# Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution

Arthur Forey, Javier Fresán, Emmanuel Kowalski

<figure><img src="image_1.png" /></figure>

## To cite this version:

Arthur Forey, Javier Fresán, Emmanuel Kowalski. Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution. Asterisque, In press. hal-05392016

HAL Id: hal-05392016

https://hal.science/hal-05392016v1

Submitted on 1 Dec 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Arithmetic Fourier transforms over finite fields

## Generic vanishing, convolution, and equidistribution

Arthur Forey

Javier Fresán

Emmanuel Kowalski

(A. Forey) UNIV. LILLE, CNRS, UMR 8524 - LABORATOIRE PAUL PAINLEVÉ, F-59000 LILLE, FRANCE

Email address: arthur.forey@univ-lille.fr

(J. Fresán) SORBONNE UNIVERSITÉ AND UNIVERSITÉ PARIS CITÉ, CNRS, IMJ-PRG, F-75005 PARIS, FRANCE

Email address: javier.fresan@imj-prg.fr

(E. Kowalski) D-MATH, ETH ZÜRICH, RÄMISTRASSE 101, CH-8092 ZÜRICH, SWITZERLAND

Email address: kowalski@math.ethz.ch
ABSTRACT. We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we construct a tannakian category with convolution as tensor operation. Using Deligne's Riemann hypothesis, we show how this leads to a general equidistribution theorem for the discrete Fourier transforms of trace functions of perverse sheaves, generalizing the work of Katz in the case of the multiplicative group. We then give some concrete examples of applications of these results and raise a number of questions.
Dedicated to Nick Katz, with the greatest admiration
<figure><img src="image_3.png" /></figure>
# Contents

Preface 1

Introduction 5

1. Statement of results 5

2. Outline 9

3. Conventions and notation 12

Part 1. Theoretical foundations 17

Chapter 1. Preliminaries 19

1.1. Specializations of perverse sheaves 19

1.2. Review of quantitative sheaf theory 19

1.3. Existence of rational points 21

1.4. Structure of commutative algebraic groups 22

1.5. Convolution 22

1.6. Character groups 23

1.7. Complexity estimates for character sheaves 25

1.8. Arithmetic Fourier transforms 27

1.9. Generic sets of characters 28

1.10. Fourier-Mellin transforms on semiabelian varieties 30

1.11. A geometric lemma 30

1.12. Geometric and arithmetic semisimplicity 31

1.13. A result from representation theory 32

Chapter 2. Generic vanishing theorems 33

2.1. Statement of the vanishing theorems 33

2.2. The case of unipotent groups 35

2.3. Perverse sheaves on tori 39

2.4. Perverse sheaves on abelian varieties 42

2.5. Proof of the general vanishing theorem 50

2.6. Proof of Theorem 2.6 53

Chapter 3. Tannakian categories of perverse sheaves 55

3.1. Introduction 55

3.2. Categories of objects defined over finite fields 56

3.3. Weakly unramified characters 56

3.4. Negligible objects 57

3.5. Tannakian categories 59

3.6. Euler-Poincaré characteristic and Grothendieck groups 62

3.7. Arithmetic fiber functors 64

3.8. The arithmetic tannakian group 66
<table>
  <tbody>
    <tr>
      <td>3.9. Frobenius conjugacy classes</td>
      <td>68</td>
    </tr>
    <tr>
      <td>3.10. Frobenius-unramified characters</td>
      <td>70</td>
    </tr>
    <tr>
      <td>3.11. Group-theoretic properties</td>
      <td>72</td>
    </tr>
    <tr>
      <td>3.12. External products</td>
      <td>74</td>
    </tr>
    <tr>
      <td>3.13. The rank 1 tannakian group</td>
      <td>75</td>
    </tr>
    <tr>
      <td>Chapter 4. Equidistribution theorems</td>
      <td>77</td>
    </tr>
    <tr>
      <td>4.1. Equidistribution on average</td>
      <td>77</td>
    </tr>
    <tr>
      <td>4.2. The basic estimate</td>
      <td>78</td>
    </tr>
    <tr>
      <td>4.3. Equidistribution for characteristic polynomials</td>
      <td>79</td>
    </tr>
    <tr>
      <td>4.4. Equidistribution for arithmetic Fourier transforms</td>
      <td>83</td>
    </tr>
    <tr>
      <td>4.5. Equidistribution for conjugacy classes</td>
      <td>85</td>
    </tr>
    <tr>
      <td>4.6. Equidistribution without average</td>
      <td>87</td>
    </tr>
    <tr>
      <td>4.7. Horizontal equidistribution</td>
      <td>88</td>
    </tr>
    <tr>
      <td>4.8. Objects of rank 1</td>
      <td>89</td>
    </tr>
    <tr>
      <td>Part 2. Applications</td>
      <td>93</td>
    </tr>
    <tr>
      <td>Description of applications</td>
      <td>95</td>
    </tr>
    <tr>
      <td>Chapter 5. Über eine neue Art von L-Reihen</td>
      <td>97</td>
    </tr>
    <tr>
      <td>5.1. L-functions</td>
      <td>97</td>
    </tr>
    <tr>
      <td>5.2. Objects with finite arithmetic tannakian groups on abelian varieties</td>
      <td>100</td>
    </tr>
    <tr>
      <td>5.3. Perverse sheaves with finitely many ramified characters</td>
      <td>103</td>
    </tr>
    <tr>
      <td>5.4. The general case</td>
      <td>105</td>
    </tr>
    <tr>
      <td>Chapter 6. Stratification and generic Fourier invertibility</td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.1. Stratification for exponential sums</td>
      <td>109</td>
    </tr>
    <tr>
      <td>6.2. Generic Fourier invertibility</td>
      <td>115</td>
    </tr>
    <tr>
      <td>Chapter 7. Independence of l</td>
      <td>117</td>
    </tr>
    <tr>
      <td>Chapter 8. Diophantine group theory</td>
      <td>123</td>
    </tr>
    <tr>
      <td>8.1. The diophantine irreducibility criterion</td>
      <td>123</td>
    </tr>
    <tr>
      <td>8.2. The Frobenius-Schur indicator</td>
      <td>123</td>
    </tr>
    <tr>
      <td>8.3. Larsen's Alternative</td>
      <td>124</td>
    </tr>
    <tr>
      <td>8.4. Sidon morphisms</td>
      <td>127</td>
    </tr>
    <tr>
      <td>8.5. Gabber's torus trick</td>
      <td>133</td>
    </tr>
    <tr>
      <td>8.6. Recognition criteria for E<sub>6</sub></td>
      <td>134</td>
    </tr>
    <tr>
      <td>8.7. Finiteness of tannakian groups on abelian varieties</td>
      <td>136</td>
    </tr>
    <tr>
      <td>Chapter 9. The product of the additive and the multiplicative groups</td>
      <td>137</td>
    </tr>
    <tr>
      <td>9.1. Introduction</td>
      <td>137</td>
    </tr>
    <tr>
      <td>9.2. Tannakian group for diagonal objects</td>
      <td>141</td>
    </tr>
    <tr>
      <td>9.3. Diagonal objects of dimension 2</td>
      <td>147</td>
    </tr>
    <tr>
      <td>9.4. Negligible objects and objects of dimension one</td>
      <td>148</td>
    </tr>
    <tr>
      <td>Chapter 10. Variance of arithmetic functions in arithmetic progressions</td>
      <td>155</td>
    </tr>
    <tr>
      <td>10.1. Introduction</td>
      <td>155</td>
    </tr>
    <tr>
      <td>10.2. Equidistribution on tori associated to polynomials</td>
      <td>156</td>
    </tr>
    <tr>
      <td>10.3. Application to von Mangoldt functions</td>
      <td>160</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>Chapter 11. Equidistribution on abelian varieties</td>
      <td>167</td>
    </tr>
    <tr>
      <td>11.1. Equidistribution in the jacobian of a curve</td>
      <td>167</td>
    </tr>
    <tr>
      <td>11.2. The intermediate jacobian of a cubic threefold</td>
      <td>171</td>
    </tr>
    <tr>
      <td>Chapter 12. "Much remains to be done"</td>
      <td>177</td>
    </tr>
    <tr>
      <td>12.1. Problems</td>
      <td>177</td>
    </tr>
    <tr>
      <td>12.2. Questions</td>
      <td>178</td>
    </tr>
    <tr>
      <td>Appendix A. Review of perverse sheaves</td>
      <td>181</td>
    </tr>
    <tr>
      <td>A.1. Complexes of l-adic sheaves</td>
      <td>181</td>
    </tr>
    <tr>
      <td>A.2. Perverse sheaves</td>
      <td>182</td>
    </tr>
    <tr>
      <td>A.3. Weights</td>
      <td>184</td>
    </tr>
    <tr>
      <td>A.4. Trace functions</td>
      <td>185</td>
    </tr>
    <tr>
      <td>Appendix B. The arithmetic Mellin transform over finite fields</td>
      <td>189</td>
    </tr>
    <tr>
      <td>B.1. The category P</td>
      <td>189</td>
    </tr>
    <tr>
      <td>B.2. Deligne's fiber functor and Frobenius conjugacy classes</td>
      <td>189</td>
    </tr>
    <tr>
      <td>B.3. Finite tannakian groups</td>
      <td>190</td>
    </tr>
    <tr>
      <td>B.4. Hypergeometric complexes and sheaves</td>
      <td>190</td>
    </tr>
    <tr>
      <td>Appendix C. The product formula for epsilon factors</td>
      <td>193</td>
    </tr>
    <tr>
      <td>C.1. The product formula</td>
      <td>193</td>
    </tr>
    <tr>
      <td>C.2. Local epsilon factors</td>
      <td>194</td>
    </tr>
    <tr>
      <td>C.3. The Euler-Poincaré characteristic formula</td>
      <td>195</td>
    </tr>
    <tr>
      <td>Appendix D. Deligne's letter to Kazdan</td>
      <td>197</td>
    </tr>
    <tr>
      <td>Appendix E. Intuition for analytic number theorists</td>
      <td>201</td>
    </tr>
    <tr>
      <td>E.1. Trace functions</td>
      <td>201</td>
    </tr>
    <tr>
      <td>E.2. Weights and purity: lisse sheaves</td>
      <td>203</td>
    </tr>
    <tr>
      <td>E.3. Weights and purity: constructible sheaves and complexes</td>
      <td>204</td>
    </tr>
    <tr>
      <td>E.4. Perverse sheaves</td>
      <td>207</td>
    </tr>
    <tr>
      <td>E.5. Tannakian categories</td>
      <td>209</td>
    </tr>
    <tr>
      <td>E.6. Frequently asked questions</td>
      <td>210</td>
    </tr>
    <tr>
      <td>Index</td>
      <td>213</td>
    </tr>
    <tr>
      <td>Glossary</td>
      <td>215</td>
    </tr>
    <tr>
      <td>Bibliography</td>
      <td>219</td>
    </tr>
  </tbody>
</table>
<figure><img src="image_4.png" /></figure>
# Preface

The Fourier transform, and the whole collection of its variants whose study is summarized under the heading of “harmonic analysis”, is one of the most important tools of mathematics. In its many forms, its applications cover the whole range not only of mathematics, but also physics, computer science, chemistry and indeed of all sciences where quantitative tools are applied.

In 1976, P. Deligne observed in a letter to D. Kazhdan (which is reproduced in Appendix D) that the formalism of algebraic geometry, and especially of $ℓ$-adic cohomology and the derived category of $ℓ$-adic sheaves, provided a new “geometric” form of the Fourier transform. Instead of the familiar integral formula

$$ \hat{f}(y) = \int_{\mathbb{R}} f(x)e^{-2i\pi xy}dx $$

associating to a function $f$ (say $f: \mathbb{R} \to \mathbb{C}$ in the Schwartz space) its Fourier transform $\hat{f}$, Deligne’s version takes as input an $ℓ$-adic constructible sheaf $M$, or a complex of those, on the one-dimensional affine space over a finite field $k$ of characteristic $p$, and outputs a Fourier transform $\hat{M}$ which is of the same kind.

We note that although the most general and convenient category of input objects $M$, which we will also call “coefficients”, is given by the formalism of derived categories of $ℓ$-adic complexes with $ℓ$ prime different from $p$, there is a simpler definition in the case considered here, where $M$ can (in almost all cases) be thought of as being a continuous finite-dimensional representation

$$ \rho: \mathrm{Gal}(k(T)^{\mathrm{sep}}/k(T)) \to \mathbf{GL}_r(\overline{\mathbb{Q}}_\ell) $$

of the absolute Galois group of the field $k(T)$ of rational functions with coefficients in $k$.

The crucial point for the interpretation of this construction as a Fourier transform is that to each object $M$ is associated classically a sequence of “trace functions”, which are functions

$$ t_M(\cdot; k_n) : k_n \to \mathbb{C} \simeq \overline{\mathbb{Q}}_\ell $$

defined on the finite extensions $k_n$ of $k$ of degree $n$, for all integers $n \ge 1$, and Deligne’s Fourier transform then satisfies

$$ t_{\hat{M}}(y; k_n) = \sum_{x \in k_n} t_M(x; k_n) e^{2i\pi \mathrm{Tr}_{k_n/\mathbb{F}_p}(xy)/p}. $$

Thus, the trace functions of $\hat{M}$ coincide with the discrete Fourier transforms of those of $M$.

Deligne’s Fourier transform shares many features with the classical euclidean Fourier transform, once properly interpreted in terms of the coefficients $M$. For instance:

- It satisfies a form of the Fourier inversion formula

$$ f(x) = \int_{\mathbb{R}} \hat{f}(y)e^{2i\pi xy}dy, $$

in the sense that applying the (similarly defined) analogue of the inverse Fourier transform to $\hat{M}$ recovers $M$.
- It satisfies analogues of the Plancherel formula, which are however less obvious: one interpretation is that if the representation $\rho$ above is irreducible, then so is the representation associated to $\hat{M}$.

- It satisfies a geometric analogue of the fundamental algebraic relation $\hat{f} * \hat{g} = \hat{f} \hat{g}$, which relates the Fourier transform and the convolution product

$$ (f * g)(x) = \int_{\mathbb{R}} f(y)g(x - y)dy $$

of functions (this property is often taken as the key feature of Fourier analysis and especially Pontryagin duality [14]). Indeed, to two coefficients $M_1$ and $M_2$, another geometric construction associates a third one $M_3$, such that the trace function of $M_3$ is given by

$$ t_{M_3}(x; k_n) = \sum_{y \in k_n} t_{M_1}(y; k_n) t_{M_2}(x - y; k_n), $$

the discrete convolution of those of $M_1$ and $M_2$.

- And there is a subtle analogue, due to Laumon, of the stationary phase principle for estimating oscillatory integrals.

There are however also special features related to the geometric nature of trace functions:

- Deligne's Fourier transform preserves a particularly important subcategory of coefficients, that of *perverse sheaves* – this extremely important fact has no obvious classical analogue.

- If a coefficient object $M$ is a perverse sheaf, and hence also its transform $\hat{M}$, then one can associate to it a natural intrinsic symmetry group, also called its *monodromy group*, which is an algebraic group over $\mathbb{Q}_\ell$ (or over $\mathbb{C}$). The definition of this group can be seen as a wide-ranging generalization of that of the Galois group of a polynomial. (In the one-dimensional case, where $\hat{M}$ can be identified, in most cases, with a Galois representation $\rho$: $\mathrm{Gal}(k(T)^{\mathrm{sep}}/k(T)) \to \mathrm{GL}_r(\mathbb{Q}_\ell)$ as above, this symmetry group is nothing but the Zariski-closure of the image of $\rho$.)

Deligne's Fourier transform has found a number of very important applications in arithmetic and algebraic geometry, as well as number theory. In the former direction, Laumon [98] used it to obtain a product formula for the epsilon factors of Artin-type L-functions on curves over finite fields. In number theory, Katz used it extensively to study in depth the distribution properties of families of exponential sums, which are obtained as discrete Fourier transforms of simple trace functions (see, for instance, [69] and [70]); the symmetry group of the Fourier transform $\hat{M}$ plays an essential role here. A prominent example of such sums are the Kloosterman sums

$$ \mathrm{Kl}_2(a; p) = \frac{1}{\sqrt{p}} \sum_{x \in \mathbb{F}_p^\times} e\left(\frac{\bar{x} + ax}{p}\right) $$

which are the values of the trace function of the Fourier transforms of a one-dimensional Galois representation, and are omnipresent in modern analytic number theory (here and below, we use the notation $e(z) = \exp(2i\pi z)$, and $\bar{x}$ is the inverse of $x$ modulo $p$). Results about these and similar sums, which often rely on properties of the $\ell$-adic Fourier transform, have by now become essential in many fundamental results of analytic number theory – some concrete examples appear in Zhang's famous work on bounded gaps between primes [128, Lemma 12], and systematic use of the Fourier transform begins in various papers of Fouvry, Kowalski and Michel (see, for instance, [41]).

Deligne's transform is the geometric analogue of the classical euclidean Fourier transform on $\mathbb{R}$ and can be generalized to $n$ variables. But, in recent years, a number of applications have led
to questions concerning similar properties of other discrete Fourier transforms, for instance those related to the multiplicative group $k_n^\times$, which are functions on the group of multiplicative characters $\chi: k_n^\times \to \overline{\mathbb{Q}}_\ell$. The study of the distribution, or average properties, of these sums is outside of the realm of applications of Deligne's Fourier transform, and these functions cannot be expressed as trace functions of complexes of $\ell$-adic sheaves on an algebraic variety over $k$.

The fundamental motivation for this book is the search for a definition of the analogue of Deligne's Fourier transform on an arbitrary commutative algebraic group over a finite field, and for the general theory and applications of this form of harmonic analysis. In particular, we believe that these arithmetic Fourier transforms can be interpreted in the context of much more general arithmetic or geometric avatars of harmonic and functional analysis.

The basic examples of commutative algebraic groups are the multiplicative groups (or tori), and abelian varieties, and these can be combined (together with additive groups) in various ways. The choice of an input object $M$ on such a group $G$ leads to its arithmetic Fourier transforms, which are the functions of the form

$$ \hat{t}_M(\chi; k_n) = \sum_{x \in G(k_n)} \chi(x) t_M(x; k_n), $$

defined for any $n \ge 1$, where the parameter $\chi$ ranges over characters of the finite group $G(k_n)$.

The simplest example beyond the additive case is that of $G(k_n) = k_n^\times$, in which case the characters are multiplicative characters of $k_n$, and $\hat{t}_M$ is called an arithmetic Mellin transform. N. Katz, in a striking breakthrough, succeeded a few years ago in finding an interpretation of these functions in his book [75]. He exploited the formalism of tannakian categories, and the fact that the convolution product extends to any commutative algebraic group: given coefficients $M_1$ and $M_2$ on $G$, there exists a geometrically-defined object $M_3$ such that, for all $n \ge 1$ and $x \in G(k_n)$, their respective trace functions satisfy

$$ t_{M_3}(x; k_n) = \sum_{y \in G(k_n)} t_{M_1}(y; k_n) t_{M_2}(xy^{-1}; k_n). $$

Although Katz's interpretation of the arithmetic Mellin transforms is not fully geometric (there is no analogue of the object $\hat{M}$ which "is" Deligne's Fourier transform for the additive group), Katz shows that it is enough to define a symmetry group for the arithmetic Mellin transform. In combination with another fundamental tool, Deligne's general form of the Riemann hypothesis over finite fields [28], this allowed Katz to prove an equidistribution theorem which controls the distributions of arithmetic Mellin transforms. A number of significant applications followed, including the work of Keating and Rudnick [81] and Hall, Keating and Roditty-Gershon [56].

One of the main theoretical achievements of this book is the extension of these ideas of Katz to any connected commutative algebraic group. This is far from routine, since certain necessary tools, such as generic cohomological vanishing, or estimates for Betti numbers, which are very elementary in the case considered by Katz, were not known previously for groups of dimension at least 2. Indeed, we rely in an essential way on the very recent quantitative sheaf theory due to Sawin [115] (which was partly motivated by this work and drafted in final form jointly with the authors).

For any suitable coefficient object on the group $G$, our construction provides the fundamental invariant of its arithmetic Fourier transform, its intrinsic symmetry group. Combined again with other tools such as Deligne's Riemann hypothesis over finite fields, this is already sufficient to prove a very general form of equidistribution theorem, which encompasses the previously known cases of Deligne and Katz (and in fact sharpens these in certain aspects). In turn, we can use this equidistribution theorem for a number of first applications, including strengthening and simplifying
the results of [56]. But there remain also many open questions and problems, both on the theoretical side and on that of applications – we will discuss briefly some of these at the end of this book.

After this preface, the book will continue with a more technical introduction, which contains precise statements of some of the key results and a quick description of some of the crucial points which are involved in the proofs. We then split the remainder of the book in two parts, one containing the main theoretical results, and the other devoted to a variety of applications. These are complemented by appendices recalling important material, and Deligne's letter to Kazhdan. A more precise outline of each chapter will be found at the end of the introduction.

*Readers with a background in analytic number theory who are not familiar with the theory of trace functions and the underlying geometric objects are invited to first read Appendix E, where we attempt to present them in a concrete and intuitive way.*

## Acknowledgements

We are particularly thankful to Will Sawin, not only for many insightful comments on parts of this work, but especially for sharing his ideas on the complexity of $l$-adic sheaves on algebraic varieties, and for allowing us to assist in the final write-up of his work, resulting in the paper [115].

We thank the referees who read various parts of the book, and gently pointed out some incorrect or confusing statements, as well as some incomplete arguments, in the first draft.

We thank K. Soundararajan for pointing out to us the definition of Sidon sets. We thank Thomas Krämer for useful comments concerning some of his papers, especially [92]. We thank Arnaud Beauville for his help with some questions concerning cubic hypersurfaces, especially for the proof of Lemma 11.8. We also thank Sophie Morel for providing us with a proof of Lemma A.15. We thank Florent Jouve for his careful reading of Appendix E; we also thank Jon Keating for his encouragements, and Zeev Rudnick for his constructive criticism, concerning this same appendix.

We thank N. Katz for forwarding to us the notes of his talk [74], and for his comments on various parts of the text.

We also thank the organizers of the various seminars where we have been able to present parts of this work over the last few years.

Finally, we thank Pierre Deligne for kindly allowing us to reproduce his letter to Kazhdan in an appendix.

During the preparation of this work, A. F. and E. K. were partially supported by the DFG-SNF lead agency program grant 200020L.175755, A. F. was partially supported by the SNF Ambizione grant PZ00P2_193354, and J. F. was partially supported by the grant ANR-18-CE40-0017 of the Agence Nationale de la Recherche.
# Introduction

## 1. Statement of results

Since Deligne's proof of his equidistribution theorem for traces of Frobenius of $\ell$-adic local systems on varieties over finite fields [28], 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 $\ell$-adic sheaf that underlies the family of exponential sums.

The best known result of this kind is probably the computation by Katz [69] of these monodromy groups in the case of Kloosterman sums in several variables over finite fields, which are defined for some fixed non-trivial additive character $\psi: \mathbf{F}_q \to \mathbf{C}^\times$ and $a \in \mathbf{F}_q^\times$ as

$$ \mathrm{Kl}_m(a; q) = \frac{1}{q^{(m-1)/2}} \sum_{\substack{(x_1, \dots, x_m) \in (\mathbf{F}_q^\times)^m \\ x_1 \cdots x_m = a}} \psi(x_1 + \cdots + x_m). $$

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 sets $\{\mathrm{Kl}_2(a; q) | a \in \mathbf{F}_q^\times\}$ with respect to the Sato-Tate measure on the interval $[-2, 2]$ as $q \to +\infty$ among prime powers. Further deep investigations by Katz, especially in his monograph [70], provide a cornucopia of examples of equidistribution statements.

Among other things, this framework allows for the study of exponential sums of the form

$$ S(\mathrm{M}, \psi) = \sum_{x \in \mathbf{F}_{q^n}} t_{\mathrm{M}}(x; \mathbf{F}_{q^n}) \psi(x), $$

where $t_{\mathrm{M}}$ is the trace function of a perverse sheaf M on the additive group $\mathbf{G}_a$ over $\mathbf{F}_q$ and $\psi$ ranges over characters of $\mathbf{F}_{q^n}$. These sums are the discrete Fourier transform $\psi \mapsto S(\mathrm{M}, \psi)$ of the function $x \mapsto t_{\mathrm{M}}(x, \mathbf{F}_{q^n})$ on the finite group $\mathbf{F}_{q^n} = \mathbf{G}_a(\mathbf{F}_{q^n})$, and the key point is that they are themselves the trace functions of another perverse sheaf on the dual group parameterizing additive characters, namely Deligne's Fourier transform of M.

In a more recent conceptual breakthrough, Katz [75] 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 $\mathbf{F}_q$ does not naturally arise as the set of $\mathbf{F}_q$-points of an algebraic variety. In analogy with the above, such sums are of the form

$$ S(\mathrm{M}, \chi) = \sum_{x \in \mathbf{F}_{q^n}^\times} t_{\mathrm{M}}(x; \mathbf{F}_{q^n}) \chi(x), $$

except that M is now a perverse sheaf on the multiplicative group $\mathbf{G}_m$ over $\mathbf{F}_q$ and $\chi$ ranges over characters of $\mathbf{F}_{q^n}^\times$. 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 [43] for an accessible survey). Further work of Katz generalized this to elliptic curves [77] and certain abelian varieties (unpublished).

**The primary goal of this book is to extend these ideas to exponential sums (arithmetic Fourier transforms) parameterized by the characters of the points of any connected commutative algebraic group over a finite field.**

More precisely, let $k$ be a finite field and $\bar{k}$ an algebraic closure of $k$. For each $n \ge 1$, we denote by $k_n$ the extension of $k$ of degree $n$ inside $\bar{k}$. Let $\ell$ be a prime number different from the characteristic of $k$ and $\overline{Q}_\ell$ an algebraic closure of the field of $\ell$-adic numbers. Let $G$ be a connected commutative algebraic group over $k$. We denote by $\hat{G}(k_n)$ the group of $\overline{Q}_\ell$-valued characters of $G(k_n)$ and, for each $\chi \in \hat{G}(k_n)$, by $\mathcal{L}_\chi$ the $\ell$-adic lisse character sheaf of rank one associated to $\chi$ by means of the Lang torsor construction, as briefly recalled in Section 1.6. By perverse sheaves, we always understand $\overline{Q}_\ell$-perverse sheaves.

