphism πdR\pi_{\mathrm{dR}} sends the Hodge decomposition to the associated Hodge filtration

H0(E,ΩE1)H1(E(C),C)=C2.H^{0}(E,\Omega^{1}_{E})\subset H^{1}\bigl(E(\mathbb{C}),\mathbb{C}\bigr)=\mathbb{C}^{2}.

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 pp-adic Hodge theory by Tate and Faltings. Let pp be a prime and let CC be the pp-adic completion of an algebraic closure of Qp\mathbb{Q}_{p}, which will play a role analogous to that of C\mathbb{C} in what follows. If E/CE/C is an elliptic curve, its étale cohomology admits a Hodge–Tate filtration:

0H1(E,OE)Heˊt1(E,Zp)ZpCH0(E,ΩE1)(1)0.0\to H^{1}(E,\mathcal{O}_{E})\to H^{1}_{\mathrm{\acute{e}t}}(E,\mathbb{Z}_{p})\otimes_{\mathbb{\mathbb{Z}}_{p}}C\to H^{0}(E,\Omega^{1}_{E})(-1)\to 0.

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 pp-adic Hodge theory and more details on the Hodge–Tate filtration. Instead of viewing the curve XΓX_{\Gamma} as a Riemann surface, we view it as an adic space XΓ\mathcal{X}_{\Gamma}, a kind of pp-adic analytic space introduced by Huber. Then there exists a diagram

where XΓ(p)\mathcal{X}_{\Gamma}(p^{\infty}), which is roughly the inverse limit of modular curves XΓ(pn)\mathcal{X}_{\Gamma(p^{n})} with increasing level at pp, is a perfectoid space. Over a point of XΓ(p)\mathcal{X}_{\Gamma(p^{\infty})} corresponding to an elliptic curve E/CE/C, we have a trivialisation of Heˊt1(E,Zp)Zp2H^{1}_{\mathrm{\acute{e}t}}(E,\mathbb{Z}_{p})\simeq\mathbb{Z}_{p}^{2}. This point gets sent under πHT\pi_{\mathrm{HT}} to the line

H1(E,OE)Heˊt1(E,Zp)ZpCC2.H^{1}(E,\mathcal{O}_{E})\subset H^{1}_{\mathrm{\acute{e}t}}(E,\mathbb{Z}_{p})\otimes_{\mathbb{Z}_{p}}C\simeq C^{2}.

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 XΓX_{\Gamma} be a Shimura variety of Hodge type associated to a connected reductive group GG. Let μ\mu denote the conjugacy class of Hodge cocharacters and let FG,μ:=G/Pμ\mathscr{F}\ell_{G,\mu}:=G/P_{\mu} denote the corresponding flag variety, considered as an adic space over a pp-adic completion of the reflex field.

  1. There exists a unique perfectoid space XΓ(p)\mathcal{X}_{\Gamma(p^{\infty})} which can be identified with the inverse limit of the adic spaces (XΓ(pn))n(\mathcal{X}_{\Gamma(p^{n})})_{n}.

  2. There exists a Hodge–Tate period morphism

    πHT:XΓ(p)FG,μ,\pi_{\mathrm{HT}}:\mathcal{X}_{\Gamma(p^{\infty})}\to\mathscr{F}\ell_{G,\mu},

    which is G(Qp)G(\mathbb{Q}_{p})-equivariant.

  3. There exists a Newton stratification

    FG,μ=bB(G,μ)FG,μb\mathscr{F}\ell_{G,\mu}=\bigsqcup_{b\in B(G,\mu)}\mathscr{F}\ell^{b}_{G,\mu}

    into locally closed strata.

  4. If XΓ\mathcal{X}_{\Gamma} is compact and of PEL type, and xˉ\bar{x} is a geometric point of the Newton stratum FG,μb\mathscr{F}\ell_{G,\mu}^{b}, we identify the fiber πHT1(xˉ)\pi_{\mathrm{HT}}^{-1}(\bar{x}) with a “perfectoid” version of an Igusa variety Igb\mathrm{Ig}^{b}.

