P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (87–96)

ON THE WORK OF AKSHAY VENKATESH

P S

Abstract Number theory while one of the oldest subjects in mathematics, continues to be one of the most active areas of research. One reason for this is that it borrows from and contributes to many other fields, sometimes quite unexpectedly. Uncovering deeper number theoretic truths often involves advancing techniques across these disciplines resulting in new avenues of research and theories. Venkatesh’s work fits squarely into this mold. His resolution and advancement of a number of long standing problems start with fundamental new insights connecting theories and have led to active research areas. He is both a problem solver and a theory builder of the highest caliber. Due to space limitation and the nature of this report, I have chosen to describe three examples of Venkatesh’s works and end by simply listing a number of his other striking results. Also, in this brief description of some of the results I have omitted some technicalities and the reader should go to the references for precise statements.

1 Subconvexity for GL(2), equidistribution and Hilbert’s 11-th Problem

The Riemann Hypothesis for the zeta function

1 s (1) (s) = X n ; n=1 implies an optimally sharp upper bound on the critical line. 1 (2) ( + it) (1 + t ) ; for any  > 0:1 2  j j

1 The bound of (1 + t ) 4 follows from a simple convexity argument and Weyl’s j j method for estimating exponential sums led to the first subconvex bound for (s) namely 1 1 with the exponent 6 replacing 4 Titchmarsh [1986]. While for (s) itself, the applica- tions and interest in this technical estimation problem might be limited, the question of estimating general automorphic L-functions L(s; ) on their critical lines has many far reaching applications (we explicate one below). Here  is an automorphic cusp form for GLn over a number field F and L(s; ) is its standard L-function [J ]. They

1 A B means A CB, with a constant depending only on   j j Ä 87 88 PETER SARNAK are the complete generalizations of Riemann’s, Dirichlet’s, and Hecke’s L-functions all of which correspond to GL1. Associated to such a  is its analytic conductor C (; t) (Iwaniec and Sarnak [2000]) which measures its complexity. The convex bound is

Â1 Ã 1 + (3) L + it;  C (; t) 4 ;  > 0: 2 

1 Any improvement in the exponent of 4 is called a subconvex bound and very often it is decisive in applications Iwaniec and Sarnak [ibid.]. For GL1 L-functions subconvex bounds were established in the 1970’s while for various L-functions of modular forms on the upper half plane the problem was resolved in a series of papers Duke, Friedlan- der, and Iwaniec [2002] by Duke–Friedlander–Iwaniec in the 1980’s and 1990’s. In Venkatesh [2010] Venkatesh made a major breakthrough in both techniques and gen- erality. These include a direct use of ergodic theoretic mixing rates for equidistribu- 2 tion of orbits in homogeneous spaces G(F ) G(AF ), and in particular passing these n to smaller orbits “sparse equidistribution” and systematically exploiting the device of varying test vectors of local representations in various period formula for special values of L-functions. This led him to many new cases of subconvexity culminating in the paper Michel and Venkatesh [2010] where he and Michel establish subconvexity in full generality for GL2: There is a universal ı > 0 such that for F fixed and  an automorphic cuspidal representation of GL2(AF )

Â1 Ã 1 ı (4) L ;  C () 4 : 2 F

Venkatesh followed this work with a series of results concerning equidistribution in arithmetic. Hilbert’s 11-th problem asks about solvability of quadratic equations in the integers or rationals of a number field F ;

(5) X t AX = B

Here A is an n n symmetric integral (or F -rational) matrix, B is m m integral   (rational) and (5) is to be solved for an n m integral (rational) X. The rational question  was resolved by Hasse; the Hasse–Minkowski Theorem asserting that (5) has a solution in F iff it can be solved at each localization Fv of F . The integral case is much more difficult and Siegel made substantial progress on it with his mass formula Siegel [1935]. The most difficult case is when A is definite at each archimedean place of F (the class number of the spin genus of A is then typically large) in which case there are only finitely many solutions to (5) and one seeks a local to global principle; that is if (5) is solvable integrally locally at all localizations, is it solvable integrally globally? The most studied case is the m = 1 scalar case. One has to allow here for a finite number of exceptions, that is an integral local to global principle for Norm(B) large (and also some other mild assumptions on the factors of B when n 4), and this “stable integral local to global Ä principle” has been proven for n > 3 (see Duke and Schulze-Pillot [1990] and Blomer