In rough outline, we establish the following types of theoretical results:

- We prove generic and stratified vanishing theorems for the cohomology of twists of perverse sheaves on $G$ by the sheaves $\mathcal{L}_\chi$ associated to characters $\chi \in \hat{G}(k_n)$.

- Using the stratified vanishing theorems, we construct a tannakian category of perverse sheaves on $G$ with the convolution coming from the group law as tensor product.

- 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 sums

$$S(M, \chi) = \sum_{x \in G(k_n)} t_M(x; k_n)\chi(x),$$

where $\chi$ ranges over the set $\hat{G}(k_n)$. Under some assumptions on $G$ (e.g., for tori and abelian varieties), we prove the stronger result that the unitary conjugacy classes of which these sums are traces become equidistributed in a maximal compact subgroup of the tannakian group as $n \to +\infty$, as is customary since Deligne's work.

Once this is done, we provide a number of applications, both of a general nature and for concrete groups and perverse sheaves.

The following statements are special cases of our main results, which we formulate in simplified form in order to make it possible to present self-contained statements at this stage.

**THEOREM 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

$$ \mathcal{X}(k_n) = \{ \chi \in \hat{G}(k_n) \mid \mathrm{H}_c^i(\mathrm{G}_{\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi) = \mathrm{H}^i(\mathrm{G}_{\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi) = 0 \text{ for all } i \neq 0 \\ \text{and } \mathrm{H}_c^0(\mathrm{G}_{\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi) \text{ is isomorphic to } \mathrm{H}^0(\mathrm{G}_{\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi) \} $$

are generic, in the sense that the estimate

$$|\hat{G}(k_n) - \mathcal{X}(k_n)| \ll |k_n|^{d-1}$$

holds for $n \ge 1$, with an implied constant that only depends on $M$.

(2) (Stratified vanishing) For $-d \le i \le d$ and $n \ge 1$, the estimate

$$|\{\chi \in \hat{G}(k_n) \mid \mathrm{H}_c^i(\mathrm{G}_{\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi) \neq 0 \text{ or } \mathrm{H}^i(\mathrm{G}_{\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi) \neq 0\}| \ll |k_n|^{d-i}$$

holds, with an implied constant that only depends on $M$.
The most general vanishing statements that we prove appear as Theorems 2.1 and 2.3. Applications to “stratification” estimates for exponential sums are then given in Chapter 6.

REMARK 1. (1) With variations in the definition of generic set of characters, such statements were proved by Katz-Laumon [78] for powers of the additive group, Saibi [112] for unipotent groups, Gabber-Loeser [50] for tori, Krämer-Weissauer [94], Weissauer [127] for abelian varieties and Krämer [91] for semiabelian varieties (see Remark 2.2 for more precise references).

(2) In characteristic zero, and especially over the complex numbers, theorems of this type have also been proved for abelian and semiabelian varieties by Schnell [116], Bhatt-Scholze-Schnell [9] and Liu-Maxim-Wang [102] (see also [101] for a survey of some applications of such results). Over arbitrary algebraically closed fields, there has also been recent works of Esnault and Kerz [36].

(3) In contrast with the case of abelian varieties, lack of properness and wild ramification phenomena are the reason one must formulate conditions on cohomology groups both with and without compact support.

Using the vanishing theorems, and ideas going back to Gabber-Loeser and Katz, 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 trace functions on G, i.e., for families of exponential sums parameterized by characters of G.

THEOREM 2 (Equidistribution on average for arithmetic Fourier transforms). Let G be a connected commutative algebraic group over $k$. Let M be a geometrically simple $\ell$-adic perverse sheaf on G that is pure of weight zero, with complex-valued trace functions $t_M(\cdot; k_n): G(k_n) \to \mathbf{C}$ for $n \ge 1$. There exists an integer $r \ge 0$ and a compact subgroup $K \subset \mathbf{U}_r(\mathbf{C})$ of the unitary group such that the sums

$$S(M, \chi) = \sum_{x \in G(k_n)} t_M(x; k_n)\chi(x)$$

for complex-valued characters $\chi$ of $G(k_n)$ become equidistributed on average in $\mathbf{C}$ with respect to the image by the trace of the Haar probability measure $\mu$ on K. That is, for any bounded continuous function $f: \mathbf{C} \to \mathbf{C}$, the following equality holds:

$$(1) \quad \lim_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|G(k_n)|} \sum_{\chi} f(S(M, \chi)) = \int_K f(\mathrm{Tr}(x)) d\mu(x),$$

where $\chi$ runs over all characters of $G(k_n)$.

The general version of this theorem appears as Theorem 4.8. Under an additional assumption (which holds for tori, abelian varieties and $G_a$, at least), we can also deduce it from Theorem 4.11, which is a more precise equidistribution result for unitary conjugacy classes of Frobenius in the compact group K. (The difference between these two statements is similar to that between the Frobenius equidistribution theorem for cycle types of Frobenius classes in the Galois group of a polynomial, viewed as a permutation group, and the more precise Chebotarev density theorem.)

REMARK 2. (1) In the classical setting of $G_a$ and the Fourier transform, the group K is a maximal compact subgroup of the arithmetic monodromy group of the (lisse sheaf underlying the) $\ell$-adic Fourier transform of M (see Proposition 3.34).

Note that this is in contrast with more usual versions of Deligne's equidistribution theorem, without the extra Cesàro average over $n$, where the focus is on the geometric monodromy group (see, e.g., the versions of Katz [69, Ch. 3] and Katz-Sarnak [79, Ch. 9]). This slight change of emphasis extends to the general situation, and means that we can avoid additional (necessary)
assumptions such as the equality of the geometric and arithmetic monodromy groups, which occur frequently otherwise (see, e.g., [69, § 3.3]), and are not always easy to check.

The Cesàro average can of course be interpreted as a form of “smoothing” (a “summation method” in the classical terminology). Although it is quite natural, it can be replaced by many others (see Remark 4.7).

(3) We will also discuss a “horizontal” version, where we consider suitable families $(M_p)_p$ of perverse sheaves over $\mathbb{F}_p$ for primes $p \to +\infty$. However, such results depend on a more quantitative version of the stratified vanishing theorem, which we have not established in full generality yet.

(4) As already mentioned, this equidistribution theorem is essentially Deligne's equidistribution theorem on average for the $\ell$-adic Fourier transform of $M$ when $G = \mathbb{G}_a$. When $G$ is the multiplicative group (or its non-split form), one obtains (an average version of) Katz's equidistribution theorem [75]. In [77], Katz proves a similar theorem for elliptic curves.

(5) The assumption that $G$ is connected arises from the fact that the Lang torsor construction is only applicable in this case. For the purpose of equidistribution results, however, one can easily handle a non-connected algebraic group by considering one by one the restrictions to the neutral component of $G$ of the objects ($[x \mapsto c^{-1}x]^*M$), where $c$ runs over representatives of the connected components of $G$. (Note that different connected components might give rise to exponential sums with different distributions.)

**EXAMPLE 1.** A simple concrete class of examples where we obtain equidistribution statements is the following (in the case when $G$ is not an abelian variety): assume that $k = \mathbb{F}_p$, and let $d$ be the dimension of $G$; then for any non-constant function $f: G \to \mathbb{A}^1$, there exists a perverse sheaf $M_f$ on $G$ with trace functions

$$t_{M_f}(x; \mathbb{F}_{p^n}) = \frac{(-1)^d}{p^{nd/2}} e\left(\frac{\mathrm{Tr}_{\mathbb{F}_{p^n}/\mathbb{F}_p}(f(x))}{p}\right)$$

for all $n \ge 1$ and $x \in G(\mathbb{F}_{p^n})$ (where $e(z) = \exp(2i\pi z)$), so that Theorem 2 shows that the exponential sums

$$\frac{1}{p^{nd/2}} \sum_{x \in G(\mathbb{F}_{p^n})} \chi(x) e\left(\frac{\mathrm{Tr}_{\mathbb{F}_{p^n}/\mathbb{F}_p}(f(x))}{p}\right)$$

(which are intuitively sums over $d$ variables) become equidistributed on average, with limiting measure of a very specific kind. Specializing even more to $G = \mathbb{G}_m^d$, the function $f$ is a Laurent polynomial in $x_1, \dots, x_d$ and their inverses, and these exponential sums become the sums

$$\frac{1}{p^{nd/2}} \sum_{x_1, \dots, x_d \in \mathbb{F}_{p^n}^\times} \chi_1(x_1) \cdots \chi_d(x_d) e\left(\frac{\mathrm{Tr}_{\mathbb{F}_{p^n}/\mathbb{F}_p}(f(x))}{p}\right)$$

parameterized by a tuple $(\chi_1, \dots, \chi_d)$ of characters of $\mathbb{F}_{p^n}^\times$.

As a further concrete application, we will see how to deduce statements like the following, which considerably strengthens earlier work of Hall, Keating and Roddity-Gershon [56].

**THEOREM 3** (Variance of the von Mangoldt function of the Legendre elliptic curve). Let $k$ be a finite field of characteristic $\ge 3$. Let $\mathcal{E}$ be the Legendre elliptic curve with affine model

$$y^2 = x(x-1)(x-t)$$

over the field $k(t)$. Let $\Lambda_{\mathcal{E}/k(t)}$ be the von Mangoldt function of $\mathcal{E}$, defined by the generating series

$$-T \frac{\mathrm{L}'(\mathcal{E}/k(t), \mathrm{T})}{\mathrm{L}(\mathcal{E}/k(t), \mathrm{T})} = \sum_g \Lambda_{\mathcal{E}/k(t)}(g) \mathrm{T}^{\mathrm{deg}(g)}$$
over monic polynomials $g \in k[t]$.

Let $f \in k[t]$ be a square-free polynomial of degree $\ge 4$ and set $B = k[t]/fk[t]$. Let $m \ge 1$ be an integer. For any $a \in B^\times$, consider the sum

$$ \psi_{\mathcal{E}}(m; f, a) = \sum_{\substack{\deg(g)=m \\ g \equiv a \pmod{f}}} \Lambda_{\mathcal{E}/k(t)}(g) $$

over monic polynomials $g \in k[t]$ of degree $m$. Then the following equality holds:

$$ \lim_{|k| \to +\infty} \frac{1}{|k|^2} \frac{1}{|B^\times|} \sum_{a \in B^\times} \left| \psi_{\mathcal{E}}(m; f, a) - \frac{1}{|B^\times|} \sum_{b \in B^\times} \psi_{\mathcal{E}}(m; f, b) \right|^2 \\ = \min(m, 2\deg(f) - 2 + \deg \gcd(t(t-1), f)). $$

The meaning of the above limit is that we replace $k$ by its extensions $k_n$ of degree $n \ge 1$, compute the variance for the base change of $\mathcal{E}$ to $k_n$ (note that $B$ depends on $k$, so it is also replaced by $k_n[t]/fk_n[t]$), and let $n \to +\infty$. This theorem is proved at the end of Chapter 10.

**REMARK 3.** The version in [56] requires the assumptions $\deg(f) > 900$ and $\gcd(t(t-1), f) = t$. We have greatly relaxed the former condition and fully removed the latter, which was recognized as being quite artificial (see [56, Rem. 11.0.2]). These improvements are due to the consideration of the problem in its natural setting, involving characters of a torus of dimension $\deg(f)$, whereas the authors of [56] used cosets of a one-dimensional torus together with Katz's work on $G_m$.

We also give a proof of an unpublished theorem of Katz [74] answering a question of Tsimerman about equidistribution of Artin L-functions on curves over finite fields.

**THEOREM 4 (Katz).** Let $C$ be a smooth projective geometrically connected curve of genus $g \ge 2$ over a finite field $k$ and let $D = \sum n_i x_i$ be a divisor of degree one on $C$. For each geometrically non-trivial character $\varrho: \pi_1(C)^{ab} \to C^\times$ of finite order satisfying $\prod \varrho(\mathrm{Fr}_{k(x_i),x_i})^{n_i} = 1$, we write its normalized Artin L-function as

$$ L(\varrho, \mathrm{T}/\sqrt{|k|}) = \det(1 - \mathrm{T}\Theta_{C/k, \varrho}) $$

for a conjugacy class $\Theta_{C/k, \varrho}$ in the unitary group $U_{2g-2}(C)$.

(1) If $C$ is non-hyperelliptic and $(2g-2)\mathrm{D}$ is a canonical divisor on $C$, then the classes $\Theta_{C/k, \varrho}$ lie in $SU_{2g-2}(C)$ and become equidistributed with respect to the image on the space of conjugacy classes of the Haar probability measure of $SU_{2g-2}(C)$.

(2) If $C$ is hyperelliptic, the hyperelliptic involution has a fixed point $O \in C(k)$ and $D = O$, then the classes $\Theta_{C/k, \varrho}$ lie in $USP_{2g-2}(C)$ and become equidistributed with respect to the image on the space of conjugacy classes of the Haar probability measure on $USP_{2g-2}(C)$.

See Chapter 11 for the proof of this result, as well as some more general statements (including, in Theorem 11.5, a result where the algebraic group $G$ occurring may involve abelian, toric and unipotent parts).

## 2. Outline

In this section, we present the plan of the book, and we sketch one of the main ideas of the proof of Theorem 2, in order to point out the key difficulties for groups of dimension bigger than one, which are solved using Sawin's quantitative sheaf theory [115].

The book is organized as follows:
- In Chapter 1, we state some preliminary results; these include a survey of the formalism of quantitative sheaf theory [115], as well as basic structural results concerning commutative algebraic groups and character sheaves.

- In Chapter 2, we prove the generic and stratified vanishing theorems for commutative algebraic groups over finite fields. The very rough idea is to prove a *relative version* of the vanishing theorems for the various basic types of commutative groups, with a good control of the implicit constants. These relative statements are of independent interest. For example, in the case of tori, Gabber-Loeser [50] prove the stratified vanishing theorem as stated above only under the assumption that resolution of singularities over $k$ holds for varieties of dimension up to that of the torus. We remove this assumption using alterations. For abelian varieties, we extend Weissauer's work [127] by proving a relative version of the theorem, which relies on Orgogozo's work [109] on constructibility and moderation.

- In Chapter 3, we construct a suitable tannakian category of perverse sheaves on a commutative group over a finite field with convolution as tensor operation, and establish its basic properties, as well as those of the corresponding tannakian monodromy group. We will see that some subtleties arise when defining “Frobenius conjugacy classes” corresponding to characters of $G$.

- In Chapter 4, we combine these two ingredients to establish a number of “vertical” equidistribution theorems; there are some issues when we want to refine the statements at the level of conjugacy classes (related to those of the previous sections), which we are not currently able to solve in full generality, although we can always establish equidistribution for the characteristic polynomials.

- The beginning of Part 2 introduces a selection of first applications of a general nature. These include the following:

(1) the definition of the analogue of the L-function for arithmetic Fourier transforms, which is used to study finite tannakian groups over abelian varieties (Chapter 5);

(2) a stratification result for exponential sums, similar to those of Katz, Laumon and Fouvry, although currently often restricted to the “vertical” direction (Chapter 6);

(3) a “generic Fourier invertibility” result (Chapter 6);

(4) some preliminary results of independence of $\ell$ for the tannakian group when working with perverse sheaves which are part of a compatible system (Chapter 7);

(5) various results of “Diophantine group theory”, where averages of exponential sums are related to invariants of the tannakian group; this includes in particular Larsen’s alternative, but also some criteria to recognize the exceptional group $\mathbf{E}_6$ (Chapter 8).

- Chapters 9, 10 and 11 contain applications to concrete cases. The algebraic groups involved are, respectively, the product $\mathbf{G}_a \times \mathbf{G}_m$, higher-dimensional tori, and jacobians of curves, as well as the intermediate jacobian of a smooth cubic threefold (where the relevant tannakian group is $\mathbf{E}_6$, as first shown in the complex setting by Krämer).

- In Chapter 12, we list some open questions and problems. The title “Much remains to be done” paraphrases Katz ([75, p. 18]).

- Finally, we include appendices to survey the basic theory of perverse sheaves (Appendix A), as well as to recall the most important results of Katz concerning the arithmetic Mellin transform on $\mathbf{G}_m$ (Appendix B) and the product formula of Laumon for the epsilon factor of L-functions over finite fields (Appendix C). In Appendix D, we reproduce, with Deligne's
permission, the letter to Kazhdan in which the ℓ-adic Fourier transform was first discussed. To conclude, we attempt to sketch the intuitive nature of the theory of general trace functions, to provide some intuition for analytic number theorists in Appendix E, and we collected some remarks on the early history of Frobenius equidistribution (Appendix ??).

We now survey the key analytic step in the proof of Theorem 2 (see Proposition 4.12).

By fixing an isomorphism $\iota_0: \overline{\mathbb{Q}}_\ell \to \mathbb{C}$, we can work with trace functions and characters with values in $\overline{\mathbb{Q}}_\ell$. The first step, following from the generic vanishing theorem, will be to prove that there exist subsets $\mathcal{Y}(k_n) \subset \tilde{\mathbb{G}}(k_n)$ of characters and conjugacy classes $\Theta_{M,k_n}(\chi)$ in some unitary group $\mathbf{U}_r(\mathbb{C})$ such that $\mathrm{Tr}(\Theta_{M,k_n}(\chi)) = \mathrm{S}(M, \chi)$ holds for all $\chi \in \mathcal{Y}(k_n)$ and

$$|\mathcal{Y}(k_n)| \sim |\mathbb{G}(k_n)|$$

as $n \to +\infty$. The second step (an application of the theory of tannakian categories) will be an intrinsic a priori definition of the compact group K for which equidistribution should hold.

By (essentially) the Weyl criterion for equidistribution, Theorem 2 will follow from the proof that, for every non-trivial irreducible representation $\varrho$ of the unitary group $\mathbf{U}_r(\mathbb{C})$, the limit

$$\lim_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|\mathbb{G}(k_n)|} \sum_{\chi \in \mathcal{Y}(k_n)} \mathrm{Tr} \, \varrho(\Theta_{M,k_n}(\chi))$$

exists and is equal to the multiplicity of the trivial representation in the restriction of $\varrho$ to K. Now, the tannakian formalism associates to each $\varrho$ a perverse sheaf $\varrho(M)$ on G such that the equality

$$\mathrm{Tr} \, \varrho(\Theta_{M,k_n}(\chi)) = \sum_{x \in \mathbb{G}(k_n)} \chi(x) t_{\varrho(M)}(x; k_n)$$

holds for $n \ge 1$ and $\chi \in \mathcal{Y}(k_n)$. The Grothendieck–Lefschetz trace formula yields then the equality

$$(2) \qquad \sum_{x \in \mathbb{G}(k_n)} \chi(x) t_{\varrho(M)}(x; k_n) = \sum_{|j| \le d} (-1)^j \mathrm{Tr} \left( \mathrm{Fr}_{k_n} \mid \mathrm{H}_c^j(\mathbb{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi}) \right)$$

for $n \ge 1$ and any character $\chi$ of $\mathbb{G}(k_n)$, where $\mathrm{Fr}_{k_n}$ is the geometric Frobenius automorphism.

The definition of the set $\mathcal{Y}(k_n)$ implies the property that for $\chi \in \mathcal{Y}(k_n)$, the only possibly non-zero term in the right-hand side of (2) is the one with $j = 0$. Thus we have

$$\sum_{\chi \in \mathcal{Y}(k_n)} \mathrm{Tr} \, \varrho(\Theta_{M,k_n}(\chi)) = \sum_{\chi \in \mathcal{Y}(k_n)} \mathrm{Tr} \left( \mathrm{Fr}_{k_n} \mid \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi}) \right).$$

If we add to the right-hand side of this last expression the two sums

$$S_1 = \sum_{\chi \notin \mathcal{Y}(k_n)} \mathrm{Tr} \left( \mathrm{Fr}_{k_n} \mid \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi}) \right), \\ S_2 = \sum_{1 \le |j| \le d} \sum_{\chi \notin \mathcal{Y}(k_n)} (-1)^j \mathrm{Tr} \left( \mathrm{Fr}_{k_n} \mid \mathrm{H}_c^j(\mathbb{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi}) \right),$$

then the resulting quantity is

$$\sum_{|j| \le d} \sum_{\chi \in \hat{\mathbb{G}}(k_n)} (-1)^j \mathrm{Tr} \left( \mathrm{Fr}_{k_n} \mid \mathrm{H}_c^j(\mathbb{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi}) \right) = \sum_{\chi \in \hat{\mathbb{G}}(k_n)} \sum_{x \in \mathbb{G}(k_n)} \chi(x) t_{\varrho(M)}(x; k_n) \\ = \sum_{x \in \mathbb{G}(k_n)} t_{\varrho(M)}(x; k_n) \sum_{\chi \in \hat{\mathbb{G}}(k_n)} \chi(x) \\ = |\mathbb{G}(k_n)| t_{\varrho(M)}(1; k_n)$$
by the trace formula again, followed by an exchange of the sums and an application of the orthog-
onality of characters of finite abelian groups. This is a single value of the trace function, and it is
relatively straightforward to show that it gives the desired multiplicity as limit. So *the key difficulty*
*is to control the two auxiliary sums S<sub>1</sub> and S<sub>2</sub>.*

This can be done if:

(1) We have some bound on the size of the individual traces Tr (Fr$_{k_n}$ | H$_c^j$(G$_{\bar{k}}$, ϱ(M) ⊗ L$_χ$));

(2) We have some bound on the number of characters $\chi$ such that $H_c^j(G_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi})$ can be non-zero in a given degree $j$.

The second bound is given by the stratified vanishing theorem for $\varrho(M)$. For the first, Deligne's
Riemann hypothesis (see Theorem A.19) implies the inequality

$$
| \mathrm{Tr} ( \mathrm{Fr}_{k_n} | \mathrm{H}_c^j ( \mathrm{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi} ) ) | \le | k_n |^{(j-d)/2} \dim \mathrm{H}_c^j ( \mathrm{G}_{\bar{k}}, \varrho(M) \otimes \mathcal{L}_{\chi} ),
$$

and we see that we require a bound on the dimension of the cohomology spaces, which should be
independent of $\chi$. We obtain such bounds as special cases of Sawin's quantitative sheaf theory [115],
which is a quantitative form of the finiteness theorems for the six operations on the derived category
of $\ell$-adic sheaves on quasi-projective algebraic varieties.

REMARK 4. If G is one-dimensional, then the Euler-Poincaré characteristic formula (see Theorem C.2) easily implies precise bounds on the dimension of the cohomology spaces that arise, and hence this critical issue does not arise for the additive or multiplicative groups, or for elliptic curves (for such groups, Theorem 1 is also straightforward). It also does not arise if the set of “good” characters $\mathcal{Y}(k_n)$ is the whole group $\hat{G}(k_n)$, which is the case in some instances considered by Katz for higher-dimensional abelian varieties.

**3. Conventions and notation**

We summarize the notation that we use, as well as some typographical conventions that we
follow consistently unless otherwise specified.

Given complex-valued functions $f$ and $g$ defined on a set $S$, we write $f \ll g$ if there exists a
real number $C \ge 0$ (called an “implicit constant”) such that the inequality $|f(s)| \le Cg(s)$ holds for
all $s \in S$. We write $f \asymp g$ if $f \ll g$ and $g \ll f$. If $f$ and $g$ are defined on a topological space $X$,
and $\mathfrak{F}$ is a filter on $X$, then we say that $f \sim g$ along $\mathfrak{F}$ if $\lim_{\mathfrak{F}} f(x)/g(x) = 1$.

For any complex number $z$, we write $e(z) = \exp(2i\pi z)$; the value $e(a/q)$ is well-defined for $q \ge 1$
and $a \in \mathbb{Z}/q\mathbb{Z}$.

By a *variety over a field k*, we mean a reduced separated *k*-scheme of finite type. In particular,
an *algebraic group*, as opposed to a group scheme, is always supposed to be reduced, and hence
smooth if the field *k* is perfect.

Let $S$ be a scheme. We say that a pair $(X, u)$ is a *quasi-projective scheme over S* if $X$ is a
scheme over $S$ and $u$ is a locally-closed immersion $u: X \to \mathbf{P}_S^n$ for some integer $n \ge 0$. We call $n$ the
embedding dimension of $(X, u)$, or simply of $u$, and we say that $u$ is a *quasi-projective embedding*
of $X$. When $S$ is the spectrum of a field $k$ and $X$ is a variety over $k$, we will speak of quasi-projective
varieties over $k$. In some cases, we omit the mention of $u$, when it is clear from the context which
locally-closed immersion is used. By a morphism $f: (X, u) \to (Y, v)$ of quasi-projective schemes
over $S$, we mean an $S$-morphism of the underlying schemes.

An algebraic group **G** over an algebraically closed field of characteristic zero is called *reductive*
if all its finite-dimensional representations are completely reducible (that is, we do not require **G**
to be connected).
Let X be a scheme and $\ell$ a prime number invertible on X.

By a $\overline{\mathbb{Q}}_\ell$-sheaf on X, we always mean a constructible étale $\overline{\mathbb{Q}}_\ell$-sheaf. Perverse sheaves (when X is an algebraic variety defined over a field $k$) are always considered with respect to the middle perversity. We include a short survey of the most important properties of perverse sheaves in Appendix A, but recall here some of the definitions. An $\ell$-adic complex is said to be semiperverse if the inequality

$$ \dim \mathrm{supp} \mathcal{H}^i(M) \le -i $$

holds for any integer $i$. This is equivalent to asking that the perverse cohomology sheaves $\mathbb{P}\mathcal{H}^i(M)$ are zero for $i \ge 1$ (see [8, Prop. 1.3.7]).