Remark 13. The first two parts of Theorem 12 are due to Scholze5Up to the precise identification of the target of the Hodge–Tate period morphism as the flag variety FG,μ\mathscr{F}\ell_{G,\mu} 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 GLn\mathrm{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 πHT\pi_{\mathrm{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)\mathrm{U}(n,n)-Shimura varieties, which are non-compact. We compute the fibers of πHT\pi_{\mathrm{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(XΓ,F)mH^{*}(\mathcal{X}_{\Gamma},\mathbb{F}_{\ell})_{\mathfrak{m}}, where the maximal ideal mT\mathfrak{m}\subset\mathbb{T} is equivalent to a mod \ell system of Hecke eigenvalues. Using the Hodge–Tate period morphism at an auxiliary prime pp\not=\ell6Here, we assume that the Hecke operators in T\mathbb{T} are all supported at primes different from pp., we obtain an action of T\mathbb{T} on the complex of sheaves RπHTFR\pi_{\mathrm{HT}*}\mathbb{F}_{\ell} living over FG,μ\mathscr{F}\ell_{G,\mu}, and we are reduced to understanding the localisation (RπHTF)m(R\pi_{\mathrm{HT}*}\mathbb{F}_{\ell})_{\mathfrak{m}}. By the properties of πHT\pi_{\mathrm{HT}}, this behaves similarly to a perverse sheaf, which is the key to controlling the degrees in which (RπHTF)m(R\pi_{\mathrm{HT}*}\mathbb{F}_{\ell})_{\mathfrak{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+EF=F^{+}\cdot E be a CM field, with maximal totally real field F+QF^{+}\not=\mathbb{Q} and EE imaginary quadratic. Let GG be a unitary group preserving a skew-Hermitian form on FmF^{m}. Assume that GG is quasi-split at all finite places. Let mT\mathfrak{m}\subset\mathbb{T} be a system of Hecke eigenvalues that occurs in Hi(XΓ,F)H^{i}(X_{\Gamma},\mathbb{F}_{\ell}). Assume m\mathfrak{m} is generic at an auxiliary prime pp\not=\ell7See [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 \ell analogue of temperedness.. This condition guarantees that all lifts of m\mathfrak{m} to characteristic 00 are as simple as possible at pp, from a representation-theoretic point of view: they are generic principal series representations of G(Qp)G(\mathbb{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 XΓ\mathcal{X}_{\Gamma} is compact and m\mathfrak{m} is generic, then Hi(XΓ(C),F)mH^{i}(X_{\Gamma}(\mathbb{C}),\mathbb{F}_{\ell})_{\mathfrak{m}} is concentrated in the middle degree i=dimCXΓi=\dim_{\mathbb{C}}X_{\Gamma}.

In the non-compact case, genericity, which is a local condition at an auxiliary prime pp\not=\ell, 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 ρˉm\bar{\rho}_{\mathfrak{m}} associated to the system of eigenvalues m\mathfrak{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 ρˉm\bar{\rho}_{\mathfrak{m}} is an instance of the global Langlands correspondence in the torsion setting. We want to assume that ρˉm\bar{\rho}_{\mathfrak{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 XΓX_{\Gamma} is a U(n,n)\mathrm{U}(n,n)-Shimura variety (so mm is even and GG is quasi-split at the infinite places as well), m\mathfrak{m} is generic, and ρˉm\bar{\rho}_{\mathfrak{m}} has at most two absolutely irreducible factors, then:

  1. Hci(XΓ(C),F)mH^{i}_{c}(X_{\Gamma}(\mathbb{C}),\mathbb{F}_{\ell})_{\mathfrak{m}} is concentrated in degrees idimCXΓi\leq\dim_{\mathbb{C}}X_{\Gamma}, and

  2. Hi(XΓ(C),F)mH^{i}(X_{\Gamma}(\mathbb{C}),\mathbb{F}_{\ell})_{\mathfrak{m}} is concentrated in degrees idimCXΓi\geq\dim_{\mathbb{C}}X_{\Gamma}.

