GENERIC VANISHING, TANNAKIAN CATEGORIES, AND EQUIDISTRIBUTION PRELIMINARY VERSION { 20.3.2021 ARTHUR FOREY, JAVIER FRESAN,´ AND EMMANUEL KOWALSKI Abstract. We prove a generic vanishing theorem for twists of perverse sheaves on a commutative algebraic group G over a finite field. Using this tool, we construct a tannakian category with convolution on G as tensor operation. Using Deligne's Riemann Hypothesis, we show how this leads to equidistribution theorems for discrete Fourier transforms of trace functions of perverse sheaves on G, generalizing the work of Katz in the case of the multiplicative group. We give some concrete examples of applications of these results. Contents 1. Introduction 1 2. Preliminaries 9 3. Generic vanishing theorems 19 4. Tannakian categories of perverse sheaves 36 5. Equidistribution theorems 47 6. Applications 55 7. Open problems 81 References 82 1. Introduction 1.1. Statement of results. Since Deligne's proof of his equidistribution theorem for traces of Frobenius of `-adic local systems on varieties over finite fields [10], it has been known that any family of exponential sums parameterized by an algebraic variety satisfies some form of equidistribution, and that the concrete expression of this equidistribution statement depends on the determination of the geometric monodromy group of the `-adic sheaf that underlies the family of exponential sums. The best known result of this kind is probably the computation by Katz [25] of these monodromy groups in the case of Kloosterman sums in many variables over finite fields, which are defined for × some fixed non-trivial additive character of Fq and a 2 Fq as (−1)n−1 X Kl (a; q) = (x + ··· + x ): n q(n−1)=2 1 n × n (x1;:::;xn)2(Fq ) x1···xn=a A. F. and E. K. are partially supported by the DFG-SNF lead agency program grant 200020L 175755. J. F. is partially supported by the grant ANR-18-CE40-0017 of the Agence Nationale de la Recherche. 1 This computation led him in particular to the proof of the average version of the Sato-Tate law for classical Kloosterman sums, namely the equidistribution of the set (Kl (a; q)) × as q ! +1 2 a2Fq with respect to the Sato-Tate measure on [−2; 2]. Further deep investigations by Katz, especially in his monograph [26], provide a cornucopia of examples of equidistribution statements. Among other things, this framework allows for the study of exponential sums of the form X S(M; ) = tM(x; Fqn ) (x); x2Fqn where tM is the trace function of a perverse sheaf M on the additive group Ga over Fq and ranges over characters of Fqn . These sums are the discrete Fourier transform 7! S(M; ) of the function x 7! tM(x; Fqn ) on the finite group Ga(Fqn ), and the key point is that they are themselves the trace functions of another perverse sheaf on the dual group parameterizing additive characters. In a more recent conceptual breakthrough, Katz [31] succeeded in proving equidistribution results for families of exponential sums parameterized by multiplicative characters, despite the fact that the set of multiplicative characters of a finite field Fq does not naturally arise as the set of Fq-points of an algebraic variety. In analogy with the above, such sums are of the form X S(M; χ) = tM(x; Fqn )χ(x); × x2Fqn except that M is now a perverse sheaf on the multiplicative group Gm over Fq and χ ranges × over characters of Fqn . Katz's beautiful insight was to replace points of algebraic varieties by fiber functors of tannakian categories as parameter spaces, and produce the groups governing equidistribution by means of the tannakian formalism (see [15] for an accessible survey). Further work of Katz generalized this to elliptic curves [33] and certain abelian varieties (unpublished). The goal of this paper is to extend these ideas to exponential sums parameterized by the characters of the points of any connected commutative algebraic group over a finite field. ¯ Let k be a finite field and k an algebraic closure of k. For each n > 1, we denote by kn the extension of k of degree n inside k¯. Let ` be a prime number different from the characteristic of k and Q` an algebraic closure of the field of `-adic numbers. Let G be a connected commutative algebraic group over k. We denote by G(b kn) the group of Q`-valued characters of G(kn) and, for each χ 2 G(b kn), by Lχ the `-adic lisse character sheaf of rank one associated to χ by means of the Lang torsor construction. By perverse sheaves, we always understand