We say that a complex M in $\mathrm{D}_c^b(X, \overline{\mathbb{Q}}_\ell)$ has *perverse amplitude* $[a, b]$ if its perverse cohomology sheaves $\mathbb{P}\mathcal{H}^i(M)$ are zero for $i \notin [a, b]$.

A stratification $\mathcal{X}$ of X is a finite set-theoretic partition of the associated reduced scheme $X^{\mathrm{red}}$ into non-empty reduced locally-closed subschemes of X, called the *strata* of $\mathcal{X}$.

Let $\mathcal{X}$ be a stratification of X, and let $\mathcal{F}$ be an $\ell$-adic sheaf on X. The sheaf $\mathcal{F}$ is said to be *tame and constructible along* $\mathcal{X}$ if it is tamely ramified, as in [109, §1.3.1], and if its restriction to any strat of $\mathcal{X}$ is a lisse sheaf. More generally, a complex $M \in \mathrm{D}_c^b(X, \overline{\mathbb{Q}}_\ell)$ is said to be tame and constructible along $\mathcal{X}$ if all its cohomology sheaves are tame and constructible along $\mathcal{X}$.

Let $f: X \to Y$ be a morphism of schemes. For an object M of $\mathrm{D}_c^b(X, \overline{\mathbb{Q}}_\ell)$, we write $Rf_!M = Rf_*M$ to indicate that the canonical “forget supports” morphism $Rf_!M \to Rf_*M$ is an isomorphism (and similarly for equality of cohomology groups with and without compact support).

Let $q \ge 1$ and $w \in \mathbb{Z}$ be integers. A complex number $\alpha$ is called a *q-Weil number* of weight $w$ if $\alpha$ is algebraic over $\mathbb{Q}$ and all its Galois conjugates have modulus $q^{w/2}$. If $k$ is a finite field, then a $k$-Weil number is a $|k|$-Weil number.

Throughout, for any prime $\ell$, we consider a fixed isomorphism $\iota_0: \overline{\mathbb{Q}}_\ell \to \mathbb{C}$. Trace functions of $\ell$-adic perverse sheaves are thus always identified with complex-valued functions through $\iota_0$, and similarly $\ell$-adic characters are identified with complex characters. On the other hand, purity of perverse sheaves (or lisse sheaves or $\ell$-adic complexes) refers to purity in the sense of Deligne, i.e., pointwise purity means that the eigenvalues of Frobenius are Weil numbers of some weight; see the survey in Section A.3.

The following notation is used consistently in all the book, although frequently with reminders (some objects, such as character sheaves, will be defined later).

- $X-Y$: difference set (elements of X that are not in Y); also used in scheme-theoretic settings.

- $M|_X$ or $M|_X$: restriction of an object M (or a section of a sheaf) to a subset or subscheme X.

- $|X|$: cardinality of a set X.

- $\tau(\chi, \psi)$: (unnormalized) Gauss sum attached to a multiplicative character $\chi: k^\times \to \overline{\mathbb{Q}}_\ell^\times$ and an additive character $\psi: k \to \overline{\mathbb{Q}}_\ell^\times$ for a finite field $k$, i.e.

$$ (3) \qquad \tau(\chi, \xi) = \sum_{x \in k^\times} \chi(x)\psi(x). $$

- $\mathrm{D}_c^b(X) = \mathrm{D}_c^b(X, \overline{\mathbb{Q}}_\ell)$: category of bounded constructible complexes of $\overline{\mathbb{Q}}_\ell$-sheaves on a scheme X such that the prime $\ell$ is invertible in X.
- $K(X) = K(X, \overline{\mathbb{Q}}_\ell)$: the Grothendieck group (or ring) of $D_c^b(X)$; it has a basis consisting of classes of simple perverse sheaves (see [98, § 0.8]).

- $\alpha^{\deg}$: for $k$ a finite field and $\alpha$ an $\ell$-adic unit, the $\ell$-adic sheaf of rank 1 on $\text{Spec}(k)$ on which the geometric Frobenius acts by multiplication by $\alpha$; more generally, for $f: X \to \text{Spec}(k)$ a scheme over $k$, the pullback of $\alpha^{\deg}$ to X.

- $M \otimes N$: derived tensor product of objects of $D_c^b(X)$.

- $M \boxtimes N$: for $M$ an object of $D_c^b(X)$ and $N$ an object of $D_c^b(Y)$, the object $p_1^*M \otimes p_2^*N$ on $X \times Y$, where $p_1$ and $p_2$ are the two projections.

- $\text{Perv}(X) = \text{Perv}(X, \overline{\mathbb{Q}}_\ell)$: the category of $\ell$-adic perverse sheaves on X. A simple perverse sheaf will also sometimes be called an irreducible perverse sheaf.

- $\text{D}(M)$: the Verdier dual of a complex M.

- $\mathcal{H}^i(M)$: for $M \in D_c^b(X)$, the $i$-th cohomology sheaf of M.

- $\mathfrak{p}\mathcal{H}^i(M)$: for $M \in D_c^b(X)$, the $i$-th perverse cohomology sheaf of M.

- $H^i(M) = H^i(X_{\bar{k}}, M)$: the étale cohomology groups of the pull-back of $M$ to $X \times_k \bar{k}$.

- $H_c^i(M) = H_c^i(X_{\bar{k}}, M)$: the étale cohomology groups with compact support of M.

- $h^i(X_{\bar{k}}, M) = \text{dim } H^i(X_{\bar{k}}, M)$.

- $h_c^i(X_{\bar{k}}, M) = \text{dim } H_c^i(X_{\bar{k}}, M)$.

- $H^*(X_{\bar{k}}, M)$ or $H_c^*(X_{\bar{k}}, M)$: the graded vector space which is the direct sum of all cohomology spaces $H^i(X_{\bar{k}}, M)$ or $H_c^i(X_{\bar{k}}, M)$.

- $\chi(X_{\bar{k}}, M)$ or $\chi_c(X_{\bar{k}}, M)$: Euler-Poincaré characteristic for cohomology or cohomology with compact support.

- $t_M(x; k_n)$: Frobenius trace function of an object $M$ of $D_c^b(X)$ for $x \in X(k_n)$. For $x \in X(k)$, we usually abbreviate it by $t_M(x) = t_M(x; k)$.

- $\langle M \rangle$: tannakian category generated by M.

- $\mathbf{G}^{\text{ari的}}_M$ (resp. $\mathbf{G}^{\text{geo}}_M$): arithmetic (resp. geometric) tannakian group associated with a perverse sheaf M.

- $\hat{\mathbb{G}}(k_n)$: group of $\overline{\mathbb{Q}}_\ell$-characters of the finite group $\mathbb{G}(k_n)$.

- $\hat{\mathbb{G}}$: disjoint union of the sets $\hat{\mathbb{G}}(k_n)$ for $n \ge 1$.

- $\Pi(G)$: for a semiabelian variety $G$, the $\overline{\mathbb{Q}}_\ell$-scheme of $\ell$-adic characters of $G$.

- $\mathcal{L}_\chi$: character sheaf on $G_{k_n}$ associated to a character $\chi \in \hat{\mathbb{G}}(k_n)$.

- $\mathcal{L}_{\chi(f)}$: for $f: X \to G$ and $\chi \in \hat{\mathbb{G}}(k_n)$, the sheaf $f^*\mathcal{L}_\chi$ on X.

- $M_\chi$: for an object $M$ of $D_c^b(G)$ and a character $\chi$, the object $M \otimes \mathcal{L}_\chi$.

Moreover, the following notational conventions will be used (often with reminders).

- $k$: a finite field of characteristic $p$.

- $\ell$: a prime different from $p$.

- $\bar{k}$: an algebraic closure of $k$.

- $k_n$: the extension of degree $n$ of $k$ inside $\bar{k}$.
- G: a connected commutative algebraic group (in particular of finite type) defined over $k$.

- T: a torus;

- U: a unipotent group;

- A: an abelian variety.

- $\mathcal{F}$: a $\overline{\mathbb{Q}}_\ell$-sheaf;

- $\mathcal{L}$: a $\overline{\mathbb{Q}}_\ell$-lisse sheaf of rank one.

- M, N: objects of $\mathrm{D}_c^b(X)$ or $\mathbf{Perv}(X)$.
<figure><img src="image_5.png" /></figure>
# Part 1

## Theoretical foundations
<figure><img src="image_6.png" /></figure>
CHAPTER 1

Preliminaries

In this chapter, we summarize some tools we use throughout the book, especially the properties of Sawin’s quantitative sheaf theory [115] with an emphasis on commutative algebraic groups.

1.1. Specializations of perverse sheaves

We will frequently use the following result concerning specializations of perverse sheaves.

PROPOSITION 1.1. Let $k$ be a field, $f: Y \to X$ a surjective affine morphism of varieties over $k$, and $M$ a perverse sheaf on $Y$. For all closed points $x$ outside of a closed strict subvariety of $X$, the object $M|_{f^{-1}(x)}[-1]$ is perverse on $f^{-1}(x)$.

This follows directly from [83, Ch. III, Lemma 6.3].

1.2. Review of quantitative sheaf theory

Let $k$ be a field, $\bar{k}$ an algebraic closure of $k$, and $\ell$ a prime different from the characteristic of $k$.

DEFINITION 1.2 (Complexity). Let $M_{n+1,m+1}$ be the variety of $(n+1) \times (m+1)$ matrices of maximal rank, viewed as an affine scheme over $k$. For each $0 \le m \le n$, consider a geometric generic point $a_m$ of $M_{n+1,m+1}$ defined over an algebraically closed extension $K$ of $k$, and let $l_{a_m}: \mathbf{P}_K^m \to \mathbf{P}_K^n$ denote the associated linear map.

(a) The *complexity* of an object $M$ of $\mathbf{D}_c^b(\mathbf{P}_k^n)$ is defined as

$$
c(M) = \max_{0 \le m \le n} \sum_{i \in \mathbb{Z}} h^i(\mathbf{P}_K^n, M \otimes l_{a_m} * \bar{\mathbf{Q}}_\ell) = \max_{0 \le m \le n} \sum_{i \in \mathbb{Z}} h^i(\mathbf{P}_K^m, l_{a_m}^* M),
$$

where the last equality follows from the projection formula.

(b) Let $(X, u)$ be a quasi-projective variety over $k$. For any object $M$ of $\mathbf{D}_c^b(X)$, the *complexity* of $M$ with respect to $u$ is defined as $c_u(M) = c(u!M)$.

The invariance of étale cohomology under base change between algebraically closed fields implies
that the complexity is well-defined (i.e., it does not depend on the choice of fields of definition of
the generic points $a_m$).

LEMMA 1.3. Let $(X, u)$ be a quasi-projective variety over $k$ and let $M$ be an object of $\mathbf{D}_c^b(X)$. The following inequality holds:

$$
(1.1) \qquad \sum_{i \in \mathbb{Z}} h_c^i(X_{\bar{k}}, M) \le c_u(M).
$$

PROOF. This follows from the equality $h_c^i(X_{\bar{k}}, M) = h^i(\mathbf{P}_{\bar{k}}^n, u!M)$ and the invariance of étale cohomology under extension of scalars between algebraically closed fields, combined with the fact that $l_{a_n}: \mathbf{P}_K^n \to \mathbf{P}_K^n$ is an isomorphism. $\square$
**DEFINITION 1.4.** Let $f: (X, u) \to (Y, v)$ be a morphism of quasi-projective varieties over $k$ with embedding dimensions $n_X$ and $n_Y$ respectively. For all integers $0 \le m_X \le n_X$ and $0 \le m_Y \le n_Y$, consider geometric generic points $a_{m_X}$ of $M_{n_X+1, m_X+1}$ and $b_{m_Y}$ of $M_{n_Y+1, m_Y+1}$ defined over an algebraically closed extension $K$ of $k$, and let $l_{a_{m_X}}: \mathbf{P}_K^{m_X} \to \mathbf{P}_K^{n_X}$ and $l_{b_{m_Y}}: \mathbf{P}_K^{m_Y} \to \mathbf{P}_K^{n_Y}$ denote the associated linear maps. The complexity of $f$ is defined as

$$c_{u,v}(f) = \max_{0 \le m_X \le n_X} \max_{0 \le m_Y \le n_Y} \sum_{i \in \mathbb{Z}} h_c^i(X_K, u^*l_{a_{m_X}} \overline{Q}_\ell \otimes f^*v^*l_{b_{m_Y}} \overline{Q}_\ell).$$

The main result of [115] establishes, among other things, the “continuity” of the six operations on the derived category with respect to the complexity. In this result and the remainder of this section, the implicit constants only depend on the embedding dimensions of the quasi-projective varieties, unless otherwise specified.

**THEOREM 1.5** ([115, Th. 6.8, Prop. 6.14, Prop. 6.12]). Let $f: (X, u) \to (Y, v)$ be a morphism of quasi-projective varieties over $k$. Let $M, N, P$ be objects of $\mathrm{D}_c^b(X)$ and let $Q$ be an object of $\mathrm{D}_c^b(Y)$. The following inequalities hold:

$$ (1) \quad c_u(M \oplus N) \le c_u(M) + c_u(N). $$

$$ (2) \quad c_u(M \otimes N) \ll c_u(M)c_u(N). $$

$$ (3) \quad \text{If } M \to N \to P \text{ is a distinguished triangle, then } c_u(N) \le c_u(M) + c_u(P). $$

$$ (4) \quad c_u(M[k]) = c_u(M) \text{ for any } k \in \mathbb{Z}. $$

$$ (5) \quad c_u(\mathrm{RHom}(M, N)) \le c_{u,\mathrm{id}}(u)c_u(M)c_u(N). $$

$$ (6) \quad c_v(\mathrm{Rf}_!M) \ll c_{u,v}(f)c_u(M) \text{ and } c_v(\mathrm{Rf}_*M) \ll c_{u,\mathrm{id}}(u)c_{v,\mathrm{id}}(v)c_{u,v}(f)c_u(M). $$

$$ (7) \quad c_u(f^*Q) \ll c_{u,v}(f)c_v(Q) \text{ and } c_u(f^!Q) \ll c_{u,\mathrm{id}}(u)c_{v,\mathrm{id}}(v)c_{u,v}(f)c_v(Q). $$

$$ (8) \quad c_{u \boxtimes v}(M \boxtimes Q) \ll c_{u \boxtimes v, u}(p_1)c_{u \boxtimes v, v}(p_2)c_u(M)c_v(Q). $$

In the last of these inequalities, $u \boxtimes v$ is the composition of $u \times v$ with the Segre embedding and $p_1, p_2$ are the projections $X \times Y \to X$ and $X \times Y \to Y$, respectively.

**REMARK 1.6.** Although the notion of complexity on a quasi-projective scheme $(X, u)$ depends on the quasi-projective immersion $u$, note that if $v$ is another quasi-projective immersion of $X$, then applying the property (7) to the identity morphism between $(X, u)$ and $(X, v)$, we get

$$ c_u(M) \asymp c_v(M) $$

for all objects $M$ of $\mathrm{D}_c^b(X)$, where the implied constants are essentially $c_{u,v}(\mathrm{Id})$ and $c_{v,u}(\mathrm{Id})$, up to constants depending on the embedding dimensions of $u$ and $v$. Thus, as long as we only consider on $X$ an absolutely bounded number of different quasi-projective immersions, we can think of the complexity as being essentially independent of them. (This is reminiscent of similar properties of height functions in diophantine geometry.)

The complexity can also be used to control the degree of the locus where a complex of sheaves is lisse, and of the locus where the generic base change theorem holds.

**THEOREM 1.7** ([115, Th. 6.23]). Let $(X, u)$ be an irreducible quasi-projective variety over $k$. Let $M$ be an object of $\mathrm{D}_c^b(X)$. Let $Z$ be the complement of the maximal open subset where $X$ is smooth and $M$ is lisse. Then the estimate

$$ \deg(u(Z)) \ll (3+s)c(u)c_u(M) $$

holds, where the degrees are computed in the projective space target of $u$, and $s$ is the degree of the codimension 1 part of the singular locus of $X$.
**THEOREM 1.8** ([115, Th. 6.27]). Let $(X, u)$, $(Y, v)$ and $(S, w)$ be quasi-projective algebraic varieties over $k$. Let $f: X \to Y$ and $g: Y \to S$ be morphisms.

For any object $M$ of $\mathbb{D}_c^b(X)$, there exists an integer $C \ge 0$, depending only on $c_u(M)$ and the morphisms $(f, g, u, v, w)$, and a dense open set $U \subset S$ such that:

(i) The image of the complement of $U$ has degree $\le C$.

(ii) The object $f_*M$ is of formation compatible with any base change $S' \to U \subset S$.

**PROPOSITION 1.9** ([115, Th. 6.15]). Let $(X, u)$ be a quasi-projective variety over $k$. Let $M$ be an object of $\mathbb{D}_c^b(X)$. For each integer $i$, let $M_{i,1}, \dots, M_{i,n_i}$ denote the Jordan–Hölder factors of the perverse cohomology sheaf $\mathbb{P}\mathcal{H}^i(M)$. Then the following estimate holds:

$$ \sum_{i \in \mathbb{Z}} \sum_{1 \le j \le n_i} c_u(M_{i,j}) \ll c_{u,id}(u)c_u(M). $$

We also recall the quantitative statement of the Riemann hypothesis over finite fields when interpreted as a quasi-orthogonality statement.

**THEOREM 1.10** ([115, Th. 7.13 (2)]). Let $k$ be a finite field and $\ell$ a prime different from the characteristic of $k$. Let $(X, u)$ be a geometrically irreducible quasi-projective algebraic variety over $k$. Let $M$ and $N$ be geometrically simple $\ell$-adic perverse sheaves on $X$ that are pure of weight zero, with complex trace functions $t_M$ and $t_N$ respectively. Then the estimate

$$ \sum_{x \in X(k)} t_M(x) \overline{t_N(x)} \ll c(u)c_u(M)c_u(N) |k|^{-1/2} $$

holds if $M$ and $N$ are not geometrically isomorphic, whereas

$$ \sum_{x \in X(k)} |t_M(x)|^2 = 1 + O(c(u)c_u(M)^2 |k|^{-1/2}). $$

In both estimates, the implied constants only depend on the embedding dimension and are effective.

Finally, we have pointwise bounds for the trace functions.

**PROPOSITION 1.11** ([115, Prop. 7.11 (2)]). Let $k$ be a finite field and $\ell$ a prime different from the characteristic of $k$. Let $(X, u)$ be a quasi-projective algebraic variety over $k$, and let $M$ be a non-punctual simple perverse sheaf on $X$ which is pure of weight zero. For any $n \ge 1$ and $x \in X(k_n)$, the following estimate holds:

$$ t_M(x; k_n) \ll \frac{1}{|k_n|^{1/2}}. $$

### 1.3. Existence of rational points

The following lemma is standard.

**LEMMA 1.12.** Let $(X, u)$ be a non-empty quasi-projective variety over a finite field $k$ with embedding dimension $n$. There exists a finite extension $k'$ of $k$ with degree bounded in terms of $(\dim(X), \deg(u(X)), n)$ such that $X(k')$ is non-empty.

**PROOF.** Write the variety $u(X)$ as $Z-Y$ for some closed subvarieties $Z$ and $Y$. Then the degrees of $Z$ and $Y \cap Z$ are bounded in terms of $c_u(X)$ by [115, Lemma 6.26], and the result then follows from the Lang–Weil bound (see [95, Th. 1]) applied to $Z$ and to $Y \cap Z$. $\square$
## 1.4. Structure of commutative algebraic groups

Let $k$ be a field and let $G$ be a commutative algebraic group over $k$. The algebraic variety $G$ is quasi-projective (see, e.g., [22, Prop. A.3.5] or [122, Lemma 0BF7]). We will always assume that a quasi-projective immersion $u$ of $G$ is given, and the complexity of $\ell$-adic complexes will be understood with respect to $u$ (so that we sometimes write just $c(M)$ instead of $c_u(M)$). If $G$ is either a power of $G_a$ or of $G_m$, we assume that $u$ is simply the obvious embedding in the projective space of the same dimension. We will on occasion use auxiliary quasi-projective immersions and rely on Remark 1.6 to compare complexities.

Smooth connected commutative algebraic groups can be built as successive extensions of abelian varieties, tori, unipotent<fnref n="1" /> and finite commutative group schemes. The most convenient formulation of this fact for us is the following statement, which follows from results of Barsotti-Chevalley and Rosenlicht (see for instance the account in the book of Brion, Samuel and Umae, combining [17, Cor. 5.5.2] with the structure theorem for connected affine commutative algebraic groups over perfect fields as a product of a unipotent group and a torus; see, e.g., [16, Th. 5.3.1 (2)]).

**PROPOSITION 1.13.** Let $k$ be a finite field and let $G$ be a connected commutative algebraic group over $k$. There exist an abelian variety $A$, a torus $T$, a unipotent group $U$ and a finite commutative subgroup scheme $N$ of $A \times U \times T$, all defined over $k$, such that $G$ is isomorphic to $(A \times U \times T)/N$.

We further recall that a finite commutative group scheme $N$ over a perfect field has a unique direct product decomposition $N = N_r \times N_l$ where $N_r$ is reduced and $N_l$ is local (i.e., equal to its connected component of the identity; see, e.g., [16, Prop. 2.5.4]).

## 1.5. Convolution

Let $G$ be a commutative algebraic group over a field $k$. We denote by

$$m: G \times G \to G, \quad \text{inv}: G \to G, \quad e \in G(k)$$

the group law, the inversion morphism, and the neutral element respectively.

**DEFINITION 1.14 (Convolution).** The convolution product and the convolution product with compact support on $G$ are the functors from $D_c^b(G) \times D_c^b(G)$ to $D_c^b(G)$ defined as

$$M *_{*} N = Rm_{*}(M \boxtimes N), \quad M *_{!} N = Rm_{!}(M \boxtimes N)$$

for objects $M$ and $N$ of $D_c^b(G)$.

If $G$ is projective, then so is the morphism $m$, and hence the two convolutions agree. In general, there is a canonical “forget supports” morphism

$$M *_{!} N \to M *_{*} N.$$

We will write $M *_{!} N = M *_{*} N$ when this morphism is an isomorphism.

If $u$ is a quasi-projective immersion of $G$, then we deduce from Theorem 1.5 (8) that for any objects $M$ and $N$, the following estimates hold:

$$c_u(M *_{*} N) \ll c_u(M)c_u(N), \quad c_u(M *_{!} N) \ll c_u(M)c_u(N),$$

where the implied constant depends on $G$, and is uniform in families (see [115, § 6.5]).

For an object $M$ of $D_c^b(G)$, we define

$$M^{\vee} = \text{inv}^{*} D(M),$$

<footnote n="1">¹In this book, “unipotent” only applies to commutative groups</footnote>
where $D(M)$ is the Verdier dual. Since $\mathrm{inv}^* = \mathrm{inv}^1$ commutes with $D$, the functor $M \mapsto M^\vee$ is an involution, in the sense that the functor $M \mapsto (M^\vee)^\vee$ is canonically isomorphic to the identity.

We denote by $\mathbf{1}$ the skyscraper sheaf supported at the neutral element $e$ of $G$.

The basic formal properties of the convolution products are given by the following lemma:

LEMMA 1.15. Let $M$ and $N$ be objects of $\mathbf{Perv}(G)$. There exist canonical isomorphisms

$$ (1.2) \qquad \mathrm{Hom}(\mathbf{1}, M^{\vee} *_{*} N) \simeq \mathrm{Hom}(M, N) \simeq \mathrm{Hom}(M *_{!} N^{\vee}, \mathbf{1}), $$

$$ (1.3) \qquad \mathrm{D}(M *_{*} N) \simeq \mathrm{D}(M) *_{!} \mathrm{D}(N), \quad \mathrm{D}(M *_{!} N) \simeq \mathrm{D}(M) *_{*} \mathrm{D}(N), $$

$$ (1.4) \qquad H_c^*(G_{\bar{k}}, M) \otimes_{\bar{Q}_{\ell}} H_c^*(G_{\bar{k}}, N) \simeq H_c^*(G_{\bar{k}}, M *_{!} N), $$

$$ (1.5) \qquad H^*(G_{\bar{k}}, M) \otimes_{\bar{Q}_{\ell}} H^*(G_{\bar{k}}, N) \simeq H^*(G_{\bar{k}}, M *_{*} N). $$

In the first isomorphisms, the $\mathrm{Hom}$ spaces are taken in the category $\mathrm{D}_c^b(G)$.

PROOF. All these are consequences of the formal properties of the six operations on $\mathrm{D}_c^b(G)$. More precisely, all can be found in [70, 8.1.8, 8.1.9], except for the first statement. This is proved for tori in [50, p. 533]; however, the argument applies verbatim to any $G$, since it only uses formal properties of the six operations on $\mathrm{D}_c^b(G)$. $\square$

From the adjunctions in (1.2) of Lemma 1.15, we see that for all $M \in \mathbf{Perv}(G)$, the identity morphism $\mathrm{id}_M: M \to M$ defines evaluation and coevaluation morphisms

$$ \mathrm{ev}: M *_{!} M^{\vee} \to \mathbf{1} \quad \text{and} \quad \mathrm{coev}: \mathbf{1} \to M^{\vee} *_{*} M. $$

As a consequence of these properties, we note that $\mathrm{D}_c^b(G)$ is a symmetric monoidal $\overline{\mathbb{Q}}_{\ell}$-linear category with respect to either the convolution $(A, B) \mapsto A *_{!} B$ or $(A, B) \mapsto A *_{*} B$,

## 1.6. Character groups

In this section, we denote by $k$ a finite field, by $\bar{k}$ an algebraic closure of $k$, and by $k_n$ the extension of degree $n$ of $k$ in $\bar{k}$. Let $\ell$ be a prime number distinct from the characteristic of $k$.

