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].