2 Here G is a linear algebraic group defined over a number field and AF its ring of adèles. ON THE WORK OF AKSHAY VENKATESH 89

and Harcos [2010]). The key ingredient in the proof of the threshold n = 3 case is GL2 subconvexity for certain ’s. Prior to 2008 the best stable integral local global principle for m > 2 was estab- lished in Hsia, Kitaoka, and Kneser [1978] and requires n > 2m + 3. In collaboration with Ellenberg, Venkatesh (Ellenberg and Venkatesh [2008]) made dramatic progress, establishing the principle for n > m+5. 3 They treat the case of F = Q, and in Schulze- Pillot [2009] Schulze–Pillot extends their arguments to general F and also relaxes the side conditions on the factors of Norm(disc(B)). Their breakthrough is the introduction of an ergodic theoretic action involving orthogonal groups associated to (5), specifically 1 SOn m(Qp) acting on SO(Z[ p ]) SOn(Qp). This technique has its roots in Linnik’s n work on the m = 1, n = 3 case, but unlike that case where SO2(Qp) is a torus, the SOn m(Qp)’s are generated by unipotents for many of these auxiliary p’s and this al- lows them to invoke decisively the powerful measure classification theorems of Ratner [1995], for such actions. The explicit formulae such as those of Waldspurger [1985] which relate periods of automorphic forms to special values of L-functions, are used both in the applications as well as in the proofs of subconvexity. Understanding these formulae in general is a fundamental problem and the recent monograph of Sakellaridis and Venkatesh [2017] lays out a far reaching theory for the periods associated with spherical varieties.

2 Class Groups and Cohen–Lenstra Heuristics

The class group of a number field is one of its fundamental invariants and one that is no- toriously difficult to study. Cohen and Lenstra Jr. [1984] put forth a probabilistic model for the statistics of invariants of these groups as one varies over quadratic extensions Q(pD) of Q. Their probability laws on the set of finite abelian groups are based on the heuristic that one should put in weights of Aut(A) 1 for such a group A. The numer- j j ical evidence for their conjectures is very convincing. While there are many works and extensions of these conjectures, very little has been proven. They avoid the 2-part of the class group since the number of elements of order 2 is determined by Gauss’ genus theory and the rest of the 2-part of the class group can be addressed partially using class field theory Rédei [1939]. Smith [2017] recently announced a proof that the rest of the 2-part of the class groups does in fact follow the Cohen–Lenstra law. The function field analogue is to replace Q by Fq(t), the rational functions in t with coefficients in the finite field Fq, and to consider the class groups of its quadratic exten- sions; that is Fq(t)(pD(t)) where D(t) is square-free in Fq[t], or more geometrically 2 y = D(t) is a hyperelliptic curve over Fq. If one allows q to get large first and degD after, then well developed tools from arithmetic geometry of curves and varieties over finite fields allow one to tackle many of these problems and their variants Achter [2006]. However, if q is fixed then the problem appeared to be as difficult as the case over Q. In Ellenberg, Venkatesh, and Westerland [2016] they all but settle this problem in the func- tion field. They show that for ` > 2 a prime and A a finite abelian `-group, the upper

3The least n for which this might hold is m + 2 and this brings us close to it. 90 PETER SARNAK