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. Jussieu 18, 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 FG,μbFG,μ\mathscr{F}\ell_{G,\mu}^{b}\subset\mathscr{F}\ell_{G,\mu} in the support of (RπHT,F)m(R\pi_{\mathrm{HT},*}\mathbb{F}_{\ell})_{\mathfrak{m}}. Since the complex (RπHT,F)m(R\pi_{\mathrm{HT},*}\mathbb{F}_{\ell})_{\mathfrak{m}} behaves like a perverse sheaf, its restriction to FG,μb\mathscr{F}\ell_{G,\mu}^{b} is concentrated in one degree. Therefore, (RπHT,Q)m(R\pi_{\mathrm{HT},*}\mathbb{Q}_{\ell})_{\mathfrak{m}} is also concentrated in one degree over FG,μb\mathscr{F}\ell_{G,\mu}^{b}. On the other hand, we can compute the alternating sum of cohomology groups of Igb\mathrm{Ig}^{b} with Q\mathbb{Q}_{\ell}-coefficients, using the trace formula and work of Shin [38 S. W. Shin, A stable trace formula for Igusa varieties. J. Inst. Math. Jussieu 9, 847–895 (2010) ]. In the end, the genericity condition is contradicted unless bb corresponds to the zero-dimensional ordinary stratum. The upshot is that (RπHT,F)m(R\pi_{\mathrm{HT},*}\mathbb{F}_{\ell})_{\mathfrak{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 pp. The resulting structure is called completed cohomology and was introduced by Emerton as a general framework for studying congruences modulo pkp^{k} between automorphic forms. Motivated by heuristics coming from the pp-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 pp. The only assumption is that the group GG giving rise to the Shimura variety is split over Qp\mathbb{Q}_{p}. This result is stronger than what Calegari–Emerton conjectured, and it also points towards analogues of Theorems 16 and 17 for =p\ell=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 FF is a number field, the locally symmetric spaces for GLn/F\mathrm{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 FF, conjectural on a precise understanding of the cohomology of locally symmetric spaces for GLn/F\mathrm{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 pkp^{k} versions of automorphic forms and treated on equal footing with their characteristic 00 counterparts. Another part of their insight was to reinterpret the failure of the Taylor–Wiles numerical criterion in terms of certain non-negative integers q0q_{0}, l0l_{0} seen on the automorphic side.

The Calegari–Geraghty method gives an automorphy lifting result for GLn/F\mathrm{GL}_{n}/F as long as the following prerequisites are in place:

  1. The construction of Galois representations associated to classes in the cohomology with Zp\mathbb{Z}_{p} coefficients of the locally symmetric spaces for GLn/F\mathrm{GL}_{n}/F.

  2. Local-global compatibility for these Galois representations at all primes of FF, including at primes above pp.

  3. A vanishing conjecture for the cohomology with Zp\mathbb{Z}_{p} coefficients outside the range of degrees [q0,q0+l0][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 q0q_{0} is the middle degree of cohomology and l0=0l_{0}=0. For 33-dimensional Bianchi manifolds, the third problem says that the non-degenerate part of cohomology is concentrated in degrees 11 and 22; 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 FF 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 00 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 GLn/F\mathrm{GL}_{n}/F in the boundary of the Borel–Serre compactification of U(n,n)\mathrm{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)\mathrm{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)\mathrm{U}(n,n)-Shimura variety. This increases the level at primes of FF dividing pp, 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 FF dividing pp. 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)\mathrm{U}(n,n)-Shimura variety with Qp\mathbb{Q}_{p}-coefficients, while remembering the level and weight at primes of FF dividing pp.

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 Zp\mathbb{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 Qp\mathbb{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 pp. We resolve this challenge by considering reduction modulo pp from a derived perspective. Outside low-dimensional cases, such as Bianchi manifolds, or Shimura varieties, the third problem remains open for Zp\mathbb{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.

Ana Caraiani is a Royal Society University Research Fellow and Reader in Pure Mathematics at Imperial College London. a.caraiani@imperial.ac.uk

  1. 1

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

  2. 2

    These give rise to automorphic representations for the group SL2/Q\mathrm{SL}_{2}/\mathbb{Q}.

  3. 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. 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\Gamma\backslash X. We ignore this subtlety here and later on in the text.

  5. 5

    Up to the precise identification of the target of the Hodge–Tate period morphism as the flag variety FG,μ\mathscr{F}\ell_{G,\mu} 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. 6

    Here, we assume that the Hecke operators in T\mathbb{T} are all supported at primes different from pp.

  7. 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 \ell analogue of temperedness.

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