Let $G$ be a connected commutative algebraic group defined over $k$. For each $n \ge 1$, the *norm map* is the group homomorphism $N_{k_n/k}: G(k_n) \to G(k)$ defined as

$$ N_{k_n/k}(x) = \prod_{i=0}^{n-1} \mathrm{Fr}_{k_n}^i(x). $$

For any $n \ge 1$, let $\hat{G}(k_n)$ be the group of characters $\chi: G(k_n) \to \overline{\mathbb{Q}}_{\ell}^\times$. We denote by $\hat{G}$ the disjoint union

$$ \hat{G} = \bigsqcup_{n \ge 1} \hat{G}(k_n) $$

(note that this is not a group; we also omit the dependency on $\ell$ from this notation).

Given any set $S \subset \hat{G}$, we also define $S(k_n) = S \cap \hat{G}(k_n)$, so that $S$ is the disjoint union of the subsets $S(k_n)$.

Since $G$ is geometrically irreducible (see, e.g., [106, Cor. 1.35]), the estimate

$$ |\hat{G}(k_n)| = |G(k_n)| = |k|^{n \dim(G)} + O(|k|^{(n-1/2) \dim(G)}) $$

holds for $n \ge 1$ by the Lang-Weil estimates. If $G$ is an abelian variety, we have more precisely

$$ (|k|^{1/2} - 1)^{2n \dim(G)} \le |\hat{G}(k_n)| \le (|k|^{1/2} + 1)^{2n \dim(G)} $$
and if G is a torus, then

$$ (|k| - 1)^{n \dim(G)} \leq |\hat{G}(k_n)| \leq (|k| + 1)^{n \dim(G)}. $$

These can be derived from the computation of the étale cohomology of abelian varieties combined with the trace formula, or from Steinberg's formula for tori; see, for instance, [105, Th. 15.1, Th. 19.1] for the case of abelian varieties and [19, Prop. 3.3.5] for the case of tori.

We now recall from [27, Sommes trig., 1.4] the *Lang torsor* construction and the basic properties of the associated character sheaves. There is an exact sequence of commutative algebraic groups<fnref n="2" />

$$ 1 \to G(k) \to G \xrightarrow{\mathcal{L}} G \to 1, $$

where $\mathcal{L}$ is the Lang isogeny $x \mapsto \text{Fr}_k(x) \cdot x^{-1}$. The Lang isogeny is a Galois étale covering with Galois group $G(k)$, and hence induces a surjective map $\pi_1^{\text{ét}}(G, e) \to G(k)$. Given a character $\chi \in \hat{G}(k)$, we denote by $\mathcal{L}_\chi$ the $\ell$-adic lisse sheaf of rank one on $G$ obtained by composing this map with $\chi^{-1}$ and we say that $\mathcal{L}_\chi$ is the *character sheaf* on $G$ associated to $\chi$.

For $x \in G(k)$, the geometric Frobenius automorphism at $x$ acts on the stalk of $\mathcal{L}_\chi$ at $x$ by multiplication by $\chi(x)$. In particular, the lisse sheaf $\mathcal{L}_\chi$ is pure of weight zero.

If $\chi$ is the trivial character, then $\mathcal{L}_\chi$ is the constant sheaf $\overline{\mathcal{Q}}_\ell^\times$.

The dual $\text{D}(\mathcal{L}_\chi)$ of a character sheaf is isomorphic to $\mathcal{L}_{\chi^{-1}}$, and for any two characters $\chi_1$ and $\chi_2$ there are canonical isomorphisms

$$ \mathcal{L}_{\chi_1} \otimes \mathcal{L}_{\chi_2} \simeq \mathcal{L}_{\chi_1 \chi_2}. $$

If $n \geq 1$ and $\chi \in \hat{G}(k_n)$ is non-trivial, then for all $i \in \mathbb{Z}$, the cohomology space $H_c^i(\hat{G}, \mathcal{L}_\chi)$ vanishes (see [27, Sommes trig., Th. 2.7*]). More generally, we have the following relative version.

**LEMMA 1.16.** Let $f: G \to H$ be a surjective morphism of commutative algebraic groups over $k$. Let $\chi \in \hat{G}(k)$. The complex $\text{Rf}_!\mathcal{L}_\chi$ vanishes unless $\mathcal{L}_\chi|_{\ker(f)^\circ}$ is the constant sheaf, i.e., unless $\chi$ is trivial on $\ker(f)^\circ$.

**PROOF.** Let $M = \text{Rf}_!\mathcal{L}_\chi$. Let $y \in H$ and let $z \in G$ be such that $f(z) = y$. By the proper base change theorem, the stalk of $M$ at $y$ is given by

$$ M_y = H_c^*(f^{-1}(y)_{\bar{k}}, \mathcal{L}_\chi) = H_c^*((z \ker(f))_{\bar{k}}, \mathcal{L}_\chi) = H_c^*(\ker(f)_{\bar{k}}, [x \mapsto xz]^* \mathcal{L}_\chi|_{\ker(f)}). $$

We write $\ker(f)$ as the disjoint union of cosets $u \ker(f)^\circ$ where $u$ runs over a set of representatives of the group of connected components of $\ker(f)$. Thus,

$$ H_c^*(\ker(f)_{\bar{k}}, [x \mapsto xz]^* \mathcal{L}_\chi|_{\ker(f)}) = \bigoplus_u H_c^*(\ker(f)_{\bar{k}}^\circ, [x \mapsto xuz]^* \mathcal{L}_\chi|_{\ker(f)^\circ}). $$

Since $\mathcal{L}_\chi$ is a character sheaf, the sheaf $[x \mapsto xuz]^*\mathcal{L}_\chi$ is geometrically isomorphic to $\mathcal{L}_\chi$, so that we have an isomorphism

$$ M_y \simeq \bigoplus_u H_c^*(\ker(f)_{\bar{k}}^\circ, \mathcal{L}_\chi|_{\ker(f)^\circ}), $$

and the result now follows from [27, Sommes trig., Th. 2.7*] as recalled above. $\square$

<footnote n="2">² Note that it is here that the assumption that G is connected plays a role, since in general the image of the morphism $\mathcal{L}$ is equal to the connected component of the neutral element.</footnote>
Let $n \ge 1$ and $\chi \in \hat{G}(k)$. The base change of $\mathcal{L}_\chi$ to $G_{k_n}$ is the character sheaf on $G_{k_n}$ associated to the character $\chi \circ N_{k_n/k}$ of $G(k_n)$. In particular, the trace function of $\mathcal{L}_\chi$ on $k_n$ is given by

$$t_{\mathcal{L}_{\chi}}(x; k_n) = \chi(N_{k_n/k}(x)).$$

When there is no risk of confusion, we will still denote by $\mathcal{L}_\chi$ the pullback of the character sheaf associated to $\chi$ to $\bar{k}$. The previous remark shows that $\chi$ and $\chi \circ N_{k_n/k}$ give rise to the same base change to $\bar{k}$.

Let $f: G \to H$ be a homomorphism of commutative algebraic groups defined over $k$. For any integer $n \ge 1$, let us denote by $f_n$ the induced morphism $G(k_n) \to H(k_n)$; then we have dual homomorphisms $\hat{f}_n: \hat{H}(k_n) \to \hat{G}(k_n)$ defined by $\chi \mapsto \chi \circ f_n$. The combination of all these maps gives a map $\hat{f}: \hat{H} \to \hat{G}$, which we will often denote simply by $\chi \mapsto \chi \circ f$. We will sometimes say that a character $\chi \in \hat{G}$ arises from $H$ if $\chi$ belongs to the image of $\hat{f}$.

For $\chi \in \hat{H}(k_n)$, there is a canonical isomorphism $\mathcal{L}_{\hat{f}(\chi)} \simeq f^*\mathcal{L}_\chi$.

For any object $M$ of $\mathcal{D}_c^b(G)$ and any character $\chi$ of $G(k)$, we denote by

$$M_{\chi} = M \otimes \mathcal{L}_{\chi}$$

the “twist” of $M$ by the character sheaf $\mathcal{L}_\chi$.

For all $\chi \in \hat{G}$, and all objects $M$ and $N$ of $\mathcal{D}_c^b(G)$ (or $\mathcal{D}_c^b(G_{\bar{k}})$), there are canonical isomorphisms
(1.6) $$D(M_{\chi}) \simeq D(M)_{\chi^{-1}},$$

$$(1.7) \qquad (M_{\chi})^{\vee} \simeq (M^{\vee})_{\chi},$$

$$(1.8) \qquad (M *_{\chi} N)_{\chi} \simeq (M_{\chi} *_{\chi} N_{\chi}), \quad (M *_{\chi} N)_{\chi} \simeq (M_{\chi} *_{\chi} N_{\chi}).$$

The first two properties follow from duality from $D(\mathcal{L}_{\chi}) = \mathcal{L}_{\chi^{-1}}$, and the third from the projection formula combined with the canonical isomorphism $m^*\mathcal{L}_{\chi} \simeq \mathcal{L}_{\chi} \boxtimes \mathcal{L}_{\chi}$, where $p_1$ and $p_2$ are the projections $G \times G \to G$ (see [70, 8.1.10 (4)]).

More generally, for any algebraic variety $X$ over $k$, any morphism $f: X \to G$, and any object $M$ of $\mathcal{D}_c^b(X)$, we set

$$M_{\chi} = M \otimes f^{*}\mathcal{L}_{\chi},$$

and we use the same notation for objects in $\mathcal{D}_c^b(G_{\bar{k}})$ and $\mathcal{D}_c^b(X_{\bar{k}})$, or in $\mathcal{D}_c^b(G_{k_n})$ and $\mathcal{D}_c^b(X_{k_n})$.

We will extensively (and often without comment) use the following standard lemma.

**LEMMA 1.17.** Let $f: X \to G$ be a morphism from an algebraic variety $X$ to a connected commutative algebraic group $G$, both defined over $k$. Let $\chi \in \hat{G}$ be a character. Then the functor $M \mapsto M_{\chi}$ on $\mathcal{D}_c^b(X)$ or $\mathcal{D}_c^b(X_{\bar{k}})$ is t-exact for the standard and perverse t-structures. In particular, if $M$ is perverse (resp. semiperverse) then so is $M_{\chi}$.

**PROOF.** Let $i \in \mathbb{Z}$. Since $\mathcal{L}_{\chi}$ is a lisse sheaf on $G$, the pullback $f^*\mathcal{L}_{\chi}$ is lisse on $X$, and hence tensoring with $f^*\mathcal{L}_{\chi}$ is exact for the standard t-structure on $\mathcal{D}_c^b(X)$ or $\mathcal{D}_c^b(X_{\bar{k}})$ (i.e., the t-structure whose heart is the category of constructible sheaves concentrated in degree 0). There are thus canonical isomorphisms $\mathcal{H}^i(M \otimes f^*\mathcal{L}_{\chi}) \simeq \mathcal{H}^i(M) \otimes f^*\mathcal{L}_{\chi}$ for all $i$. Hence, looking at the support, we see that the functor $M \mapsto M_{\chi}$ is right t-exact for the perverse t-structure. It is also left t-exact since $D(M_{\chi})$ is isomorphic to $D(M)_{\chi^{-1}}$, hence the result. $\square$

## 1.7. Complexity estimates for character sheaves

We keep the notation of the previous section. The first essential new ingredient for our work is the fact that the complexity of character sheaves on $G$ is uniformly bounded.
PROPOSITION 1.18. Let G be a connected commutative algebraic group over k together with a quasi-projective immersion u. There exists a real number C ≥ 0 such that, for every n ≥ 1 and for every character χ ∈ $\hat{G}(k_n)$, the inequality $c_u(\mathcal{L}_\chi) \le C$ holds.

PROOF. We will proceed in several steps, first noting that we may assume that $n = 1$.

(1) If the result is true for the groups $G_1$ and $G_2$, then it is true for their product $G = G_1 \times G_2$. Indeed, let $p_i: G \to G_i$ denote the two projections. Since any character $\chi$ of $G(k)$ takes the form $(x_1, x_2) \mapsto \chi_1(x_1)\chi_2(x_2)$ for some characters $\chi_i$ of $G_i(k)$, the corresponding character sheaf is the external product $\mathcal{L}_\chi = \mathcal{L}_{\chi_1} \boxtimes \mathcal{L}_{\chi_2}$, which has complexity bounded in terms of the complexity of $\mathcal{L}_{\chi_1}$ and that of $\mathcal{L}_{\chi_2}$, and hence bounded uniformly by assumption.

(More precisely, this is one case where we use Remark 1.6, since we most easily bound the complexity of $\mathcal{L}_{\chi_1} \boxtimes \mathcal{L}_{\chi_2}$ with respect to the composition $v$ of the given quasi-projective immersions $u_1$ and $u_2$ of $G_1$ and $G_2$ and the Segre embedding using Theorem 1.5, as in [115, Prop. 6.12].)

(2) If the result holds for a group G, then for any finite subgroup scheme H (defined over $k$), the results holds for the quotient G/H (if this quotient is an algebraic group). To see this, we can further decompose $H = H_r \times H_l$ where $H_r$ is reduced and $H_l$ is local, so that we may assume that H is either reduced or local. Let $v$ be a quasi-projective embedding of G/H and let $\pi: G \to G/H$ be the quotient morphism.

If H is reduced, then $\pi$ is a finite étale covering, so for any lisse sheaf $\mathcal{L}$ on G/H, the sheaf $\mathcal{L}$ is a direct factor of $\pi_*\pi^*\mathcal{L}$, and we deduce

$$c_v(\mathcal{L}) \le c_v(\pi_*\pi^*\mathcal{L}) \ll c_u(\pi^*\mathcal{L}).$$

This implies the result since $\pi^*\mathcal{L}$ is a character sheaf on G if $\mathcal{L}$ is a character sheaf on G/H.

If H is local, then the quotient morphism $\pi$ is finite and radicial, and hence the adjunction map $\mathcal{L} \to \pi_*\pi^*\mathcal{L} = \pi_1\pi^*\mathcal{L}$ is an isomorphism (see, e.g., [47, Cor. 5.3.10]). By Theorem 1.5 (6), the complexity $c_v(\mathcal{L}) = c_v(\pi_1\pi^*\mathcal{L})$ is hence $\ll c_u(\pi^*\mathcal{L})$, and the result again follows.

(3) The result is valid for tori and unipotent groups. For the former, since complexity is a geometric invariant, we may assume that we have a split torus, and the result then follows from (1) and the case of $G = G_m$, which is established in [115, Prop. 7.5].

Assume then that G is a unipotent group. Let $G^\vee$ be its Serre dual (or more precisely, an algebraic group model of it; see Section 2.2 for details). There exists a lisse $\ell$-adic sheaf $\mathcal{L}$ of rank one on $G^\vee \times G$ such that the character sheaves associated to characters of $G(k)$ are in bijection with the points $a \in G^\vee(k)$ by mapping $a \in G^\vee(k)$ to the restriction of the sheaf $\mathcal{L}$ to $\{a\} \times G$. Hence, by Theorem 1.5, the complexity of any character sheaf of G is bounded in terms of the complexity of the single sheaf $\mathcal{L}$.

(4) The result holds for abelian varieties by [115, Prop. 7.9], since abelian varieties are projective and any character sheaf is lisse on G.

(5) The general case now follows using the previous results and the dévissage of Proposition 1.13. This completes the proof of the proposition. $\square$

REMARK 1.19. A potential alternative (more conceptual) approach to this result would be the following. For a character sheaf $\mathcal{L}$ on G, there is an isomorphism

$$m^*\mathcal{L} \simeq p_1^*\mathcal{L} \otimes p_2^*\mathcal{L}$$

(recall that $m$ is the multiplication map $G \times G \to G$). If one could prove directly the estimate

$$(1.9) \qquad c(\mathcal{L})^2 \ll c(p_1^*\mathcal{L} \otimes p_2^*\mathcal{L}),$$
then we would deduce from Theorem 1.5 that

$$c(\mathcal{L})^2 \ll c(m^*\mathcal{L}) \ll c(\mathcal{L}),$$

and hence $c(\mathcal{L}) \ll 1$. Note that Proposition 1.18 shows that (1.9) is indeed true, and it is maybe not out of the question that one could provide a direct proof.

## 1.8. Arithmetic Fourier transforms

We continue with the notation of the previous section. Given an $\ell$-adic complex M in $D_c^b(G)$, we can consider for any fixed $n \ge 1$ the discrete Fourier transform of the trace function $x \mapsto t_M(x; k_n)$ on $G(k_n)$, which we normalize to be the function from $\hat{G}(k_n)$ to $\overline{\mathbf{Q}}_\ell$, or $\mathbf{C}$, defined by

$$\chi \mapsto S(M, \chi) = \sum_{x \in G(k_n)} \chi(x) t_M(x; k_n).$$

This Fourier transform satisfies the usual formalism of commutative harmonic analysis (see, e.g., [14]). For instance the Fourier inversion formula

$$(1.10) \qquad t_M(x; k_n) = \frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} S(M, \chi) \bar{\chi}(x)$$

holds for any $x \in G(k_n)$, and there is also a Plancherel formula

$$\sum_{x \in G(k_n)} |t_M(x; k_n)|^2 = \frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} |S(M, \chi)|^2.$$

Putting together the data of these discrete Fourier transforms on $G(k_n)$ for all $n \ge 1$, we obtain what we call the arithmetic Fourier transform of the complex M, an element of the product set

$$\mathcal{C}(\hat{G}, \overline{\mathbf{Q}}_\ell) = \prod_{n \ge 1} \mathcal{C}(\hat{G}(k_n), \overline{\mathbf{Q}}_\ell),$$

where, for any set X and ring A, we denote by $\mathcal{C}(X, A)$ the A-module of functions $f: X \to A$.

Combining the Fourier inversion formula (1.10) with the known injectivity theorem for trace functions (see Proposition A.22), we deduce a corresponding injectivity property of the discrete Fourier transform of complexes:

**PROPOSITION 1.20.** Let $M_1$ and $M_2$ be complexes in $D_c^b(G)$ such that for all $n \ge 1$ and all characters $\chi \in \hat{G}(k_n)$, the equality

$$\sum_{x \in G(k_n)} \chi(x) t_{M_1}(x; k_n) = \sum_{x \in G(k_n)} \chi(x) t_{M_2}(x; k_n)$$

holds. Then the classes of $M_1$ and $M_2$ in the Grothendieck group $K(G) = K(G, \overline{\mathbf{Q}}_\ell)$ are equal.

**REMARK 1.21.** In Chapter 6, we will establish a more refined statement where the equality of discrete Fourier transforms is only assumed to hold for characters in a “generic” set, as described in the next section.
## 1.9. Generic sets of characters

For an arbitrary connected commutative algebraic group, there is no obvious topology (or measure) on the set $\hat{G}$ of characters which would lead to a natural notion of sets containing “almost all” characters. We will use instead the following definition of a generic set of characters.

**DEFINITION 1.22.** Let $k$ be a finite field and let $G$ be a connected commutative algebraic group of dimension $d$ over $k$. Let $S$ be a subset of $\hat{G}$.

Let $i \ge 0$ be an integer. We say that $S$ has character codimension at least $i$, which we denote sometimes by $\mathrm{ccodim}(S) \ge i$, if the estimate

$$ (1.11) \qquad |S(k_n)| \ll |k|^{n(d-i)} $$

holds for all integers $n \ge 1$.

We say that $S$ is *generic* if $\hat{G} - S$ has character codimension at least 1, i.e., if the estimate

$$ (1.12) \qquad |\hat{G}(k_n) - S(k_n)| \ll |k|^{n(d-1)} $$

holds for all integers $n \ge 1$.

We now discuss the relation between the definition of generic sets and other notions that appear in the literature, in the case of unipotent and semiabelian varieties.

If $G$ is unipotent, then the set of characters can be identified with the $\bar{k}$-points of a $k$-scheme $G^\vee$; see again Section 2.2. If $S \subset \hat{G}$ is algebraic (i.e., the disjoint union of the sets $\tilde{S}(k_n)$ for some subvariety $\tilde{S}$ of $G$), then the condition $\mathrm{ccodim}(S) \ge i$ implies that the codimension of $\tilde{S}$ in $G^\vee$ is at least $i$. Conversely, if $\tilde{S}$ is a closed subvariety of $G^\vee$ over $k$, then $\mathrm{ccodim}(\tilde{S}(\bar{k})) \ge \mathrm{codim}_{G^\vee}(\tilde{S})$.

Let $G$ be a semiabelian variety over $k$. Let $\ell$ be a prime different from the characteristic of $k$. The set of $\ell$-adic characters of $G$ can be naturally identified with the set of $\overline{\mathbb{Q}}_\ell$-points of a $\overline{\mathbb{Q}}_\ell$-scheme, as we now recall. Let $\pi_1^t(G_{\bar{k}})$ be the geometric tame étale fundamental group of $G$ (see, for instance, the paper [82] of Kerz and Schmidt for various equivalent definitions; note that it is well-known that semiabelian varieties have good compactifications), and let $\Pi(G, \overline{\mathbb{Q}}_\ell)$ be the group of continuous characters $\chi: \pi_1^t(G_{\bar{k}}) \to \overline{\mathbb{Q}}_\ell^\times$. For any $n \ge 1$ and $\chi \in \hat{G}(k_n)$, the character sheaf $\mathcal{L}_\chi$ is tamely ramified (indeed, only the case of tori requires proof; since the question is geometric, we may assume that $G = \mathbb{G}_m^d$ for some integer $d \ge 0$, and the result follows by induction from the well-known case of $\mathbb{G}_m$ and the multiplicativity of the tame fundamental group [108, Th. 5.1]), and hence corresponds to a point in $\Pi(G, \overline{\mathbb{Q}}_\ell)$. For each $n \ge 1$, this leads to a natural injective map

$$ \hat{G}(k_n) \hookrightarrow \Pi(G, \overline{\mathbb{Q}}_\ell), $$

and we will identify $\hat{G}(k_n)$ this way with a subset of $\Pi(G, \overline{\mathbb{Q}}_\ell)$.

There is a decomposition

$$ \Pi(G, \overline{\mathbb{Q}}_\ell) = \Pi(G, \overline{\mathbb{Q}}_\ell)_{\ell'} \times \Pi(G, \overline{\mathbb{Q}}_\ell)_\ell, $$

where $\Pi(G, \overline{\mathbb{Q}}_\ell)_{\ell'}$ is the group of torsion characters of order prime to $\ell$ and $\Pi(G, \overline{\mathbb{Q}}_\ell)_\ell$ is the group of characters that factor through the maximal pro-$\ell$ quotient $\pi_1^t(G_{\bar{k}})_\ell$ of $\pi_1^t(G_{\bar{k}})$. Since $\pi_1^t(G_{\bar{k}})_\ell$ is a free $\mathbb{Z}_\ell$-module of finite rank, by a result of Brion and Szamuely [18], we can identify $\Pi(G, \overline{\mathbb{Q}}_\ell)_\ell$ with the $\overline{\mathbb{Q}}_\ell$-points of a scheme $\Pi(G)_\ell$, following the arguments of Gabber and Loeser [50, Section 3.3].

Letting $\Pi(G)$ be the disjoint union of the schemes $\Pi(G)_\ell$ indexed by $\chi \in \Pi(G, \overline{\mathbb{Q}}_\ell)_{\ell'}$, we get

$$ \Pi(G, \overline{\mathbb{Q}}_\ell) = \Pi(G)(\overline{\mathbb{Q}}_\ell), $$

and as above we will identify $\hat{G}$ with a subset of $\Pi(G)(\overline{\mathbb{Q}}_\ell)$.
Let $G'$ be a semiabelian variety over $k$ and $f: G \to G'$ a homomorphism. There is a dual morphism $\Pi(G') \to \Pi(G)$, denoted by $\chi \mapsto \chi \circ f$; if $f$ is an inclusion, we also write simply $\chi \circ f = \chi|_G$. The restriction of this map to the subset $\hat{G}'$ is the map $\hat{f}: \hat{G}' \to \hat{G}$ previously defined.

**DEFINITION 1.23.** Let $G$ be a semiabelian variety over a finite field $k$, and let $\ell$ be a prime different from the characteristic of $k$.

(1) A subset $S \subset \Pi(G)(\overline{\mathbb{Q}}_\ell)$ is a translate of an algebraic cotorus (abbreviated tac) if there exists a surjective morphism $\pi: G_{\bar{k}} \to G'$ of semiabelian varieties over $\bar{k}$, with non-trivial connected kernel, and a character $\chi_0 \in \Pi(G)(\overline{\mathbb{Q}}_\ell)$ such that

$$S = \{\chi_0 \cdot (\chi' \circ \pi) \in \Pi(G)(\overline{\mathbb{Q}}_\ell) \mid \chi' \in \Pi(G')(\overline{\mathbb{Q}}_\ell)\}.$$

We then say that $S$ is *defined by the quotient* $G_{\bar{k}} \to G'$ *and the character* $\chi_0$, and that $S$ *has dimension* $\dim(G')$. The kernel of $\pi$ is also called the *kernel of the tac*. If $G'$ and $\pi$ are defined over a finite extension $k'$ of $k$, then we say that $S$ is a *tac of* $G_{k'}$.

(2) We say that a subset $S \subset \Pi(G)(\overline{\mathbb{Q}}_\ell)$ contains *most characters* if the complement of $S$ is contained in a finite union of tacs.

(3) We say that a subset $S \subset \Pi(G)(\overline{\mathbb{Q}}_\ell)$ is *weakly generic* if it is a generic set in the sense of the Zariski topology in $\Pi(G)$, i.e it contains a dense open subset of $\Pi(G)$.

By extension, we shall say that a subset $S \subset \hat{G}$ contains *most characters*, or is *weakly generic*, if its image in $\Pi(G)(\overline{\mathbb{Q}}_\ell)$ satisfies this property.

**REMARK 1.24.** (1) The terminology “most” is used by Krämer and Weissauer [94]; Esnault and Kerz [36] speak of “quasi-linear” subsets. What we call “weakly generic” is usually called “generic” (see, for example, the papers [94], [91] and [50]).