(resp lower) densities of imaginary quadratic extensions of Fq(t) with `-part isomor- i phic to A as degD , converge to Q (1 ` )/ Aut(A) as q with q 1(`). ! 1 i>1 j j ! 1 6Á This density is exactly that predicted by the Cohen–Lenstra law. Note that here one first establishes bounds for the densities as deg D goes to infinity and these approach the prediction as q is increased at the end (conjecturally one should allow q to be fixed). Their methods also give the densities when q 1(`) and these can be different which Á can be explained by there being `-th roots of 1 in Fq in this case Garton [2015]. The striking and novel starting point for this work is that even though q is fixed, as deg(D) goes to infinity one can exploit a stable range of the cohomology of certain Hurwitz moduli spaces of curves attached to A, specifically applying the Lefshetz fixed point theorem to execute an effective count! For this to work one needs a rather large stability range for the Betti numbers for these moduli spaces of growing dimension. While such stability has been established in recent years for the moduli space Mg of curves of genus g (see Madsen and Weiss [2007] which perhaps was one inspiration for the present work?) the requisite stability for these Hurwitz spaces poses substantial problems since these spaces aren’t even connected. A large portion of the paper is devoted to the proof of the stability range using tools from topology and combinatoral group theory. The Cohen–Lenstra law emerges from this analysis in a canonical and convincing fashion. This breakthrough has inspired a number of further works on both topological and arithmetic aspects, including non-abelian versions Boston and Wood [2017].

3 Homology of Arithmetic Groups of positive defect

Let G be a noncompact real semi-simple Lie group, K a maximal compact subgroup and S = G/K the corresponding Riemannian symmetric space. For Γ an arithmetic subgroup of G (in fact a congruence subgroup of a Q-group, here and in what follows we are a little loose with terminology) the cohomology groups H (Γ;R) with R various coefficient rings, have been studied widely. These come with an action of Hecke cor- respondences and setting aside various technicalities H (Γ; C) or H (Γ S; C) can be   n computed using automorphic forms using g, K cohomology (“Matsushima’s formula” Borel and Wallach [2000]). In particular one can use the trace formula to show that if ı(G) := rank(G) rank(K), then as Γ varies over congruence subgroups (de George and Wallach [1978])4

dim H j (Γ S; C) (6) lim n 0; Vol(Γ S) Vol(Γ S) ¤ n !1 n dim S iff ı(G) = 0 and j = : 2 The ‘defect’ ı(G) being zero is precisely Harish-Chandra’s condition for G to carry discrete series representations. In this case the maximal growth rate of Betti numbers is in the middle dimension. The much studied algebro-geometric arithmetic Shimura

4various recent works allow for this general formulation see Finis and Lapid [2018]. ON THE WORK OF AKSHAY VENKATESH 91 varieties all fall into this zero defect setting. Venkatesh’s work is concerned with the positive defect cases for which much less is known about H (Γ; Z) and in particular its torsion. Ray and Singer defined analytic torsion as regularized determinants of Laplacians on differential forms and conjectured that it computes the topological torsions in the same way that the Hodge theorem computes Betti numbers. Their conjecture was proved in- dependently by Cheeger [1977] and Müller [1978]. In work with Bergeron (Bergeron and Venkatesh [2013]) and the book with Calegari (Calegari and Venkatesh [2012]), Venkatesh introduced analytic torsion as an effective tool to study torsion in this arith- metic setting. In the first work the torsion analogue of (6), namely the maximal growth rate of torsion is conjectured and for certain cases of strongly acyclic arithmetic Γ- modules M (i.e. ones for which the eigenvalues of the Laplacians are bounded away from zero uniformly) proved:5