Q`-perverse sheaves. In rough outline, we establish the following results: { We prove generic vanishing theorems for the cohomology of twists of perverse sheaves on G by the sheaves Lχ associated to characters χ 2 G(kn). { Using these vanishing theorems, we construct a tannakian category of perverse sheaves on G in which the tensor product is given by the convolution coming from the group law. { We prove that the tannakian group of a semisimple object M of this category that is pure of weight zero controls the distribution properties of the analogue of the sums above, namely X S(M; χ) = tM(x; kn)χ(x); x2G(kn) where χ ranges over the set G(b kn). Under some assumptions on G, we prove the stronger result that the unitary conjugacy classes of which these sums are traces become equidis- tributed in a maximal compact subgroup of the tannakian group as n ! +1, as is custom- ary since Deligne's work. 2 The following theorems are special cases of our main results, which we formulate in simplified form in order to make it possible to present a self-contained statement at this stage. Theorem 1.1.1. Let M be a perverse sheaf on a connected commutative algebraic group G of dimension d over a finite field k. (1) (Generic vanishing) The sets i i X (kn) = χ 2 G(b kn) j Hc(Gk¯; M ⊗ Lχ) = H (Gk¯; M ⊗ Lχ) = 0 for all i 6= 0 0 0 and Hc (Gk¯; M ⊗ Lχ) is isomorphic to H (Gk¯; M ⊗ Lχ) are generic, in the sense that the estimate d−1 G(b kn) X (kn) jknj holds for n > 1, with an implied constant that only depends on M. (2) (Stratified vanishing) For −d 6 i 6 d and n > 1, the estimate i i d−i χ 2 G(b kn) j Hc(Gk¯; M ⊗ Lχ) 6= 0 or H (Gk¯; M ⊗ Lχ) 6= 0 jknj holds, with an implied constant that only depends on M. The most general vanishing statements that we prove appear as Theorems 3.1.1 and 3.1.3. Ap- plications to “stratification" estimates for exponential sums are then given in Section 6.1. Remark 1.1.2. With variations in the definition of generic set of characters, such statements were proved by Katz{Laumon [34] for powers of the additive group, Saibi [46] for unipotent groups, Gabber{Loeser [18] for tori, Weissauer [53] for abelian varieties and Kr¨amer[37] for semi-abelian varieties (see Remark 3.1.2). The very rough idea to obtain Theorem 1.1.1 is to prove for the various types of commutative groups a relative version of the vanishing theorem with a good control of the implicit constant. Hence, we also significantly improve the aforementioned results. For example, in the case of tori, Gabber{Loeser [18] prove the vanishing as stated above only under the assumption that resolution of singularities holds up to the dimension of the torus. We remove this assumption using alterations instead. For abelian varieties, we extend Weissauer's work [53] by proving a relative version of the theorem, which involves the use of Orgogozo's work [44] on constructibility and moderation. Using the vanishing theorems, and ideas going back to Gabber and Loeser, we can construct tannakian categories with the convolution on G as tensor operation. Using these, and Deligne's Riemann Hypothesis over finite fields, we obtain the following equidistribution theorem for the Fourier transforms of exponential sums on G: Theorem 1.1.3 (Equidistribution on average). Let G be a connected commutative algebraic group over k. Let ι: Q` ! C be an isomorphism. Let M be a geometrically irreducible `-adic perverse sheaf on G that is ι-pure of weight zero, with trace function tM(·; kn): G(kn) ! C. There exists an integer r > 1 and a compact subgroup K ⊂ Ur(C) of the unitary group such that the sums X S(M; χ) = tM(x; kn)χ(x) x2G(kn) for complex-valued characters χ of G(kn) become equidistributed on average in C with respect to the image by the trace of the probability Haar measure µ on K. That is, for any bounded continuous function f : C ! C, the following equality holds: 1 X 1 X Z (1.1) lim f(S(M; χ)) = f(Tr(x))dµ(x); N!+1 N jG(kn)j K 16n6N χ 3 where χ runs over all characters of G(kn). The general version of this theorem appears as Theorem 5.3.3. Under an additional assump- tion (which holds for all tori and all abelian varieties, at least), we could also deduce it from Theorem 5.4.1, which is a more precise equidistribution result for conjugacy classes of Frobenius in K. Remark 1.1.4. (1) In the classical setting of Ga and the Fourier transform, the group K is a maximal compact subgroup of the arithmetic monodromy group of the `-adic Fourier transform of M.