(2) Let $S \subset \Pi(G)(\overline{\mathbb{Q}}_\ell)$ be a subset that contains most characters. The Lang-Weil estimates imply that $S \cap \hat{G}$ is generic in the sense of (1.12). Also, if $S \subset \hat{G}$ is a generic set and $\Pi(G)(\overline{\mathbb{Q}}_\ell) - S$ is not Zariski-dense, then $S$ is weakly generic.

(3) The tac defined by $\pi$ and $\chi_0$ can also be interpreted as the set of characters $\chi$ such that the restriction of $\chi$ to $\ker(\pi)$ is equal to that of $\chi_0$.

(4) If a tac $S$ of $G$ has dimension $i$, then $S \cap \hat{G}$ has character codimension $\ge \dim(G) - i$ since

$$|(S \cap \hat{G})(k_n)| \le |G'(k_n)| \ll |k|^{ni}$$

if $S$ is defined by the quotient $G \to G'$ and the character $\chi_0$.

**LEMMA 1.25.** Let $G$ be a semiabelian variety over a finite field $k$. Let $\ell$ be a prime different from the characteristic of $k$. Let $I$ be a non-empty finite set and let $(S_i)_{i \in I}$ be a family of tacs in $G$, defined by quotient morphisms $\pi_i: G_{\bar{k}} \to G_{i, \bar{k}}$ and characters $\chi_i \in \Pi(G)(\overline{\mathbb{Q}}_\ell)$.

Let $K$ be the subgroup of $G_{\bar{k}}$ generated by the subgroups $\ker(\pi_i)$. The intersection $S = \bigcap S_i$ is non-empty if and only if the restriction of $\chi_i$ to $K$ is independent of $i$.

If this is the case, then $S$ is a tac, which is defined by the quotient morphism $\pi: G_{\bar{k}} \to G_{\bar{k}}/K$ and any of the characters $\chi_i$.

**PROOF.** We write $K_i = \ker(\pi_i)$ for $i \in I$. Since each $K_i$ is connected by definition, the subgroup $K$ generated by the $K_i$ is also connected.

A character $\chi \in \Pi(G)(\overline{\mathbb{Q}}_\ell)$ belongs to $S_i$ if and only if $\chi|_{K_i} = \chi_i|_{K_i}$. If $\chi \in S$, then the restriction of $\chi_i$ to $K$ must coincide with the restriction of $\chi$ to $K$, and is therefore independent of $i$.
Conversely, if this condition is satisfied, then pick any $i_0 \in I$. The tac defined by $G_{\bar{k}} \to G_{\bar{k}}/K$ and the character $\chi_{i_0}$ consists of characters $\chi$ such that $\chi|_{K} = \chi_{i_0}|_{K}$. This condition is equivalent to $\chi|_{K_i} = \chi_{i_0}|_{K_i}$ for all $i \in I$. Since $\chi_i|_{K_i} = \chi_{i_0}|_{K_i}$, this tac is exactly the intersection of the $S_i$. $\square$

## 1.10. Fourier-Mellin transforms on semiabelian varieties

Let $k$ be a finite field and $G$ a semiabelian variety over $k$. Let $\ell$ be a prime different from the characteristic of $k$. We use the notation of the previous section.

We recall here some results of Gabber and Loeser for tori [50], generalized by Krämer [91] to semiabelian varieties.

Let $R$ be the ring of integers of a finite extension of $\mathbb{Q}_\ell$ and $\Omega_G = R[\pi_1^t(G_{\bar{k}})_\ell]$. We have

$$ \Pi(G)_\ell = \mathrm{Spec}(\overline{\mathbb{Q}}_\ell \otimes_R \Omega_G). $$

Let $p: G_{\bar{k}} \to \mathrm{Spec}(\bar{k})$ be the structural morphism. We denote by $\mathrm{can}_G$ the tautological character

$$ \mathrm{can}_G: \pi_1^t(G_{\bar{k}})_\ell \to \Omega_G^\times, $$

which defines a lisse $\Omega_G$-sheaf of rank one $\mathscr{L}_G$ on $G_{\bar{k}}$. Given an object $N$ of $\mathrm{D}_c^b(G_{\bar{k}}$, $R)$, one defines the *Fourier-Mellin transforms* of $N$, with and without compact support, as the objects

$$ \mathrm{FM}_!(N) = \mathrm{R}p_!(N \otimes_R \mathscr{L}_G) $$

$$ \mathrm{FM}_*(N) = \mathrm{R}p_*(N \otimes_R \mathscr{L}_G) $$

of the category $\mathrm{D}_c^b(\bar{k}, \Omega_G) = \mathrm{D}_{\mathrm{coh}}^b(\Omega_G)$. Inverting $\ell$ and passing to the direct limit over all $R \subset \overline{\mathbb{Q}}_\ell$ and all $\chi \in \Pi(G, \overline{\mathbb{Q}}_\ell)_\ell'$, we then get two functors

$$ \mathrm{FM}_!, \mathrm{FM}_*: \mathrm{D}_c^b(G_{\bar{k}}) \to \mathrm{D}_{\mathrm{coh}}^b(\Pi(G)), $$

where $\mathrm{D}_{\mathrm{coh}}^b(\Pi(G))$ is the derived category of the category of coherent sheaves on $\Pi(G)$.

By (the generalization of) [50, Cor. 3.3.2], for an object $N$ of $\mathrm{D}_c^b(G_{\bar{k}})$ and every $\chi \in \Pi(G)(\overline{\mathbb{Q}}_\ell)$, viewed as a closed immersion $i_\chi: \{\chi\} \to \Pi(G)$, there are canonical isomorphisms

$$ \mathrm{Li}_\chi^* \mathrm{FM}_!(N) \simeq \mathrm{R}p_!(N_\chi) \quad \text{and} \quad \mathrm{Li}_\chi^* \mathrm{FM}_*(N) \simeq \mathrm{R}p_*(N_\chi), $$

where $\mathrm{Li}_\chi$ indicates left-derived functors.

## 1.11. A geometric lemma

A connected commutative algebraic group $G$ is said to be *almost simple* if it has no proper connected closed subgroup. Examples of such groups are $G_a$, $G_m$ and simple abelian varieties.

We will use the following lemma in the proof of the general higher vanishing theorem.

**LEMMA 1.26.** Let $k$ be a field. Let $s \ge 0$ be an integer. We denote $[s] = \{1, \dots, s\}$. Let

$$ G = \prod_{i=1}^{s} G_i $$

be a product of almost simple connected commutative algebraic groups over $k$. Let $d = \dim(G)$.

For any subset $I \subset [s]$, let

$$ G_I = \prod_{i \in I} G_i, $$

which we identify with a subgroup of $G$ in the obvious way.
Let $1 \le i \le d$. Let $\mathcal{E}_i$ be the set of subsets $I$ such that $\dim(G_I) > d - i$. For each $I \in \mathcal{E}_i$, let $H_I$ be a non-trivial connected subgroup of $G_I$. Then the algebraic subgroup $H$ generated by all $H_I$ has dimension at least $i$.

PROOF. We denote $d_i = \dim(G_i)$ for $1 \le i \le s$.

We work by induction on $s$, and for each $s$, by induction on $i$. The case $s = 1$ is elementary, since $\{1\} \in \mathcal{E}_i$ then, hence $H = H_{[s]} = G$ in that case. For any $s$, the result is also elementary for $i = 1$, since for $I = [s] \in \mathcal{E}_1$, we have $\dim(H) \ge \dim(H_I) \ge 1$. Assume now that $2 \le i \le g$ and that the result is known for $(s, i')$ for $i' < i$ as well as for $(s', i)$ for any $s' < s$.

The subgroup $H_{[s]} \subset G$ is non-trivial, and hence there exists some integer $j \le s$ such that the image of $H_{[s]}$ under the projection $G \to G_j$ is non-trivial; this means that this image must be equal to $G_j$ since all $G_i$ are almost simple. Up to reordering the factors, we may assume that the projection of $H_{[s]}$ on $G_s$ is surjective.

If $d_s \ge i$, then we are done since we then have $\dim(H_{[s]}) \ge \dim(G_s) = d_s \ge i$. We therefore assume now that $d_s < i$.

Let $G' = G_1 \times \cdots \times G_{s-1}$ and $i' = i - d_s$. The dimension of $G'$ is $d' = d - d_s$. We have $1 \le i' \le d'$ and $d - i = d' - i'$. Each $J \subset [s-1]$ with $\dim(G'_J) > d' - i' = d - i$ is an element of $\mathcal{E}_i$. By induction, applied to the subgroups $H_J$ for $J \in \mathcal{E}_{i'}$, the subgroup $H'$ of $G'$ generated by all $H_J$ has dimension $\ge i' = i - d_s$.

To conclude, we observe that since $H'$ is a subgroup of $G'$ with dimension $\ge i - d_s$ and $H_{[s]}$ is a subgroup of $G = G' \times G_s$ such that the projection of $H_{[s]}$ to $G_s$ is surjective, the subgroup $H$ that they generate together satisfies

$$ \dim(H) = \dim(H') + \dim(H_{[s]}) - \dim(H' \cap H_{[s]}) \\ \ge \dim(H') + \dim(H_{[s]}) - \dim(G' \cap H_{[s]}) \ge i - d_s + d_s = i $$

$$ \text{since } \dim(G' \cap H_{[s]}) + \dim(G_s) = \dim(H_{[s]}). \quad \square $$

## 1.12. Geometric and arithmetic semisimplicity

Let $k$ be a finite field, and $\bar{k}$ an algebraic closure of $k$. Let $\ell$ be a prime different from the characteristic of $k$.

For an algebraic variety $X$ over $k$ and a complex $M$ in $D_c^b(X, \overline{Q}_\ell)$, we will sometimes refer to properties of $M$ (e.g., $M$ being a simple or semisimple perverse sheaf) as *arithmetic*, and to the analogue for the base change of $M$ to $M_{\bar{k}}$ as being *geometric*. Thus we may speak of a geometrically simple perverse sheaf, or an arithmetically semisimple perverse sheaf.

We collect here some facts about certain relations between such properties.

LEMMA 1.27. Let $X$ a geometrically irreducible algebraic variety over $k$ and $\mathcal{F}$ a lisse $\ell$-adic sheaf on $X$. If $\mathcal{F}$ is arithmetically semisimple, then it is geometrically semisimple.

PROOF. Using the correspondence between lisse sheaves and representations of the étale fundamental group, this follows, e.g., from [119, Lem. 5 (a)]. $\square$

LEMMA 1.28. Let $(X, u)$ be a quasi-projective variety over $k$. Let $M$ be an arithmetically simple perverse sheaf on $X$. There exists a finite extension of $k$ of degree bounded in terms of $c_u(M)$ such that the base change of $M$ to $X_{k'}$ is a direct sum of geometrically simple sheaves on $k'$.

In particular, $M$ is geometrically semisimple.
PROOF. By [8, Prop. 5.3.9 (ii)], there exists an integer $n \ge 1$ and a geometrically simple perverse sheaf N on $X_{k_n}$ such that $M = f_{n*}N$, where $f_n: X_{k_n} \to X$ is the base change morphism. Since N is non-zero, we deduce that $n \ll c_u(M)$ by looking at the rank at a generic point of the support. The base change of M to $k_n$ is then a direct sum of geometrically simple perverse sheaves. $\square$

LEMMA 1.29. Let $k$ be a finite field and $\bar{k}$ an algebraic closure of $k$. Let $\ell$ be a prime different from the characteristic of $k$. Let $X$ be a smooth and geometrically connected quasi-projective variety over $k$. Two perverse sheaves $M$ et $N$ on $X$ which are geometrically simple are geometrically isomorphic if and only if there exists $\alpha \in \overline{\mathbb{Q}}_\ell$ such that $M \simeq \alpha^{\deg} \otimes N$.

This is a standard fact (see, e.g., [103, Lemme 4.4.4]).

1.13. A result from representation theory

The following basic fact from the representation theory of reductive groups will play a crucial role.

PROPOSITION 1.30. Let $F$ be a field of characteristic zero and let $G$ be a reductive algebraic group over $F$. Let $V$ be a finite-dimensional faithful representation of $G$ over $F$. Any finite-dimensional irreducible representation of $G$ over $F$ occurs in a tensor power $(V \oplus V^\vee)^{\otimes m}$ for some integer $m \ge 0$, where $V^\vee$ is the contragredient of $V$.

See, for instance, [29, Prop. 3.1] for the proof.
CHAPTER 2

Generic vanishing theorems

Throughout this chapter, $k$ denotes a finite field, $\bar{k}$ an algebraic closure of $k$, and $k_n$ the extension of degree $n$ of $k$ inside $\bar{k}$ for each $n \ge 1$. We also fix once for all a prime number $\ell$ different from the characteristic of $k$. All complexes of sheaves and characters are tacitly understood to be $\ell$-adic complexes and characters for this choice of $\ell$.

2.1. Statement of the vanishing theorems

We now state our main vanishing theorems.

THEOREM 2.1 (Generic vanishing). Let $G$ be a connected commutative algebraic group over $k$ and let $M$ be a perverse sheaf on $G$. The set $\mathcal{U}$ of characters $\chi \in \hat{G}$ satisfying

$$
\begin{align*}
H^i(G_{\bar{k}}, M_{\chi}) &= H_c^i(G_{\bar{k}}, M_{\chi}) = 0 \quad \text{for all } i \neq 0, \\
H_c^0(G_{\bar{k}}, M_{\chi}) &= H^0(G_{\bar{k}}, M_{\chi})
\end{align*}
$$

is generic in the sense of Definition 1.22.

This gives the first part of Theorem **1** from the introduction.

REMARK 2.2. Versions of Theorem 2.1 have been proved by the following authors:

(1) Katz-Laumon [78, Th. 2.1.3, Scholie 2.3.1] in the case of powers of the additive group and Saibi [112, Th. 3.1] in the case of general unipotent groups; in both cases, the generic set is a Zariski-dense open subset of the $k$-scheme parameterizing characters.

(2) Gabber-Loeser [50, Cor. 2.3.2] for tori, with “generic” replaced by a condition implying “weakly-generic” in the sense of Definition 1.23; see also [50, Th. 7.2.1].

(3) Weissauer [127, Vanishing Th., p. 561] for abelian varieties, with “generic” replaced by “most” characters, and Krämer [91, Th. 2.1] for semiabelian varieties, with “weakly generic” characters.

For our main applications, the second part of Theorem **1** is more important. It is provided by
the following result, which also controls the “stratification” arising from the non-vanishing of other
cohomology groups.

THEOREM 2.3 (Stratified vanishing). Let $G$ be a connected commutative algebraic group of dimension $d$ over $k$, and $M$ a perverse sheaf on $G$. There exist subsets

$$
\mathcal{S}_d \subset \dots \subset \mathcal{S}_0 = \hat{G}
$$

such that the following holds:

(1) For $0 \le i \le d$, the subset $\mathcal{S}_i$ has character codimension at least $i$.
(2) For $0 \le i \le d$, any $\chi \in \hat{G}$ such that at least one of the cohomology groups

$$ (2.1) \qquad H^i(G_{\bar{k}}, M_{\chi}), \quad H^{-i}(G_{\bar{k}}, M_{\chi}), \quad H_c^i(G_{\bar{k}}, M_{\chi}), \quad H_c^{-i}(G_{\bar{k}}, M_{\chi}) \\ \text{is non-zero belongs to } \mathcal{S}_i. $$

(3) For $\chi \in \mathcal{S}_0-\mathcal{S}_1$, the equality $H_c^0(G_{\bar{k}}, M_{\chi}) = H^0(G_{\bar{k}}, M_{\chi})$ holds.<fnref n="1" />

(4) If $G$ is a torus or an abelian variety, then $\mathcal{S}_i$ is a finite union of tacs of $G$ of dimension $\le d - i$.

(5) If $G$ is a unipotent group, then $\mathcal{S}_i$ is the set of closed points of a closed subvariety of dimension $\le d - i$ of the Serre dual $G^\vee$.

Concretely, this implies that for $0 \le i \le d$, the estimate

$$ |\{\chi \in \hat{G}(k_n) \mid H_c^i(G_{\bar{k}}, M_{\chi}) \neq 0 \text{ or } H_c^{-i}(G_{\bar{k}}, M_{\chi}) \neq 0 \\ \text{or } H^i(G_{\bar{k}}, M_{\chi}) \neq 0 \text{ or } H^{-i}(G_{\bar{k}}, M_{\chi}) \neq 0\}| \ll |k_n|^{d-i} $$

holds for all $n \ge 1$, and so this implies the second part of Theorem 1.

Note that Theorem 2.1 is a consequence of Theorem 2.3, since the set of characters satisfying the property of Theorem 2.1 contains the generic set $\mathcal{S}_0-\mathcal{S}_1$.

REMARK 2.4. We expect that this result should be true with the stronger information that the implied constants in (1.11) for the subsets $\mathcal{S}_i$ depend only on the complexity of $M$. A result of this type would be especially useful for applications to “horizontal” equidistribution theorems.

However, we can only prove this at the current time in the following cases:

(1) if $G$ is a unipotent group (use the equality of Fourier transforms of [112, Th. 3.1] combined with Theorem 1.7);

(2) if $G$ is a geometrically simple abelian variety (see Corollary 2.22).

The issues that arise in attempting to handle the general case are:

- For tori, the use of de Jong's theorem on alterations, where we do not control the number of exceptional components that appear (thus, a suitably effective version of de Jong's theorem, or an effective form of embedded resolution of singularities, would probably imply the desired conclusion in this case).

- For abelian varieties, the need to find and control the complexity of an alteration that “moderates” certain perverse sheaves, to apply results of Orgogozo.

COROLLARY 2.5. Let $G$ be a connected commutative algebraic group over $k$ and let $M$ be an object of $\mathrm{D}_c^b(G)$. Then for generic $\chi \in \hat{G}$ and any $i \in \mathbb{Z}$, there are canonical isomorphisms

$$ H_c^i(G_{\bar{k}}, M_{\chi}) \simeq H^i(G_{\bar{k}}, M_{\chi}) \simeq H_c^0(G_{\bar{k}}, \mathfrak{p}\mathcal{H}^i(M_{\chi})). $$

PROOF. The proof is similar to that of [91, Cor. 2.3]; see also Corollary 2.21 below. □

We will prove Theorems 2.1 and 2.3 in Section 2.5. Before doing this, we need to establish some preliminaries concerning perverse sheaves on the basic building blocks of Proposition 1.13, namely (in rough order of difficulty) unipotent groups, tori and abelian varieties.

Note that proving either Theorem 1 or Theorem 2 for a given group $G$ only involves the corresponding material for groups of the types that actually appear in Proposition 1.13 applied to $G$. In particular, for instance, the proof of Theorem 3 (and other similar statements) only depends on the case of tori, i.e., on Section 2.3.

<footnote n="1">¹ Recall that by convention, this means that the “forget support” map is an isomorphism.</footnote>
To facilitate orientation, we list below the key statements about each type of groups; Section 2.5 only requires these statements from the next three sections.

(1) Unipotent groups: Proposition 2.9.

(2) Tori: Corollary 2.18.

(3) Abelian varieties: Corollary 2.29 and the auxiliary Theorem 2.26, due to Orgogozo [109].

To conclude this section, we also point out that we expect that in Theorems 2.1 and 2.3, one can find a suitable set $\mathcal{U}$ of “geometric” nature. For example, if $G = A \times U \times T$, where A is an abelian variety, U a unipotent group and T a torus, we would expect that there exists a finite extension $k'$ of k, a finite union of tacs $\mathcal{S}'$ of A defined over $k'$, a finite union of tacs $\mathcal{S}$ of T defined over $k'$ and a strict closed subvariety Z of $\hat{U}$ such that $\hat{G} - \mathcal{U} \subseteq \mathcal{S}' \times Z \times \mathcal{S}$. In this direction, following a suggestion of one referee, we can prove the following:

**THEOREM 2.6.** Let $G$ be a connected commutative algebraic group over $k$ such that $G = A \times U$ (resp. $G = A \times T$) where A is an abelian variety, U is a unipotent group and T is a torus. Let $M$ be a perverse sheaf on $G$.

There exists a finite extension $k'$ of $k$, a finite union of tacs $\mathcal{S}'$ of A defined over $k'$ and a strict closed subvariety Z of $\hat{U}$ (resp. a finite union of tacs $\mathcal{S}$ of T defined over $k'$) such that for $\chi \in \hat{G} - (\mathcal{S}' \times Z)$ (resp. for $\chi \in \hat{G} - (\mathcal{S}' \times \mathcal{S}))$, we have

$$ H^i(G_{\bar{k}}, M_{\chi}) = H_c^i(G_{\bar{k}}, M_{\chi}) = 0 \quad \text{for all } i \neq 0, \\ H_c^0(G_{\bar{k}}, M_{\chi}) \text{ is isomorphic to } H^0(G_{\bar{k}}, M_{\chi}). $$

**REMARK 2.7.** We do not claim that the isomorphism between $H_c^0(G_{\bar{k}}, M_{\chi})$ and $H^0(G_{\bar{k}}, M_{\chi})$ in this statement is the “forget support” map, since this does not follow from the proof. We expect that this should be true (maybe up to enlarging the set of exceptional characters).

## 2.2. The case of unipotent groups

We begin by summarizing the duality theory of commutative unipotent groups; a good account can also be found in [15, App. F].

Let U be a connected unipotent commutative algebraic group over a finite field $k$ of characteristic $p$. The functor that sends a perfect $k$-scheme $S$ (i.e., a scheme for which the absolute Frobenius is an automorphism) to the extension group

$$ \mathrm{Ext}^1(U \times_k S, \mathbf{Q}_p/\mathbf{Z}_p) = \varinjlim_m \mathrm{Ext}^1(U \times_k S, p^{-m}\mathbf{Z}_p/\mathbf{Z}_p) $$

in the category of commutative group schemes over $S$ (with $\mathbf{Q}_p/\mathbf{Z}_p$ viewed as a constant group scheme) is representable by a connected commutative group scheme $U^*$ over $k$, called the *Serre dual* of U. This goes back to a remark by Serre [117, p.55] and was subsequently developed by Bégueri in [6, Prop. 1.2.1] and Saibi [112]. Morever, if $m_0 \ge 1$ is such that $p^{m_0}U = 0$, then the natural map yields an isomorphism

$$ \mathrm{Ext}^1(U \times_k S, p^{-m_0} \mathbf{Z}_p/\mathbf{Z}_p) \xrightarrow{\sim} \varinjlim_m \mathrm{Ext}^1(U \times_k S, p^{-m} \mathbf{Z}_p/\mathbf{Z}_p). $$

Let $F$ be a finite abelian group. For each integer $n \ge 1$, the short exact sequence

$$ 1 \to U(k_n) \to U_{k_n} \xrightarrow{x \mapsto \mathrm{Fr}_{k_n}(x) \cdot x^{-1}} U_{k_n} \to 1 $$

induces an isomorphism

$$ \mathrm{Hom}(U(k_n), F) \xrightarrow{\sim} \mathrm{Ext}^1(U_{k_n}, F) $$
(see [15, Prop. F.2]).

Let $m \ge 1$ be such that $p^m U = 0$. We take $F = p^{-m}Z_p/Z_p \simeq Z/p^mZ$. For any integer $n \ge 1$, we obtain an isomorphism

$$ \mathrm{Hom}(U(k_n), F) \xrightarrow{\sim} U^*(k_n). $$

Fix now a faithful character $\psi: p^{-m}Z_p/Z_p \to \overline{\mathbb{Q}}_\ell^\times$. We then obtain, for each integer $n \ge 1$, an isomorphism

$$ U^*(k_n) \xrightarrow{\sim} \hat{U}(k_n). $$

Saibi [112, Lemma 1.5.4.1] (see also [15, Remark F.1 (ii)]) proved that there exists a connected commutative unipotent algebraic group $U^\vee$ and a biextension $\mathcal{L}_{U,U^\vee}$ of $U^\vee \times U$ by $Q_p/Z_p$ such that $\mathcal{L}_{U,U^\vee}$ induces an isomorphism between the perfectizations of $U^\vee$ and $U^*$. Together with the above character $\psi$, this induces isomorphisms

$$ (2.2) \qquad \beta_n: U^\vee(k_n) \to \hat{U}(k_n) $$

for all $n \ge 1$. (See also [15, Remark F.4 (ii)] for a different approach to the construction of the model of finite type $U^\vee$.) We also write $\psi_x$ for the character $\beta_n(x)$.

We denote by $\mathcal{L}_{U,U^\vee,\psi}$ the lisse $\ell$-adic sheaf of rank 1 on $U^\vee \times U$ associated to $\mathcal{L}_{U,U^\vee}$; its trace functions are given by

$$ t_n(x, y; k_n) = \beta_n(x)(y) $$

for all $n \ge 1$ and $(x, y) \in U(k_n) \times U^\vee(k_n)$.

**EXAMPLE 2.8.** Fix a non-trivial additive character $\psi: k \to \overline{\mathbb{Q}}_\ell$. Suppose that $U = G_a^d$ for some $d \ge 0$. For $(x, y) \in U \times U$, write

$$ x \cdot y = \sum_{i=1}^{d} x_i y_i. $$

There exists a choice of biextension with $U^\vee = U$, and the isomorphisms

$$ \beta_n: (G_a^d)(k_n) \to \hat{G}_a^d(k_n) $$

are given by $x \mapsto \psi_x$, where

$$ \psi_x(y) = \psi(\mathrm{Tr}_{k_n/k}(x \cdot y)). $$

We now also fix a quasi-projective scheme $S$ over $k$, and shall work relative to $S$. We denote by $\mathcal{L}_{U,U^\vee,\psi,S}$ the pull-back of $\mathcal{L}_{U,U^\vee,\psi}$ to $U \times_k U^\vee \times_k S$.