log Hj (Γ S;M )tor (7) lim j n j 0 Vol(Γ S) Vol(Γ S) ¤ n !1 n iff ı(G) = 1 and j = (dim S 1)/2: 3 3 For example for arithmetic hyperbolic 3-manifolds Γ H ; H = SL2(C)/SU (2), n ı = 1 and H1(Γ; Z) should be primarily torsion growing exponentially (with a univer- sal exponent) in the volume. The second work goes deeper into the arithmetic. The well known Jacquet–Langlands correspondence Jacquet and Langlands [1970] gives an explicit relation between the automorphic spectrum of a compact arithmetic 3-manifold Y = Γ H3 defined over an imaginary quadratic field F and non-compact arithmetic n 3 3-manifolds Y = Γ H which are congruence quotients corresponding to GL2(F ). 0 0n One may interpret the analysis in the Calegari–Venkatesh discovery as using the trace formula to compute the analytic torsion of Y and of Y 0 in terms of orbital integrals in- volving conjugacy classes of Γ and Γ0, and comparing these to show that aspects of the Jacquet–Langlands correspondence extend to torsion homology! Matsushima’s transcendental formula shows that when the defect ı is positive, the tempered (these being the representations that occur in the decomposition of L2(G)) dim S ı dim S+ı cohomology lives in degrees j0 = 2 , j0 + 1; : : : ; j0 + ı = 2 , and moreover the Betti numbers are a fixed multiple of those of a ı-dimensional torus. Understand- ing this relation algebraically directly in terms of cohomology classes and operations that raise degrees is central to developing analogues of the powerful tools that have been developed in the Shimura variety (and specifically ı = 0) setting. Venkatesh j raised and answered this question brilliantly for H (Γ; Qp) in his derived Hecke alge- bra paper Venkatesh [2016]. One would like to cup classes in H j (Γ; Q) with a class in H 1(Γ; Q) to raise the degree, however the latter group is typically 0. On the other hand, one can produce elementary torsion classes in H 1(Γ; Z/pnZ) by passing to suit- able congruence subgroups of Γ. Using these via cup product he defines derived Hecke correspondences which act on H (Γ; Qp) and raise degrees. The construction is remi- niscent of the Taylor–Wiles patching method and aspects of that technique are used to show conditionally on expected properties of Galois representations, that this derived 5formulated for homology rather than cohomology 92 PETER SARNAK

Hecke algebra acts non-trivially and that the full range ı of degrees is generated from the lowest degree in this range. This inspiring paper was followed by a flurry of activity, specifically Galatius and Venkatesh [2018] introduce derived deformation rings associated to Galois representa- tions in this positive defect setting. Prasanna and Venkatesh [2016] developed a de- rived action in the context of H (Γ; R), using differential forms, and study the ratio- nality properties of this action. Their conjectures have concrete implications that can be checked and proven in special cases. Not surprisingly the latter involves explicit period formulae in terms of special values of L-functions and computations with ana- lytic torsion. Suffice it to say that these ideas, techniques and conjectures of Venkatesh are already a central part of a rapidly developing arithmetic cohomology theory in the positive defect setting.

The following is a list of some further striking works by Venkatesh: 1. Established a Weyl law for the cuspidal spectrum of non co-compact congruence locally symmetric spaces, joint with Lindenstrauss (Lindenstrauss and Venkatesh [2007]).

2. Equidistribution of torus orbits in P GL3(Z) P GL3(R) corresponding to ideal n classes of totally real cubic number fields F as disc(F ) goes to infinity, joint j j with Einsiedler, Lindenstrauss and Michel (Einsiedler, Lindenstrauss, Michel, and Venkatesh [2011]). 3. Effective equidistribution of large periodic orbits in locally symmetric spaces, joint with Einsiedler and Margulis (Einsiedler, Margulis, and Venkatesh [2009]). 4. Integral points on elliptic curves and the 3-torsion of class groups, joint with Helf- gott (Helfgott and Venkatesh [2006]). 5. Upper bounds for the number of extensions of number fields of fixed degree and discriminant below X, with Ellenberg (Ellenberg and Venkatesh [2006]). 6. First improvement in forty years on the lower bound for sphere packing in high dimensions (Venkatesh [2013]). 7. Functoriality, Smith theory and the Brauer homomorphism, with Truemann (Treumann and Venkatesh [2016]). Thanks to his innovative use of modern tools to study the theory of numbers, Ven- katesh is shaping fields ranging from the theory of L-functions, Diophantine analysis, automorphic forms, representation theory, topology and locally symmetric spaces, to homogeneous dynamics and especially the very fruitful interactions between these ar- eas.

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Received 2018-07-30.

P S D M P U and T I A S P, NJ [email protected]