Let $p: U \times_k U^\vee \times_k S \to U \times_k S$, $p^\vee: U \times_k U^\vee \times_k S \to U^\vee \times_k S$ and $q: U \times_k S \to S$ denote the projections. The *Fourier transform* is the equivalence of categories

$$ \mathrm{FT}_{\psi,S}: \mathrm{D}_c^b(U \times_k S) \to \mathrm{D}_c^b(U^\vee \times_k S) $$

defined by

$$ \mathrm{FT}_{\psi,S}(M) = \mathrm{R}p_1^\vee(p^*(M) \otimes \mathcal{L}_{U,U^\vee,\psi,S}) = \mathrm{R}p_*^\vee(p^*(M) \otimes \mathcal{L}_{U,U^\vee,\psi,S}), $$

where the second equality (more precisely, the fact that the natural transformation “forget supports” from the left-hand side to the right-hand side is an isomorphism) is [112, Th. 3.1]. A corollary of this is that the Fourier transform is compatible with Verdier duality, in that there is a canonical functorial isomorphism

$$ \mathrm{D}(\mathrm{FT}_{\psi,S}(M)) \simeq \mathrm{FT}_{\psi^{-1},S}(\mathrm{D}(M))(\mathrm{dim} U) $$

for each object $M$ of $\mathrm{D}_c^b(U \times_k S)$; see [112, Cor. 3.2.1]. We refer the reader to Saibi's article [112] for the other main properties of the $\ell$-adic Fourier transform on unipotent groups, such as the formula for the inverse Fourier transform.
By the proper base change theorem and the definition of Fourier transform using $Rp_1^\vee$, for all $a \in U^\vee(\bar{k})$ and $i \in \mathbb{Z}$, there are natural isomorphisms

$$ (2.3) \qquad R^i q_!(M_{\psi_a}) = \mathcal{H}^i(\mathrm{FT}_\psi(M))_a. $$

Since unipotent groups are affine, it follows from Artin's vanishing theorem that the Fourier transform shifts the perverse degree by the dimension of $U$. In particular, if $M$ is perverse, then so is

$$ \mathrm{FT}_{\psi,S}(M)[\mathrm{dim}(U)]. $$

**PROPOSITION 2.9.** Let $U$ be a connected unipotent commutative algebraic group of dimension $d$ over $k$. Fix a locally-closed immersion $u$ (resp. $u^\vee$) of $U$ (resp. $U^\vee$) into some projective space to compute the complexity. Let $M$ be an object of $\mathrm{D}_c^b(U)$ of perverse amplitude $[a, b]$.

There exists an integer $C \ge 0$, depending only on $c_u(M)$, and a stratification $(S_i)$ of $U^\vee$ such that every strat $S_i$ is either empty or has dimension $d - i$, with the following properties:

(1) The sum of the degrees of the irreducible components of $u(S_i)$ is at most $C$.

(2) For each $\xi \in S_i(\bar{k})$, the vanishing $H_c^j(U_{\bar{k}}, M_{\psi_\xi}) = 0$ holds for all $j \notin [a, b + i]$.

In particular, the estimate

$$ (2.4) \qquad |S_i(k_n)| \ll |k_n|^{d-i} $$

holds for all $n \ge 1$, with an implicit constant that only depends on $c_u(M)$.

Moreover, for any $\xi \in S_0(\bar{k})$ and any $j \in \mathbb{Z}$, we have

$$ H_c^j(U_{\bar{k}}, M_{\psi_\xi}) = H^j(U_{\bar{k}}, M_{\psi_\xi}). $$

**PROOF.** Since the Fourier transform shifts the perverse degree by $d$, the complex $\mathrm{FT}_\psi(M)$ has perverse amplitude $[a+d, b+d]$. By Theorem 1.5, the complexity $c_u(\mathrm{FT}_\psi(M))$ is bounded in terms of $c_u(M)$.

By Theorems 1.7 and 1.8, there exists a smooth open subscheme $S_0 \subset U^\vee$, with closed complement $Y_0$ of degree bounded in terms of $c_u(\mathrm{FT}_\psi(M))$, and hence in terms of $c_u(M)$, such that the restriction of $\mathrm{FT}_\psi(M)$ to $S_0$ has lisse cohomology sheaves and such that $\mathrm{FT}_\psi(M)$ is of formation compatible with any base change $S' \to S_0 \subset U^\vee$ (this follows from the formula for the Fourier transform in terms of $Rp_*^\vee$). Up to replacing $S_0$ by a smaller open subset, we may assume that $S_0$ is affine (and this does not increase the complexity of the complement).

In particular, using (2.3) and this compatibility, we obtain the following equality for $a \in S_0(\bar{k})$:

$$ (2.5) \qquad H_c^i(U_{\bar{k}}, M_{\psi_a}) = \mathcal{H}^i(\mathrm{FT}_\psi(M))_a = H^i(U_{\bar{k}}, M_{\psi_a}). $$

By a slight generalization of [8, Cor. 4.1.10. ii], the pullback by a closed immersion with affine complement of a complex of perverse amplitude $[a, b]$ has perverse amplitude $[a - 1, b]$. Therefore, the restriction of $\mathrm{FT}_\psi(M)$ to $Y_0$ has perverse amplitude $[a + d - 1, b + d]$. Proceeding by induction, we construct a stratification $(S_i)_{0 \le i \le d}$ of $U^\vee$ into strats $S_i$ such that

(1) each $S_i$ is smooth, empty or equidimensional of dimension $d - i$;

(2) the closure of each $S_i$ has degree bounded in terms of $c_u(M)$;

(3) the restriction of $\mathrm{FT}_\psi(M)$ to each $S_i$ has lisse cohomology sheaves and is of perverse amplitude $[a + d - i, b + d]$.

Let $0 \le i \le d$. On each connected component of $S_i$, the support of the cohomology sheaves of $\mathrm{FT}_\psi(M)$ is either empty or equal to $S_i$ (since these sheaves are lisse). However, the definition of
perversity implies the inequality

$$ \dim \mathrm{supp} \mathcal{H}^j(\mathrm{FT}_\psi(\mathrm{M})|_{S_i}) \le -j + b + d $$

for all integers $j$. Since $S_i$ has dimension $d - i$, the non-vanishing of $\mathcal{H}^j(\mathrm{FT}_\psi(\mathrm{M})|_{S_i})$ implies therefore the inequality

$$ d - i \le -j + b + d, \quad \text{i.e. } j \le b + i. $$

Since $S_i$ is smooth of dimension $d - i$ (so the dualizing complex on $S_i$ is $\overline{\mathbb{Q}}_\ell(d - i)[d - i]$ and the Verdier dual of a lisse sheaf is the naive dual, up to shift) and the cohomology sheaves on $S_i$ are lisse, duality implies that $\mathrm{D}(\mathrm{FT}_\psi(\mathrm{M})|_{S_i})$ also has lisse cohomology sheaves, given by the formula

$$ \mathcal{H}^j(\mathrm{D}(\mathrm{FT}_\psi(\mathrm{M}))|_{S_i}) = (\mathcal{H}^{-j-2d+2i}(\mathrm{FT}_\psi(\mathrm{M})|_{S_i}))^\vee(d - i) $$

for all $j$. Thus, arguing as above, the perversity condition shows that $\mathcal{H}^j(\mathrm{FT}_\psi(\mathrm{M})|_{S_i}) \ne 0$ implies

$$ d - i \le j + 2d - 2i - a - d + i, \quad \text{i.e. } j \ge a. $$

We conclude that the cohomology sheaves of the complex $\mathrm{FT}_\psi(\mathrm{M})|_{S_i}$ are concentrated in degrees $[a, b + i]$. By (2.5), this implies assertion (2) of the proposition and concludes the proof. $\square$

**REMARK 2.10.** This result is a generalization to all unipotent groups, and a quantification by means of the complexity, of some of the Fouvry–Katz–Laumon stratification results for additive exponential sums [78, 39]. It may have interesting applications to analytic number theory, since the quantitative form means that it may be used over varying finite fields, e.g. $\mathbb{F}_p$ as $p \to +\infty$ as in Chapter 6 (although a referee pointed out that the complexity of the Fourier transform of a perverse sheaf on a non-additive unipotent group, such as Witt vectors of length 2, will usually have to depend on $p$).

For the proof of Theorem 2.6, we will also require a relative version of the “first step” of Proposition 2.9.

**PROPOSITION 2.11.** Let $U$ be a connected unipotent commutative algebraic group of dimension $d$ over $k$, and $S$ a quasi-projective scheme over $k$. Fix a locally-closed immersion $u$ (resp. $u^\vee$) of $U \times S$ (resp. $U^\vee \times S$) into some projective space to compute the complexity. Let $q: U \times S \to S$ be the projection.

There exists an integer $C \ge 0$, depending only on $c_u(M)$, and a dense open subset $U_0^\vee$ of $U^\vee$ with the following properties:

(1) The sum of the degrees of the irreducible components of $u^\vee(U^\vee - U_0^\vee)$ is at most $C$.

(2) For any $\xi \in U_0^\vee(\bar{k})$, we have

$$ Rq!(M_{\psi_{\xi}}) = Rq_*(M_{\psi_{\xi}}), $$

where $\psi_\xi$ denotes the character corresponding to $\xi$ by means of (2.2).

In particular, the estimate

$$ (2.6) \qquad |(U^\vee - U_0^\vee)(k_n)| \ll |k_n|^{d-1} $$

holds for all $n \ge 1$, with an implicit constant that only depends on $c_u(M)$.

Moreover, if $M$ is perverse, then $Rq!(M_{\psi_\xi}) = Rq_*(M_{\psi_\xi})$ is perverse for every $\xi \in U_0^\vee(\bar{k})$.

**PROOF.** By Theorem 1.8, there exists a smooth open subscheme $U_0^\vee \subset U^\vee$, with closed complement $Y_0$ of degree bounded in terms of $c_u(\mathrm{FT}_{\psi,S}(\mathrm{M}))$, and hence in terms of $c_u(M)$, such that $\mathrm{FT}_{\psi,S}(\mathrm{M})$ is of formation compatible with any base change $S' \to U_0^\vee \subset U^\vee$ (this follows from
the formula for the Fourier transform in terms of $R_{p_*}^\vee$). Combining this with proper base change, we get for every $\xi \in U_0^\vee(\bar{k})$, the equality

$$Rq_!(M_{\psi_\xi}) = Rq_*(M_{\psi_\xi}).$$

If M is perverse, then Artin's vanishing theorem implies that $Rq_!(M_{\psi_\xi})$ is perverse since $q$ is affine.

Finally, the estimates on the degrees of the irreducible components of $U^\vee - U_0^\vee$ follow from Theorem 1.8 (i) and the counting estimate follows from Theorem 1.10. □

## 2.3. Perverse sheaves on tori

In this section, we generalize some of the results of Gabber and Loeser [50] about perverse sheaves on tori. We begin with a generalization of [50, Th. 4.1.1'], which is proved in loc. cit. under the assumption that resolution of singularities and simplification of ideals hold for varieties of dimension at most the dimension of the torus in question. The structure of our proof is the same, but we are able to replace the appeal to resolution of singularities with de Jong's theorem on alterations [24].

**THEOREM 2.12.** Let $T$ be a torus over $\bar{k}$ and let $M$ be an object of $D_c^b(T)$. For all characters $\chi \in \Pi(T)(\bar{Q}_\ell)$ outside of a finite union of tacs, the equality

$$H^i(T, M_\chi) = H_c^i(T, M_\chi)$$

holds for all $i \in \mathbb{Z}$.

As in [50], the proof of Theorem 2.12 relies on the auxiliary Proposition 2.13 stated below. We pick a smooth compactification of $T$ by a simple normal crossing divisor $j: T \to \bar{T}$ (for example, the projective space), and denote by $i: \bar{T} - T \to \bar{T}$ the complementary closed immersion. Given any morphism $\varphi: W \to \bar{T}$ of varieties over $k$, denote by $j_W: \varphi^{-1}(T) \to W$ and $i_W: \varphi^{-1}(\bar{T} - T) \to W$ the corresponding open and closed immersions. Recall the $\Omega_T$-sheaf of rank one $\mathcal{L}_T$ on $T$ from Section 1.10. In particular, for an object $N$ of $D_c^b(\varphi^{-1}(T))$ and a point $\xi$ of $\varphi^{-1}(\bar{T} - T)$, the stalk $j_{W_*}(N \otimes \varphi^*(\mathcal{L}_T))_\xi$ is a complex of coherent sheaves on $\Pi(T)$.

**PROPOSITION 2.13.** With notation as above, let $N$ be an object of $D_c^b(\varphi^{-1}(T))$. There exists a finite union $\mathcal{S}$ of tacs in $\bar{T}$ such that, for any $r \ge 0$ and any $\xi \in \varphi^{-1}(\bar{T} - T)$, the support of the module $(R^r j_{W_*}(N \otimes \varphi^*(\mathcal{L}_T)))_\xi$ is contained in $\mathcal{S}$.

**PROOF.** The idea of the proof is to reduce to the situation of [50, Prop. 4.3.1'].

We use induction on the dimension of $W$. We can then readily assume that $N$ is a lisse sheaf on a locally-closed irreducible subvariety $U$ of $\varphi^{-1}(T)$, extended by zero to $\varphi^{-1}(T)$. We can assume further that $U$ is dense in $W$. Now the monodromy of $N$ can be assumed to be pro-$\ell$. Indeed, consider the finite étale cover $f: U' \to U$ associated to the $\ell$-Sylow subgroup of the monodromy group of $N$, and let $W'$ be the normalization of $W$ in the function field of $U$. The sheaf $N$ is a direct factor of $f_*f^*N$, and it suffices to prove the theorem for $f^*N$ and $W'$. Hence, we assume that the monodromy of $N$ is pro-$\ell$.

By de Jong's theorem [24, Th. 4.1], there exists an alteration $f: W' \to W$ such that $W'$ is smooth and the reduction of the complement of $f^{-1}(U)$ in $W'$ is a strict normal crossing divisor. Since we are working over a perfect field, we can further assume that the alteration $f$ is generically étale. Hence, there exists a dense open subset $U_0$ of $U$ such that $f$ is finite étale over $f^{-1}(U_0)$. By induction, it is enough to prove the result for $U_0$ and $N|_{U_0}$, and hence by the same argument as above, it is enough to prove it for $f_*f^*N|_{U_0}$. By proper base change, it is then enough to prove the result for $W'$ and $f^*N|_{U_0}$. By a last dévissage, it is finally enough to prove it for $f^*N$.
We are now in a situation where we can suppose that W is smooth, that the complements of $\varphi^{-1}(T)$ and U in W are strict normal crossing divisors, and that the monodromy of N is pro-$\ell$. This is exactly the situation at the end of the proof of [50, Prop. 4.3.1', starting from p. 544, line -4] (with N replacing A there) and the remaining argument is identical to that of loc. cit. □

PROOF OF THEOREM 2.12. The fact that Proposition 2.13 implies Theorem 2.12 is completely similar to the fact that Proposition 4.3.1' implies Théorème 4.1.1' in [50]. We keep the notation introduced before the statement of Proposition 2.13, and apply Proposition 2.13 with $W = \bar{T}$, with $\varphi$ the identity morphism and $N = M$, so that $j_W = j$ and $i_W = i$.

Let $\chi \in \hat{T}$ such that $\chi$ does not belong to the finite number of tacs of $\hat{T}$ given by Proposition 2.13. According to [50, Prop. 4.5.1 (2)], this implies that the object $i^*Rj_*(M_\chi) \in D_c^b(\bar{T} - T)$ is trivial, and hence its cohomology complex

$$R\Gamma(\bar{T} - T, i^*Rj_*(M_\chi))$$

is also trivial. But this last complex is isomorphic to the cone of the morphism

$$R_{S!}(M_\chi) \to R_{S*}(M_\chi),$$

where $s: T_{\bar{k}} \to \text{Spec}(\bar{k})$ is the structure morphism, hence the theorem. □

We now use Proposition 2.13 to deduce a relative version of Theorem 2.12.

THEOREM 2.14. Let $T$ be a torus over $\bar{k}$, let $S$ be an arbitrary scheme over $\bar{k}$, and let $G = S \times_T T$. Denote by $p: G \to S$ the projection. Let $N$ be an object of $D_c^b(G)$.

For $\chi \in \Pi(T)(\bar{Q}_\ell)$ away from a finite union of tacs $\mathcal{S}$, we have $R_{p!}(N_\chi) = R_{p*}(N_\chi)$.

In particular, if $N$ is a perverse sheaf, then for $\chi$ not in $\mathcal{S}$, the complex $R_{p!}(N_\chi) = R_{p*}(N_\chi)$ is a perverse sheaf on $S$.

PROOF. This is similar to Theorem 2.12. We apply Proposition 2.13 with $W = S \times_{\bar{k}} \bar{T}$, and check that, for each character $\chi$ away from the finite union of tacs given by the proposition, the object $i_W^*Rj_{W*}(N_\chi)$ is trivial, which follows from the immediate extension of [50, Prop. 4.7.2 (ii)] to an arbitrary base scheme $S$ (instead of just tori). □

THEOREM 2.15. Let $T$ be a $d$-dimensional torus over $k$, let $S$ be an arbitrary scheme over $k$, and define $X = T \times_S S$. Let $i$ be an integer such that $1 \le i \le d$.

Let $M$ be a perverse sheaf on $X$. There exist a finite extension $k'$ of $k$ and a family $(S_f)_{f \in \mathcal{F}}$ of tacs of $T_{k'}$ of dimension $\le d - i$ with the property that for any $\chi \in \hat{T}_{k'}$ that does not belong to the union of the $S_f$ there exists a quotient torus $q: T_{k'} \to Z$ of dimension $i - 1$ such that

$$R_{qS!}M_\chi = R_{qS*}M_\chi,$$

with $q_S$ denoting the projection $q_S: T_{k'} \times_S S \to Z_{k'} \times_S S$, and this complex is perverse on $Z \times_{k'} S_{k'}$.

PROOF. Up to replacing $k$ by a finite extension, we can assume that $T_k$ is split, and thus reduce to $T = G_m^d$. Now let $1 \le i \le d$. For each subset $I$ of $[d] = \{1, \dots, d\}$ of size $i - 1$, we apply Theorem 2.14 with $(T, S) = (G_m^{[d]-I}, G_m^I \times_S S)$ over $\bar{k}$, so that the projection $p$ in the theorem is then the canonical projection

$$q_I: G_m^d \times_S S = G_m^{[d]} \times_S S \to G_m^I \times_S S.$$

We obtain a finite union of tacs of $G_m^{[d]-I}$ such that for characters $\chi$ of $G_m^{[d]-I}$ outside of this finite union, we have

$$R_{qI!}(M_\chi) = R_{qI*}(M_\chi)$$

and this complex is perverse.
Let

$$ (\pi_{I,j}: \mathbf{G}_m^{[d]-I} \to Y_{I,j}, \chi_{I,j})_{j \in X_I} $$

be the quotient morphisms and characters defining this finite family of tacs. For $j \in X_I$, we define $K_{I,j} = \ker(\pi_{I,j})$; this is a non-trivial subtorus of $\mathbf{G}_m^{[d]-I}$, which we identify with a subtorus of $\mathbf{G}_m^d$ using the canonical embedding $\mathbf{G}_m^{[d]-I} \to \mathbf{G}_m^d$. In addition, we define $\chi'_{I,j} \in \Pi(\mathbf{G}_m^d)(\bar{\mathbb{Q}}_\ell)$ to be the character that is trivial on $\mathbf{G}_m^I$ and coincides with $\chi_{I,j}$ on $\mathbf{G}_m^{[d]-I}$.

Let $\mathcal{F}$ be the set of all maps $f$ from the subsets of $[d]$ of size $i-1$ to the disjoint union of the $X_I$ that send a subset $I$ to an element $j \in X_I$ for each $I$; this set is finite. For $f \in \mathcal{F}$, let $S_f$ be the intersection of the tacs of $\mathbf{G}_m^d$ defined by

$$ (\mathbf{G}_m^d \to \mathbf{G}_m^d/K_{I,f(I)}, \chi'_{I,f(I)}). $$

We claim that the family $(S_f)_{f \in \mathcal{F}}$ (to be precise, the subfamily where $S_f$ is not empty) satisfies the assertions of the theorem.

Indeed, first of all Lemma 1.25 shows that $S_f$ is either empty or is again a tac; moreover, in the second case, it is defined by the projection $\mathbf{G}_d^m \to \mathbf{G}_m^d/T_f$ where $T_f$ is the subtorus of $\mathbf{G}_m^d$ generated by the $K_{I,f(I)}$ (as subtori of $\mathbf{G}_m^d$). By Lemma 1.26 applied to $G_i = \mathbf{G}_m$ for all $i$ and the subgroups $K_{I,f(I)}$, we have $\dim(T_f) \ge i$ for all such $f$, and hence the quotient

$$ p_f: \mathbf{G}_m^d \to Y_f = \mathbf{G}_m^d/T_f $$

has image of dimension $\le d-i$, as desired.

Finally, let $\chi \in \hat{\mathbf{G}}_m^d$ be a character that does not belong to any of the tacs $S_f$. This implies that there exist some $f \in \mathcal{F}$, some subset $I \subset [d]$ of size $i-1$ and some $j \in X_I$ such that the restriction $\chi_I$ of $\chi$ to $\mathbf{G}_m^{[d]-I}$ is not equal to $\chi_{I,j}$.

We can write $\chi = \chi_1\chi'$ where $\chi'$ is a character of $\mathbf{G}_m^I$. Considering the quotient $q: \mathbf{G}_m^d \to \mathbf{G}_m^I$, the base change $q_S$ is the canonical projection $q_I$ and from the application of Theorem 2.14 to $q_I$, we obtain

$$ R_{q_S}(M_\chi) = R_{q_S}(M_{\chi_I}) \otimes \mathcal{L}_{\chi'} = R_{q_S}(M_{\chi_I}) \otimes \mathcal{L}_{\chi'} = R_{q_S}(M_\chi), $$

and the fact that this object is perverse. $\square$

We deduce two corollaries that are sometimes more convenient for applications. The first one is Theorem 2.3 for tori.

**COROLLARY 2.16.** Let $T$ be a torus of dimension $d$ over $k$ and let $M$ be a perverse sheaf on $T$. For $-d \le i \le d$, the sets

$$ \{\chi \in \hat{T} \mid H^i(T_{\bar{k}}, M_\chi) \neq 0\} \quad \text{and} \quad \{\chi \in \hat{T} \mid H_c^i(T_{\bar{k}}, M_\chi) \neq 0\} $$

are contained in a finite union of tacs of $T$ of dimension $\le d - |i|$, and in particular they have character codimension at least $|i|$.

**PROOF.** We apply Theorem 2.15 to $|i|$ and claim that the characters in either of these sets belong to the union of the tacs $S_f$ that arise. Indeed, if $\chi$ is not in any $S_f$, then there exists a quotient torus $T_{k'} \to Z$ of dimension $i-1$ such that $R_{q_S}M_\chi = R_{q_S}M_\chi$, and hence

$$ H^i(T_{\bar{k}}, M_\chi) = H^i(B_{\bar{k}}, R_{q_S}M_\chi) = 0 $$

since $R_{q_S}M_\chi$ is a perverse sheaf and $\dim(B) = i-1$. The argument is similar for the cohomology with compact support. $\square$
### REMARK 2.17. We recall that, concretely, this corollary implies that for $|i| \le d$, the estimate

$$ \left| \{ \chi \in \hat{T}(k_n) \mid H^i(\hat{T}_{\bar{k}}, M_{\chi}) \neq 0 \text{ or } H_c^i(\hat{T}_{\bar{k}}, M_{\chi}) \neq 0 \} \right| \ll |k_n|^{d-|i|} $$

holds for all $n \ge 1$.

The following “stratified” statement is also a useful formulation of the result.

### COROLLARY 2.18. Let $T$ be a torus of dimension $d$ over $k$ and $S$ a scheme over $k$. Set $X = T \times S$ and let $q$ denote the projection $q: X \to S$. Let $M$ be a perverse sheaf on $X$. There exists a finite extension $k'$ of $k$ and a partition of $\hat{T}_{k'}$ into subsets $(S_i)_{0 \le i \le d}$ of character codimension $\ge i$ such that, for any $i$ and $\chi \in S_i$, the object $Rq!(M_\chi)$ of $D_c^b(S)$ has perverse amplitude $[0, i]$.

### PROOF. Using the notation of the proof of Theorem 2.15, for any integer $i$ with $1 \le i \le d$, let $k'_i$ be the finite extension arising from its application to $i$ and let $\mathcal{F}_i$ be the corresponding family of tacs. Define $\tilde{S}_i$ to be the union of the $S_f$ for $f \in \mathcal{F}_i$ for $1 \le i \le d$.

Let $k'$ be the compositum of all $k'_i$. Define $S_0 = \hat{T} - \tilde{S}_1$ and $S_i = \tilde{S}_i - \tilde{S}_{i+1}$ for $1 \le i \le d$. These sets form a partition of $\hat{T}_{k'}$, and since $S_i \subset \tilde{S}_i$ for $i \ge 1$, they have character codimension $\ge i$. This property is also clear for $i = 0$.

Let $0 \le i \le d$, and let $\chi \in S_i$. Then $\chi \notin \tilde{S}_{i+1}$, and hence the theorem provides a projection $qs: G_m^d \times S \to Z \times S$ with $\dim(Z) = i$ such that $Rqs!M_\chi$ is perverse. Composing with the projection $r: Z \times S \to S$, which is affine and hence such that $Rr!$ preserves objects with perverse amplitude $[0, +\infty]$ (by Artin's vanishing theorem), it follows that $Rq!M_\chi$ has perverse amplitude $[0, i]$. $\square$

## 2.4. Perverse sheaves on abelian varieties

In this section, we will review and extend some results of Krämer and Weissauer on perverse sheaves on abelian varieties.

### 2.4.1. Statement of the results and corollaries.
Let $k$ be a finite field, and $\bar{k}$ an algebraic closure of $k$. Let $X$ be an abelian variety over $k$. We fix a projective embedding $u$ of $X$. For subvarieties of $X$, the degree means the degree of the image by $u$; for a tac of $S$ defined by $\pi: X \to A$ and $\chi$, we will say that the *degree* of $S$ is the degree of the image $u(\ker(\pi))$.

For a perverse sheaf $M$ on $X$, a combination of the main result of Weissauer [127] and of the machinery developed by Krämer and Weissauer [94] implies that for most characters $\chi \in \hat{X}$, we have $H^i(\hat{X}_{\bar{k}}, M_\chi) = 0$ for all $i \neq 0$; we will show here that this result can be made quantitative using the complexity of $M$, and will then establish a relative version (see Section 2.4.3).

### THEOREM 2.19. Let $X$ be an abelian variety over $k$ and let $M$ be a perverse sheaf on $X$.

There exist an integer $c \ge 0$ depending only on $c_u(M)$, a finite extension $k'$ of $k$ of degree $\le c$, and a finite family $(S_f)_{f \in F}$ of tacs of $X_{k'}$ with $|F| \le c$, each of degree at most $c$, such that any $\chi \in \hat{X}_{k'}$ that does not belong to the union of the $S_f$ satisfies

$$ H^i(\hat{X}_{\bar{k}}, M_{\chi}) = 0 $$

for all $i \neq 0$.

We will prove this below, but first we establish some corollaries.

### COROLLARY 2.20. Let $M$ be an object of $D_c^b(X)$.
There exist an integer $c \ge 0$, depending only on $c_u(M)$, a finite extension $k'$ of $k$ of degree $\le c$, and a finite family $(S_f)_{f \in F}$ of tacs of $X_{k'}$, each of degree at most $c$, with $|F| \le c$, such that for any $\chi \in \hat{X}_{k'}$ that does not belong to the union of the $S_f$, there is a canonical isomorphism

$$H^i(X_{\bar{k}}, M_{\chi}) \simeq H^0(X_{\bar{k}}, \mathfrak{p}\mathcal{H}^i(M)_{\chi})$$

for all $i \in \mathbb{Z}$.

PROOF. This is the same argument as in the proof of Corollary 2.5; the dependency on $c_u(M)$ is obtained by means of Proposition 1.9 to control the perverse cohomology sheaves of M. $\square$

Alternatively, the next corollary may be more convenient for applications.

COROLLARY 2.21. Let $M$ be an object of $\mathrm{D}^b_c(X_k)$. The set $\mathcal{S}$ of characters $\chi \in \hat{X}$ such that there are isomorphisms

$$H^i(X_{\bar{k}}, M_{\chi}) \simeq H^0(X_{\bar{k}}, \mathfrak{p}\mathcal{H}^i(M)_{\chi})$$

for all $i \in \mathbb{Z}$ is generic, and the implicit constant in (1.12) depends only on $c_u(M)$.

In particular, if $M$ is a perverse sheaf, then the set of $\chi$ such that $H^i(X_{\bar{k}}, M_{\chi}) = 0$ for all $i \ne 0$ is generic and the implicit constant in (1.12) only depends on $c_u(M)$.

PROOF. Assume first that $M$ is a perverse sheaf. We apply Theorem 2.19 to $M$, and use the notation there. For $n \ge 1$, let $k'_n = k'k_n$. For any $\chi \in \hat{X}(k_n) - \mathcal{S}(k_n)$, the corresponding character in $\hat{X}(k'_n)$ belongs to $S_f(k'_n)$ for some $f \in F$. Let $A_f$ be the abelian variety such that $S_f$ is defined by $\pi_f: X_{k'} \to A_f$; we have

$$|\hat{X}(k_n) - \mathcal{S}(k_n)| \le \sum_{f \in F} |A_f(k'_n)| \le |F| (|k'_n|^{1/2} + 1)^{2 \dim(A_f)} \ll |k_n|^{\dim(X)-1},$$

where the implied constant depends only on $c_u(M)$ by the theorem.

Now in the general case, recalling that $\mathfrak{p}\mathcal{H}^i(M_{\chi})$ is canonically isomorphic to $\mathfrak{p}\mathcal{H}^i(M)_{\chi}$ for all $i$ and all $\chi$, we have the convergent perverse spectral sequences

$$E_2^{i,j} = H^i(X_{\bar{k}}, \mathfrak{p}\mathcal{H}^j(M)_{\chi}) \Rightarrow H^{i+j}(X_{\bar{k}}, M_{\chi}).$$

By the previous case applied to each of the finitely many perverse cohomology sheaves, the set of $\chi$ such that $H^i(X_{\bar{k}}, \mathfrak{p}\mathcal{H}^j(M)_{\chi}) = 0$ for all $i \ne 0$ and all $j$ is generic; for any such character, the spectral sequence degenerates and we obtain isomorphisms

$$H^i(X_{\bar{k}}, M_{\chi}) \simeq H^0(X_{\bar{k}}, \mathfrak{p}\mathcal{H}^i(M)_{\chi}).$$

Applying Proposition 1.9, we see that the last statement concerning the implicit constant in (1.12) holds. $\square$

COROLLARY 2.22. Let $X$ be a geometrically simple abelian variety over $k$. Let $M$ be a perverse sheaf on $X$. Then there exists a constant $c$ depending only on $c_u(M)$ and a finite set $\mathcal{S} \subset \Pi(X)(\overline{\mathbb{Q}}_\ell)$ of cardinality at most $c$ such that for $\chi \in \Pi(X)(\overline{\mathbb{Q}}_\ell) - \mathcal{S}$,

$$H^i(X_{\bar{k}}, M_{\chi}) = 0 \text{ for } i \ne 0.$$

PROOF. Since $X$ is a geometrically simple abelian variety, then a tac of $X$ contains a single character. Hence, the result follows from Theorem 2.19. $\square$
### 2.4.2. Proof of the results.
We now proceed with the proof of Theorem 2.19. As we indicated, the first ingredient is a quantitative version of a result of Weissauer [127].

**PROPOSITION 2.23.** Let $X$ be an abelian variety over $k$ with a projective embedding $u$, and let $M$ be a geometrically simple perverse sheaf on $X$ such that $\chi(X_{\bar{k}}, M) = 0$.

There exists a tac S on $X$ with kernel an abelian subvariety $A$ of degree bounded in terms of $c_u(M)$, such that

$$ \bigoplus_i H^i(X_{\bar{k}}, M_\chi) \neq 0 $$

if and only if $\chi$ is in S.

Moreover, $M$ is invariant by translation by $A$.

**PROOF.** For any perverse sheaf $N$ on $X$, we denote by $\mathcal{S}_1(N)$ the set of characters $\chi$ such that for some $i \neq 0$ the cohomology group $H^i(X_{\bar{k}}, N)$ is non-zero.

By [127, Th. 3 and Lem. 6], there exists a maximal abelian variety $A_{\bar{k}}$ of $X_{\bar{k}}$ such that $M$ is invariant by translation by $A_{\bar{k}}$, and this abelian variety is non-trivial.

Denoting by $q: X_{\bar{k}} \to X_{\bar{k}}/A_{\bar{k}}$ the quotient morphism, this is equivalent to the fact that $M$ is isomorphic over $\bar{k}$ to a perverse sheaf of the form $\mathcal{L}_{\chi_0} \otimes q^*(\tilde{M})[\dim(A)]$ for some character $\chi_0: \pi_1(X_{\bar{k}}) \to \overline{\mathcal{Q}}_\ell^\times$ and some simple perverse sheaf $\tilde{M}$ on $X_{\bar{k}}/A_{\bar{k}}$.

We claim first that $A_{\bar{k}}$ is defined over $k$ and that the degree of $A_{\bar{k}}$ in the image of $u$ is bounded in terms of $c_u(M)$.

The fact that $A_{\bar{k}}$ is defined over $k$ is implicit in the proof of the existence of $A$ by Weissauer. We recall his argument. First, a perverse sheaf $\mathcal{P}_M$ is defined as follows ([127, p. 563]): the evaluation morphism $ev: M^\vee * M \to \mathbf{1}$ (see Section 1.5) induces morphisms of perverse sheaves $\mathcal{P}\mathcal{H}^i(M^\vee * M)[-i] \to \mathbf{1}$ for all $i$, and each $\mathcal{P}\mathcal{H}^i(M^\vee * M)[-i]$ is a direct sum of shifted irreducible perverse sheaves. Since by Lemma 1.15 $\dim \operatorname{Hom}(M^\vee * M, \mathbf{1}) = \dim \operatorname{Hom}(M, M) = 1$, there is a unique integer $\nu \geq 0$ and a unique perverse irreducible summand $\mathcal{P}_M[-\nu]$ of $\mathcal{P}\mathcal{H}^\nu(M^\vee * M)[-\nu]$ on which the restriction of $ev$ is not zero (see [127, p. 563 and Remark (2), p. 569] for details).

Since $\chi(X_{\bar{k}}, M) = 0$, we have $\nu \geq 1$, and by [127, Lemma 2], it follows that $\mathcal{S}_1(M) = \mathcal{S}_1(\mathcal{P}_M)$. Weissauer shows (see [127, Prop. 2]) that there exists an abelian subvariety $A$ of $X$ of dimension $\nu \geq 1$, with closed immersion $i: A \to X$, and a character $\chi_0$ such that there is a geometric isomorphism $\mathcal{P}_M \simeq \mathcal{L}_{\chi_0} \otimes i_* \overline{\mathcal{Q}}_\ell[\nu]$. Since $A$ can therefore be recovered as the support of $\mathcal{P}_M$, it is defined over $k$.

The perverse sheaf $\mathcal{P}_M$ is invariant by translation by $A_{\bar{k}}$, and then so is $M$ by [127, Remark (2)].

Moreover, we have $c_u(\mathcal{P}_M) \ll c_u(M)$ by the definition of $\mathcal{P}_M$ and Proposition 1.9, and $\deg(u \circ i(A)) \ll c_u(\mathcal{P}_M) \ll c_u(M)$ by Theorem 1.7.

Let $q: X \to X/A$ be the quotient morphism and $\chi$ be a character not in the tac S of $X_k$ defined by $(q, \chi_0^{-1})$. We now compute for every $i \in \mathbf{Z}$ that

$$ H^i(X_{\bar{k}}, \mathcal{P}_{M\chi}) = H^i((X/A)_{\bar{k}}, Rq_* (\mathcal{P}_{M\chi}) = H^i((X/A)_{\bar{k}}, Rq_* (\mathcal{L}_{\chi\cdot\chi_0}) \otimes \mathbf{1}[\dim(A)]). $$

Since $\chi$ is not in the tac S, the restriction of $\mathcal{L}_{\chi\cdot\chi_0}$ to $A_{\bar{k}}$ is non-trivial, and hence we have $Rq_* (\mathcal{L}_{\chi\cdot\chi_0}) = 0$ by Lemma 1.16, and therefore $H^i(X_{\bar{k}}, \mathcal{P}_{M\chi}) = 0$ for all $i$.

Conversely, if $\chi = \chi_0^{-1} \cdot (\tilde{\chi} \circ q)$, then we have

$$ H^*(X_{\bar{k}}, \mathcal{P}_{M\chi}) = H^*(A_{\bar{k}}, \overline{\mathcal{Q}}_\ell) \otimes H^*((X/A)_{\bar{k}}, \mathcal{L}_{\tilde{\chi}}\mathbf{1}[\dim(A)]), $$

by the Künneth formula, and this is non-zero. $\square$
PROOF OF THEOREM 2.19. We follow the method used by Krämer and Weissauer to prove [94, Th. 1.1], keeping track of the complexity.

Since X is an abelian variety, the two convolution products of Section 1.5 coincide; for an object M of $D_c^b(X)$ and an integer $n \ge 1$, we denote by $M^{*n}$ the $n$-th iterated convolution product of M.

We recall the axiomatic framework of [94, Section 5], specialized to our situation as in [94, Example 5.1]. Let $\mathbf{D}$ be the full subcategory of $D_c^b(X_{\bar{k}})$ whose objects are direct sums of shifts of geometrically semisimple perverse sheaves which are obtained by pullback from $X_{k_n}$ for some $n \ge 1$. Let $\mathbf{P} \subset \mathbf{Perv}(X_{\bar{k}})$ be the corresponding subcategory of perverse sheaves, namely that with objects the geometrically semisimple perverse sheaves arising by pullback from $X_{k_n}$ for some $n \ge 1$. Then the categories $\mathbf{P}$ and $\mathbf{D}$ satisfy the axioms (D1), (D2), and (D3) of [94, Section 5], namely:

(D1) The category $\mathbf{D}$ is stable under degree shift, convolution and perverse truncation functors; the category $\mathbf{P}$ is the heart of this $t$-structure, and is a semisimple abelian category.

(D2) Any object M of $\mathbf{D}$ can be written (non-canonicaly) as a direct sum

$$ \bigoplus_{n \in \mathbb{Z}} \mathfrak{P}\mathcal{H}^m(M)[-m]. $$

(D3) The Hard Lefschetz Theorem holds for objects of $\mathbf{D}$.

Let $\mathbf{N}$ be the full subcategory of $\mathbf{D}$ whose objects are the complexes $\mathbf{N}$ such that all geometrically simple constituents of all perverse cohomology sheaves $\mathfrak{P}\mathcal{H}^i(\mathbf{N})$ for $i \in \mathbb{Z}$ have Euler-Poincaré characteristic equal to 0. By [94, Cor. 6.4], the category $\mathbf{N}$ satisfies the axioms (N1), (N2), (N3) and (N4) of [94, Section 5], namely:

(N1) We have $\mathbf{N} * \mathbf{D} \subset \mathbf{N}$ and the category $\mathbf{N}$ is stable under direct sums, retracts, degree shifts, perverse truncation and duality;

(N2) If $\mathbf{N}$ is an object of $\mathbf{N}$, then for most characters $\chi$, we have $H^i(X_{\bar{k}}, \mathbf{N}_{\chi}) = 0$ for all $i$;

(N3) The category $\mathbf{N}$ contains all objects M of $\mathbf{D}$ such that $H^i(X_{\bar{k}}, \mathbf{N}) = 0$ for all $i \in \mathbb{Z}$;

(N4) The category $\mathbf{N}$ contains all simple objects of $\mathbf{P}$ with zero Euler-Poincaré characteristic.

(Note that we will not make use of this version of (N2).)

By [94, Theorem 9.1], every $M \in \mathbf{P}$ is an $\mathbf{N}$-multiplier, meaning that for all integers $i \ne 0$ and any integer $r \ge 1$, every subquotient of $\mathfrak{P}\mathcal{H}^i((M \oplus M^\vee)^{*r})$ lies in $\mathbf{N}$.

We now argue as in the proof of [94, Lemma 8.2] to prove Theorem 2.19 for a perverse sheaf M on X.

**Step 1.** We assume that M is arithmetically simple. By Lemma 1.28, the base change of M to $\bar{k}$ is an object of $\mathbf{P}$. We denote $g = \dim(X)$; by (D2), we have

$$ M_{\bar{k}}^{*(g+1)} \simeq \bigoplus_{m \in \mathbb{Z}} M_m[m], $$

for some objects $M_m$ of $\mathbf{P}$, which are in fact objects of $\mathbf{N}$ for $m \ne 0$ since M is an $\mathbf{N}$-multiplier.

By Proposition 1.9, the number of integers $m$ such that $M_m$ is non-zero is bounded in terms of $c_u(M)$, and similarly $c_u(M_m)$ is bounded in terms of $c_u(M)$. By the semisimplicity property in (D1), each $M_m$ is a direct sum of simple perverse sheaves in $\mathbf{N}$, and by Proposition 1.9, the number and the complexity of these constituents are bounded in terms of $c_u(M)$. We denote by $\mathcal{C}$ the finite set of all these simple perverse sheaves. By Lemma 1.28, there exists a finite extension $k'$ of $k$, of degree bounded in terms of $c_u(M)$, such that any element $C$ of $\mathcal{C}$ is defined over $k'$.
We apply Proposition 2.23 to each $C \in \mathcal{C}$. Let $\mathcal{S}_C$ denote the corresponding tac; it is of degree bounded in terms of $c_u(M)$.

We claim that if $\chi \in \hat{X}$ does not belong to the union of the tacs $\mathcal{S}_C$, then we have

$$H^i(X_{\bar{k}}, M_{\chi}) = 0$$

for all $i \neq 0$. This statement will conclude the proof of Theorem 2.19 for M.

Let $\chi$ be a character that is not in any of the tacs $\mathcal{S}_C$. Since $M_{\chi}^{*(g+1)}$ is isomorphic to $(M^{*(g+1)})_{\chi}$ and $H^*(X_{\bar{k}}, C_{\chi}) = 0$ for $\chi \notin \mathcal{S}_C$, we have

$$H^i(X_{\bar{k}}, M_{\chi}^{*(g+1)}) = H^i(X_{\bar{k}}, M_{0\chi}),$$

for any $i \in \mathbb{Z}$. The right-hand side vanishes if $|n| > g$ since $M_0$ is perverse. Finally, by the compatibility between convolution and the Künneth formula (see Lemma 1.15 below) we also have an isomorphism

$$H^*(X_{\bar{k}}, M_{\chi}^{*(g+1)}) \simeq H^*(X_{\bar{k}}, M_{\chi})^{\otimes(g+1)},$$

and by comparing we see that only the space $H^0(X_{\bar{k}}, M_{\chi})$ may be non-zero, which establishes the claim.

**Step 2.** Now let M be an arbitrary perverse sheaf on X. By Proposition 1.9, the number of geometric Jordan-Hölder factors of M is bounded in terms of $c_u(M)$, and hence also the number of arithmetic Jordan-Hölder factors; we then apply the first step to each of the terms of a composition series for M, and deduce the corresponding result for M. $\square$

**2.4.3. The relative version.** Our next goal is to establish a relative version of Theorem 2.19. The arguments over the complex numbers of Krämer and Weissauer in [94, Section 2] do not apply to our situation over finite fields, since they rely on Verdier stratifications. We instead use a constructibility result of Orgogozo [109], which is a stratification result, locally for the alteration topology.

**THEOREM 2.24.** Let S be a quasi-projective scheme over k, and let A be an abelian variety over k. Let $X = A \times S$, and denote by $f: X \to S$ the canonical morphism. Fix a projective embedding u of X.

Let $\alpha: X' \to X$ be an alteration defined over k, and $\mathcal{X}'$ a stratification of $X'$.

Let $a \le b$ be integers. Let M be an object of $\mathrm{D}_c^b(X)$ with perverse amplitude $[a, b]$ such that $\alpha^*M$ is tame and constructible along $\mathcal{X}'$.

There exist an integer $d \ge 1$, a finite extension $k'$ of k and a finite family $(S_f)_{f \in \mathcal{F}}$ of tacs of $A_{k'}$, such that

(1) The integer $d$ and the size of $\mathcal{F}$ are bounded in terms of $c_u(M)$ and the data $(X, X', \alpha, \mathcal{X}')$,

(2) Each tac $S_f$ has degree at most $d$,

(3) The degree of $k'$ is at most $d$,

with the property that for any $\chi \in \hat{A}_{k'}$ which does not belong to the union of the $S_f$, the object $Rf_*(M_{\chi})$ has perverse amplitude $[a, b]$.

By [109, Prop. 1.6.7], for any object M of $\mathrm{D}_c^b(X)$, there does exist an alteration $\alpha: X' \to X$ (in fact, a finite surjective morphism) and a stratification $\mathcal{X}'$ of $X'$ such that $\alpha^*M$ is tame and constructible along $\mathcal{X}'$. In particular, the following corollary follows.

**COROLLARY 2.25.** Let S be a quasi-projective scheme over k and let A be an abelian variety over k. Define $X = A \times S$ and denote $f: A \times S \to S$ the projection.
Let $a \le b$ be integers and let $M$ be an object of $\mathrm{D}_c^b(X)$ with perverse amplitude $[a, b]$. There exist a finite extension $k'$ of $k$ and a finite family $(S_f)_{f \in \mathcal{F}}$ of tacs of $A_{k'}$ such that for any character $\chi \in \hat{A}_{k'}$ that does not belong to the union of the $S_f$, the object $Rf_*(M_\chi)$ has perverse amplitude $[a, b]$.

For the proof of Theorem 2.24, we use the following special case of [109, Th. 3.1.1].

**THEOREM 2.26 (Orgogozo).** Let $f: X \to Y$ be a proper morphism defined over $k$. Let $\alpha: X' \to X$ be an alteration and $\mathcal{X}'$ a stratification of $X'$. Then there exist an alteration $\beta: Y' \to Y$ and a stratification $\mathcal{Y}'$ of $Y'$ such that for any object $M$ of $\mathrm{D}_c^b(X)$, the condition that $\alpha^*(M)$ is tame and constructible along $\mathcal{X}'$ implies that $\beta^*Rf_*M$ is tame and constructible along $\mathcal{Y}'$.

**PROOF OF THEOREM 2.24.** By shifting and Verdier duality, it is enough to prove the weaker statement where “M is of perverse amplitude $[a, b]$” is replaced by “M is semiperverse”.

Apply Theorem 2.26 to the proper morphism $f: A \times S \to S$ and to the alteration $\alpha$. We obtain an alteration $\beta: S' \to S$ and a stratification $\mathcal{S}'$ of $S'$ such that $\beta^*Rf_*M$ is tame and constructible along $\mathcal{S}'$. Note that since any $\mathcal{L}_\chi$ is lisse and tame, $\alpha^*M_\chi$ is tame and constructible along $\mathcal{X}'$ (see [109, 5.2.5] for details), and hence the complex $\beta^*Rf_*M_\chi$ is also tame and constructible along $\mathcal{S}'$ for any $\chi \in \hat{A}$.

Consider the image of the stratification $\mathcal{S}'$ by $\beta$. By Chevalley's theorem, it is a covering of $S$ by constructible sets, but not necessarily a partition. Refine this covering and remove redundant strats in order to obtain a stratification $\mathcal{S}$ of $S$ where all strats are equidimensional. Then refine the stratification $\mathcal{S}'$ in such a way that preimages by $\beta$ of strats of $\mathcal{S}$ are union of strats of $\mathcal{S}'$ and that $\beta$ induces surjective morphisms from each strat of $\mathcal{S}'$ to a strat of $\mathcal{S}$.

Let $\chi \in \hat{A}$. Even if the complex $Rf_*M_\chi$ is not necessarily constructible along $\mathcal{S}$, it has the property that for any strat $S_i$ of $\mathcal{S}$, the support of the restriction of each cohomology sheaf of $Rf_*M_\chi$ to $S_i$ is either $S_i$ or empty, since the analogue property holds for $\beta^*Rf_*M_\chi$ and the stratification $\mathcal{S}'$, and $\beta$ is surjective from a strat of $\mathcal{S}'$ to one of $\mathcal{S}$.

Consider now the preimage of the stratification $\mathcal{S}$ by $f$, and also the image of the stratification $\mathcal{X}'$ of $X'$ by $\alpha$. Choose a stratification $\mathcal{X}$ of $X$ that refines both these coverings of $X$, with the property that for any strats $X_i$ and $S_j$ of $\mathcal{X}$ and $\mathcal{S}$ such that $f(X_i) \subset S_j$, the restriction of $f$ to $X_i$ is smooth (in particular, that $X_i$ is equidimensional above $S_j$). Now refine $\mathcal{X}'$ similarly to $\mathcal{S}'$, in such a way that preimages by $\alpha$ of strats of $\mathcal{X}$ are union of strats of $\mathcal{X}'$ and $\alpha$ induces surjective morphisms from any strat of $\mathcal{X}'$ to a strat of $\mathcal{X}$.

By Lemma 1.12, up to replacing $k$ with a finite extension of degree bounded in terms of $c_u(M)$ (and the fixed data $(X, X', \alpha, \mathcal{X}')$), we can assume that each strat $S_i$ of $\mathcal{S}$ has a $k$-rational point $s_i$. We now apply Corollary 2.20 for each $i$ to the restriction $M_{s_i}$ of $M$ to $f^{-1}(s_i) \simeq A$ for each $i$, obtaining extensions $k_i$ of $k$ and families $(S_{f,i})_{f \in \mathcal{F}_i}$ of tacs of $A_{k_i}$ satisfying the properties of this corollary.

Let $k'$ be the compositum of all $k_i$, which has degree bounded in terms of $c_u(M)$ and the fixed data. We claim that for any character $\chi \in \hat{A}_{k'}$ that belongs to none of the tacs $S_{f,i}$ for any $i$, the object $Rf_*M_\chi$ is semiperverse. This will conclude the proof.

Suppose that the claim fails for some $\chi$. Then there exists an integer $k \in \mathbb{Z}$ such that

$$ \dim \mathrm{Supp}(\mathcal{H}^k(Rf_*(M_\chi))) > -k. $$

Since $\mathrm{Supp}(\mathcal{H}^k(Rf_*(M_\chi)))$ is a union of strats of $\mathcal{S}$, there is a strat $S_i \subset \mathrm{Supp}(\mathcal{H}^k(Rf_*(M_\chi)))$ of $\mathcal{S}$ of dimension $> -k$. In particular, we have $\mathcal{H}^k(Rf_*(M_\chi))_{s_i} \neq 0$. By proper base change, we have $\mathcal{H}^k(Rf_*(M_\chi))_{s_i} = \mathrm{H}^k(A_{\bar{k}} \times \{s_i\}, M_{s_i\chi})$, and hence the latter is also non-zero. From the
assumption on $\chi$ and Corollary 2.20, we have

$$H^k(A_{\bar{k}} \times \{s_i\}, M_{s_i\chi}) \simeq H^0(A_{\bar{k}} \times \{s_i\}, \mathfrak{p}\mathcal{H}^k(M_{s_i})\chi),$$

and hence $\mathfrak{p}\mathcal{H}^k(M_{s_i}) = \mathfrak{p}\mathcal{H}^0(M_{s_i}[k]) \neq 0$. By definition of the perverse t-structure, this implies that there exists some $r \in \mathbb{Z}$ such that

$$\dim \mathrm{Supp}(\mathcal{H}^r(M_{s_i})) \geq -r + k.$$

The support of $\mathcal{H}^r(M)$ is a union of strata of $\mathcal{X}$, so there exists a strat $X_j \subset \mathrm{Supp}(\mathcal{H}^r(M))$ of $\mathcal{X}$ with $\dim(X_j \cap A \times \{s_i\}) = \dim \mathrm{Supp}(\mathcal{H}^r(M_{s_i}))$. Since $X_j$ is equidimensional over $S_i$ and $\dim(S_i) > -k$, we conclude that

$$\dim \mathrm{Supp}(\mathcal{H}^r(M)) \geq \dim(X_j) \geq -r + k + \dim(S_i) > -r,$$

contradicting the semiperversity of M. $\square$

We now prove a vanishing theorem for higher cohomology groups of perverse sheaves on abelian varieties. We begin with an analogue of Theorem 2.15.

**PROPOSITION 2.27.** Let A be a $g$-dimensional algebraic variety over $k$, let S be a quasi-projective scheme over $k$, and define $X = A \times S$. Fix a projective embedding $u$ of X.

Let $\alpha: X' \to X$ be an alteration and $\mathcal{X}'$ a stratification of $X'$.

Let $i$ be an integer with $1 \leq i \leq g$. Let $a \leq b$ be integers.

Let M be an object of $\mathrm{D}_c^b(X)$ with perverse amplitude $[a, b]$ such that $\alpha^*M$ is tame and constructible along $\mathcal{X}'$. There exist a finite extension $k'$ of $k$ and a family $(S_f)_{f \in \mathcal{F}}$ of tacs of $A_{k'}$ of dimension $\leq d - i$ with the property that for any $\chi \in \hat{A}_{k'}$ which does not belong to the union of the $S_f$ there exists a quotient abelian variety $q: A_{k'} \to B$ of dimension at most $i - 1$ such that $R_{q_*M_\chi}$ has perverse amplitude $[a, b]$.

Moreover, the degree of $k'$ over $k$ and the size of $\mathcal{F}$ depend only on $c_u(M)$ and $(X, X', \alpha, \mathcal{X}')$.

**PROOF.** As in the proof of Theorem 2.26, we can work with each perverse cohomology sheaf, and it is therefore enough to prove the proposition for $a = b = 0$, which means that M is perverse.

By Poincaré's complete reducibility theorem, up to replacing $k$ with a finite extension, there exists an isogeny $f: A \to B$ over $k$ where B is a product of geometrically simple abelian varieties. We first claim that it is enough to prove the proposition for B.

To see this, we assume that the statement holds for B. Consider the base change $f_B: X \to B \times S$. Since $f$ is finite, $f_{B^*}(M)$ is perverse for every perverse sheaf M on X. By Theorem 2.26, we find an alteration $\beta: B' \to B \times S$ and a stratification of $B'$ such that $\beta^*f_{B^*}(M_\chi)$ is tame and adapted for every M such that $\alpha^*M$ is tame and adapted to $\mathcal{X}'$. Then the proposition can be applied to $f_{B^*}(M_\chi)$. Let N be the kernel of the isogeny $f$. Choose up to $|N|$ characters of A whose restrictions to N run over the character group of N. Then the proposition for A follows by applying the result for B to the objects $f_{B^*}(M_\chi)$, where $\chi$ varies among this finite set of characters. This proves the claim.

So we assume that $A = A_1 \times \cdots \times A_s$ is a product of geometrically simple abelian varieties. Set $g_j = \dim(A_j)$ for all $j$. For any subset $I \subset [s]$, let

$$A_I = \prod_{i \in I} A_i,$$

viewed as a subvariety of A, and let $A_I^\perp = A_{[s]-I}$ be the kernel of the canonical projection $A \to A_I$.
Fix an integer $1 \le i \le g = \dim(A)$. Let $\mathcal{E}$ be the set of subsets $I \subset [s]$ such that $\dim(A_I) < i$; for $I \in \mathcal{E}$, we have $\dim(A_I^\perp) > g - i$.

Fix $I \in \mathcal{E}$. Let $p: A \times S \to A_I \times S$ be the projection. We apply Theorem 2.24 to $p$ and $M$, i.e., with $(A, S)$ there equal to $(A_I^\perp, A \times S)$. Up to replacing $k$ by a finite extension $k'$, we obtain a finite family $(S_{I,j})_{j \in X_I}$ of tacs of $A_{I,k'}^\perp$ such that the object $R_{p*}(M_\chi)$ is perverse on $A_I \times S$ for any $\chi \in \hat{A}_{I,k'}^\perp$ not in the union of these tacs. Let

$$ (\pi_{I,j}, \chi_{I,j})_{j \in X_I} $$

be the projection and characters defining these tacs, and let $K_{I,j} = \ker(\pi_{I,j})$, viewed as a subgroup of $A_{k'}$.

Let $\mathcal{F}$ be the set of all maps $f$ from $\mathcal{E}$ to the disjoint union of the $\mathcal{I}_1$ that send a subset $I$ to an element $j \in X_I$ for each $I$; this set is finite. For $f \in \mathcal{F}$, let $S_f$ be the intersection of the tacs of $A_{k'}$ defined by

$$ (A_{k'} \to A_{k'}/K_{I,f(I)}, \chi'_{I,f(I)}) $$

for $I \in \mathcal{E}$.

We claim that the family $(S_f)_{f \in \mathcal{F}}$ (to be precise, the subfamily where $S_f$ is not empty) satisfies the assertions of the theorem.

Indeed, first of all Lemma 1.25 shows that $S_f$ is either empty or is again a tac; moreover, in the second case, it is defined by the projection $A_{k'} \to A_{k'}/B_f$ where $B_f$ is the abelian subvariety in $A_{k'}$ generated by the $K_{I,f(I)}$, viewed as subvarieties of $A_{k'}$. For such $f$, by Lemma 1.26 applied to $A$ and the subgroups $K_{I,f(I)}$, we have $\dim(B_f) \ge i$, and hence the quotient

$$ p_f: A_{k'} \to A_{k'}/B_f $$

has image of dimension $\le d - i$.

Finally, let $\chi \in \hat{A}_{k'}$ be a character that does not belong to any of the tacs $S_f$. This implies that there exists some $f \in \mathcal{F}$, some subset $I \subset \mathcal{E}$ and some $j \in X_I$ such that the restriction $\chi_I$ of $\chi$ to $A_{I,k'}^\perp$ is not equal to $\chi_{I,j}$.

We can write $\chi = \chi_I\chi'$ where $\chi'$ is a character of $A_{I,k'}$. Then, considering the particular quotient $q: A_{k'} \to A_{I,k'}$, the base change $q_S$ is the canonical projection $q_I$ and hence

$$ R_{q_S*}M_\chi = R_{q_S*}(M_{\chi_I}) \otimes \mathcal{L}_{\chi'} $$

is perverse.

□

As in the case of tori, we state two further consequences that are useful in applications.

**COROLLARY 2.28.** Let $A$ be an abelian variety defined over $k$ of dimension $g$. Let $M$ be a perverse sheaf on $A$. For $-g \le i \le g$, the sets

$$ \{\chi \in \hat{A} \mid \mathrm{H}^i(A_{\bar{k}}, M_\chi) \neq 0\} $$

are contained in a finite union of tacs of $A$ of dimension $\le g - |i|$, and in particular they have character codimension at least $|i|$.

**PROOF.** We argue as in the proof of Corollary 2.16 using the previous theorem (with $a = b = 0$), as we may since we have recalled that one can find an alteration $\alpha$ of $A$ such that the pull-back $\alpha^*M$ is tame. □

**COROLLARY 2.29.** Let $A$ be a $g$-dimensional algebraic variety over $k$, let $S$ be a quasi-projective scheme over $k$, and define $X = A \times S$. Fix a projective embedding $u$ of $X$ and denote by $q$ the projection $X \to S$.
Let $\alpha: X' \to X$ be an alteration and $\mathcal{X}'$ a stratification of $X'$.

Let M be a perverse sheaf on X such that $\alpha^*M$ is tame and constructible along $\mathcal{X}'$. There exists a finite extension $k'/k$ and a partition of $\hat{A}_{k'}$ into subsets $(S_i)_{0\le i\le g}$ of character codimension $\ge i$ such that for any $i$ and $\chi \in S_i$, the object $Rq_i M_\chi$ has perverse amplitude $[-i, i]$.

Moreover, for any integer $n \ge 1$, we have

$$ (2.7) \qquad |S_i(k_n)| \ll |k|^{n(g-i)}, $$

where the implied constant depends only on $(c_u(M), X, X', \alpha, \mathcal{X}')$.

**PROOF.** We argue as in the proof of Corollary 2.18 for the first part; to deduce (2.7), we simply note for each $i \le g$, the number of tacs in Proposition 2.27 is bounded in terms of the indicated data, and for each tac S of dimension $i$, the number of characters in $S(k_n)$ is $\le (|k_n|^{1/2}+1)^{2i} \ll |k|^{ni}$. $\square$

## 2.5. Proof of the general vanishing theorem

We can now prove Theorems 2.1 and 2.3.

We consider the dévissage of Proposition 1.13. Namely, let A be an abelian variety, T a torus, U a unipotent group and N a finite commutative subgroup scheme of $A \times U \times T$ such that G is isomorphic to $(A \times U \times T)/N$. Further, we write $N = N_r \times N_l$ where $N_r$ is reduced and $N_l$ is local.

Let M be a perverse sheaf on G.

**Step 1.** We claim that it is enough to prove the theorems for the group $\tilde{G} = A \times U \times T$.

Indeed, since $N = N_r \times N_l$, the quotient morphism $p: \tilde{G} \to G$ can be factored as the composition of an étale isogeny and a purely inseparable one. The latter is a universal homeomorphism, and since universal homeomorphisms preserve the étale site, and since pull-back by a finite étale map preserves perversity, it follows that the pull-back $p^*(M)$ is perverse.

Assume that the result of Theorem 2.3 holds for $p^*(M)$ on $\tilde{G}$. Then we obtain the vanishing theorem for M as follows. Let $\mathcal{S}'_i$ be the subsets of loc. cit. for $p^*(M)$ on $\tilde{G}$, and define $\mathcal{S}_i$ to be the set of $\chi \in \tilde{G}$ such that $\chi \circ p \in \mathcal{S}'_i$. Since G has the same dimension as $\tilde{G}$ and $\mathcal{S}'_i$ has character codimension $i$, do does $\mathcal{S}_i$.

If $\chi \in \tilde{G}$, then the projection formula gives isomorphisms

$$ H^i(\tilde{G}_{\bar{k}}, p^*(M_{\chi})) = H^i(\tilde{G}_{\bar{k}}, p^*(M)_{\chi \circ p}) $$

for all $i \in \mathbb{Z}$.

The vanishing of $H^i(\tilde{G}, p^*(M_\chi)_{\chi \circ p})$ implies that of $H^i(G_{\bar{k}}, M_\chi)$, since the latter space is a direct summand of the former. A similar argument applies for compactly-supported cohomology, which shows that the characters $\chi \in \tilde{G}$ such that any of the groups (2.1) is non-zero belong to $\mathcal{S}_i$.

Finally, suppose that $\chi \in \mathcal{S}_0 - \mathcal{S}_1$, so that $\chi \circ p \in \mathcal{S}'_0 - \mathcal{S}'_1$. Since the forget support map is functorial, the forget support morphism

$$ H_c^0(\tilde{G}_{\bar{k}}, p^*(M_{\chi})) \to H^0(\tilde{G}_{\bar{k}}, p^*(M_{\chi})) $$

induces by restriction the forget support morphism

$$ H_c^0(G_{\bar{k}}, M_{\chi}) \to H^0(G_{\bar{k}}, M_{\chi}), $$

and since the former is an isomorphism (from our assumption that Theorem 2.3 holds for $\tilde{G}$), so is the latter. This concludes the proof of the claim of Step 1.
**Step 2.** We now assume that $G = A \times U \times T$, and will prove Theorem 2.3. We fix a quasi-projective immersion $u$ of $G$. Let $d_A = \dim(A)$, $d_U = \dim(U)$, $d_T = \dim(T)$, and $d = d_A+d_U+d_T = \dim(G)$. We denote by $p_T: A \times U \times T \to A \times U$ the canonical projection.

Up to replacing $k$ by a finite extension, we can assume that $T$ is split. By applying Theorem 2.15 and Corollary 2.18 with $S = A \times U$, we can partition $\hat{T}$ into subsets $(S_i)_{0 \le i \le d_T}$ of character codimension $\ge i$ such that

$$ (1) \text{ if } \chi \in S_i, \text{ then the complex } \mathrm{R}p_{T!}(M_{\chi}) \in \mathrm{D}_c^b(A \times U) \text{ is of perverse amplitude } [-i, i]. $$

$$ (2) \text{ if } \chi \in \hat{T} - S_1, \text{ then } \mathrm{R}p_{T!}(M_{\chi}) = \mathrm{R}p_{T*}(M_{\chi}). $$

We now wish to apply Proposition 2.27 to $A \times U$, but we first need to find an alteration that moderates all complexes $\mathrm{R}p_{T!}(M_{\chi})$.

Let $j: T \to \bar{T} = (\mathbf{P}^1)^{d_T}$ be the obvious compactification of $T$. By [109, Prop. 1.6.7], there exists an alteration $\alpha: X \to A \times U \times \bar{T}$ and a stratification $\mathcal{X}$ of $X$ such that $\alpha^*(j!M)$ is tame and constructible along $\mathcal{X}$. For each character $\chi \in \hat{T}$, the sheaf $j!(\mathcal{L}_{\chi})$ is tame, and hence $\alpha^*(j!M_{\chi})$ is also constructible and tame along $\mathcal{X}$ (see [109, 5.2.5] for details).

We apply Theorem 2.26 to the proper projection $A \times U \times \bar{T} \to A \times U$. This provides us with an alteration $\beta: X' \to A \times U$ and a stratification $\mathcal{X}'$ of $X'$ such that the complex $\beta^*\mathrm{R}p_{T!}(M_{\chi})$ is tame and constructible along $\mathcal{X}'$ for every $\chi \in \hat{T}$. Moreover, by Proposition 1.18 and Theorem 1.5, the complexity of $\mathrm{R}p_{T!}(M_{\chi})$ is bounded independently of $\chi \in \hat{T}$.

We can now apply Corollary 2.29 to $S = U$ and the complexes $\mathrm{R}p_{T!}(M_{\chi})$. For each character $\chi \in \hat{T}$, we obtain a partition $(S_{\chi,j})_{0 \le j \le d_A}$ of $\hat{A}$ into subsets such that $S_{\chi,j}$ has character codimension at least $j$, with the property that for $(\chi, \xi) \in S_i \times S_{\chi,j}$, the complex $\mathrm{R}p_{A!}(\mathrm{R}p_{T!}(M_{\chi}))_{\xi}$ has perverse amplitude $[-i-j, i+j]$.

By Proposition 1.18 and Theorem 1.5, the complexity of the object $\mathrm{R}p_{A!}(\mathrm{R}p_{T!}(M_{\chi}))_{\xi}$ is bounded independently of $(\chi, \xi) \in S_i \times S_{\chi,j}$. Hence, by applying Proposition 2.9 to these objects we find for each $(\chi, \xi)$ a partition $(S_{\chi,\xi,m})_{0 \le m \le d_U}$ of $\hat{U}$ such that the set $S_{\chi,\xi,m}$ has character codimension at least $m$ and, moreover, we have

$$ H_c^n(\mathcal{G}_{\bar{k}}, M_{\chi\xi\psi}) = 0 $$

for each $\psi \in S_{\chi,\xi,m}$ unless $n \in [-i-j, i+j+m]$.

For $0 \le r \le d$, we now define $\tilde{\mathcal{F}}_r$ to be the set of characters $(\chi, \xi, \psi) \in \hat{G}$ such that

$$ \psi \in S_{\chi, \xi, m}, \quad \xi \in S_{\chi, j}, \quad \chi \in S_i $$

for some $i, j, m$ such that $i+j+m \ge r$.

For any integer $n \ge 1$, we have

$$ |\tilde{\mathcal{F}}_r(k_n)| = \sum_{i+j+m \ge r} \sum_{\chi \in S_i(k_n)} \sum_{\xi \in S_{\chi,j}(k_n)} |S_{\chi, \xi, m}(k_n)| \ll |k|^{n(d-(i+j+m))} \ll |k|^{n(d-r)} $$

by (2.4) and (2.7) (note that the uniformity with respect to the perverse sheaf in these estimates, and the uniform bound on the complexity, are crucial to control the sums over $\chi$ and $\xi$). Thus the set $\tilde{\mathcal{F}}_r$ has character codimension at least $r$.

By construction of the sets $S_i$, $S_{\chi,j}$ and $S_{\chi,\xi,m}$, the condition $H_c^i(\mathcal{G}_{\bar{k}}, M_{\chi\xi\psi}) \ne 0$, for $(\chi, \xi, \psi) \in \hat{G}$, implies that $(\chi, \xi, \psi) \in \mathcal{S}_{[i]}$. We apply a similar argument with $D(M)$ to obtain the analogue conclusion for ordinary cohomology and set $\mathcal{S}_i$ to be the intersection of the set $\tilde{\mathcal{F}}_i$ for $M$ and of the analogue for $D(M)$. By construction, the sets $\mathcal{S}_i$ satisfy the first two claims of Theorem 2.3.
We now establish the last claims of Theorem 2.3.

First, let $(\chi, \xi, \psi) \in \hat{\mathbb{G}} - \mathcal{S}_1$. By construction of $S_1$ through Theorem 2.15 (see point (2) above), we have $Rp_{T!}(M_\chi) = Rp_{T*}(M_\chi)$. Moreover $p_{A!} = p_{A*}$ since $p_A$ is proper, and by the last claim of Proposition 2.9, we obtain

$$H^0(G_{\bar{k}}, M_{\chi\xi\psi}) = H^0(U_{\bar{k}}, Rp_{A*}Rp_{T*}M_{\chi\xi\psi}) = H_c^0(U_{\bar{k}}, Rp_{A!}Rp_{T!}M_{\chi\xi\psi}) = H_c^0(G_{\bar{k}}, M_{\chi\xi\psi}).$$

Finally, if $G$ is a torus (resp. an abelian variety) then we use Corollary 2.16 (resp. Corollary 2.28) to prove that the sets $\mathcal{S}_i$ are contained in a finite union of tacs of $G$ of dimension $\le d - i$.

This finally concludes the proof.

□

**REMARK 2.30.** Once we have reduced the proof of Theorem 2.3 to a product, the order in which we handle the toric, unipotent and abelian variety parts of $G$ in the proof is essentially dictated by the fact that the current versions of Theorem 2.14 (the relative vanishing theorem for tori) and its corollaries are not uniform in terms of the complexity of the input object $M$.

However, if the toric part has dimension 1, it is not difficult to obtain such a statement, and thus to vary the proof. This is not entirely anecdotal, because the choice of order has implications on the structure of the sets $\mathcal{S}_i$ in Theorem 2.3.

We describe the special case of $G = \mathbf{G}_m \times \mathbf{G}_a$, which will be used in Chapter 9. We note first that if $M$ is a perverse sheaf on $\mathbf{G}_m$ over $k$, then the vanishing

$$H^i(\mathbf{G}_{m,\bar{k}}, M_\chi) = H_c^i(\mathbf{G}_{m,\bar{k}}, M_\chi) = 0$$

holds for $i \ne 0$ and $\chi$ outside of a set $\mathcal{S} \subset \hat{\mathbf{G}}_m$ such that $|\mathcal{S}(k_n)| \ll 1$ for all $n \ge 1$, where the implied constant depends only on the complexity of $M$ (one reduces to the case of a simple perverse sheaf, and then one can apply Lemma A.15, for instance). In particular, for $\chi \notin \mathcal{S}$, the complex $M_\chi$ is a perverse sheaf.

**THEOREM 2.31 (Stratified vanishing for $\mathbf{G}_m \times \mathbf{G}_a$).** Let $M$ a perverse sheaf on $G = \mathbf{G}_m \times \mathbf{G}_a$. There exist subsets

$$\mathcal{S}_2 \subset \mathcal{S}_1 \subset \mathcal{S}_0 = \hat{\mathbf{G}} \quad \text{and} \quad \mathcal{T} \subset \hat{\mathbf{G}}_a$$

such that the following holds:

(1) For $n \ge 1$, we have $|\mathcal{S}_2(k_n)| \ll 1$, $|\mathcal{T}(k_n)| \ll 1$ and $|\mathcal{S}_1(k_n)| \ll |k_n|$.

(2) For $0 \le i \le 2$, any $\chi \in \hat{\mathbf{G}}$ such that at least one of the cohomology groups

$$H^i(\mathbf{G}_{\bar{k}}, M_\chi), \quad H^{-i}(\mathbf{G}_{\bar{k}}, M_\chi), \quad H_c^i(\mathbf{G}_{\bar{k}}, M_\chi), \quad H_c^{-i}(\mathbf{G}_{\bar{k}}, M_\chi)$$

is non-zero belongs to $\mathcal{S}_i$.

(3) For $\chi \in \mathcal{S}_0 - \mathcal{S}_1$, the equality $H_c^0(\mathbf{G}_{\bar{k}}, M_\chi) = H^0(\mathbf{G}_{\bar{k}}, M_\chi)$ holds.

(4) For $\psi \in \hat{\mathbf{G}}_a - \mathcal{T}$, the set $\mathcal{T}_\psi$ of all $\chi \in \hat{\mathbf{G}}_m$ such that $\psi \boxtimes \chi \in \mathcal{S}_1$ satisfies the bound $|\mathcal{T}_\psi(k_n)| \ll 1$ for all $n \ge 1$, with an implied constant that only depends on $c(M)$.

**PROOF.** Let $p: G \to \mathbf{G}_m$ be the projection. For $\psi$ varying in $\hat{\mathbf{G}}_a$, the complexes $Rp_!(M_\psi)$ on $\mathbf{G}_m$ have bounded complexity. By Proposition 2.9, we can partition $\hat{\mathbf{G}}_a$ in subsets $S_0$ and $S_1$, with $S_1$ of character codimension $\ge 1$, such that $Rp_!(M_\psi)$ is perverse if $\psi \in S_0$.

Let $\psi \in S_0$. Then by the elementary remark before the statement, the set $S_{0,\psi}$ of $\chi \in \hat{\mathbf{G}}_m$ such that $Rp_!(M_\psi)_\chi$ is not perverse has the property that $|S_{0,\psi}(k_n)| \ll 1$ for all $n \ge 1$, where the implied
constant depends only on $c(M)$. The result now follows with

$$ \mathcal{S}_0 = \hat{G}, $$

$$ \mathcal{S}_1 = (\hat{G}_m \times S_0) \cup \{\chi \boxtimes \psi \mid \psi \in S_0 \text{ and } \chi \in S_{\psi,1}\}, $$

$$ \mathcal{T} = S_0, $$

and $\mathcal{S}_2$ the set of characters such that one of

$$ H^2(G_{\bar{k}}, M_{\chi}), \quad H^{-2}(G_{\bar{k}}, M_{\chi}), \quad H_c^2(G_{\bar{k}}, M_{\chi}), \quad H_c^{-2}(G_{\bar{k}}, M_{\chi}) $$

is non-zero (which satisfies $|\mathcal{S}_2(k_n)| \ll 1$ for all $n$ by the observation before the theorem; note that $\mathcal{S}_2 \subset \mathcal{S}_1$ because the existence for given $(\chi, \psi) \in \hat{G}$ of non-zero $H^2$ or $H_c^2$ implies that $Rp_!(M_\psi)$ is not perverse). $\square$

## 2.6. Proof of Theorem 2.6

We conclude this chapter with the proof of the partial geometric version of the vanishing theorem. This section is not needed for the rest of the results of this book, and in particular may be omitted by readers interested in equidistribution problems.

We will first prove Theorem 2.6 in the case $G = A \times T$, where A is an abelian variety and T is a torus. The case when $G = A \times U$ will be a simple adaptation of this argument.

We denote by $p_T: T \to \text{Spec}(k)$ and $p_A: A \to \text{Spec}(k)$ the structural morphisms. We will use the same notation for base-change morphisms, e.g. for the projection $p_T: G \to A$.

Let $M$ be a perverse sheaf on $G = A \times T$. By Corollary 2.25 applied to $p_A: G \to T$ and $M$, up to replacing $k$ by a finite extension, there is a finite union $\mathcal{S}'$ of tacs of A such that the complex $Rp_{A*}(M_\xi)$ is perverse for all $\xi \in \hat{A} - \mathcal{S}'$. By Theorem 2.14 applied to $p_T: G \to A$ and $M$, also up to replacing $k$ by a finite extension, there is a finite union $\mathcal{S}$ of tacs of T such that $Rp_{T!}(M_\chi) \simeq Rp_{T*}(M_\chi)$ for all $\chi \in \hat{T} - \mathcal{S}$, and in particular the object $Rp_{T!}(M_\chi)$ is then perverse.

Let $\xi \in \hat{A} - \mathcal{S}'$ and $\chi \in \hat{T} - \mathcal{S}$. We will prove that $(\xi, \chi)$ satisfies the properties of Theorem 2.6, thus concluding the proof for the case $G = A \times T$.

We start from the isomorphism $Rp_{T*}M_\chi = Rp_{T!}M_\chi$ of complexes on A, and tensor with $\mathcal{L}_\xi$. By the projection formula, and the fact that $\mathcal{L}_\xi$ is lisse, we obtain an isomorphism

$$ Rp_{T*}(M_{\xi\chi}) \simeq Rp_{T!}(M_{\xi\chi}). $$

Applying the functor $Rp_{A*} = Rp_{A*}$ and using functoriality in the square

$$ \begin{array}{ccc} A \times T & \xrightarrow{p_A} & T \\ \big|_{p_T} & & \big|_{p_T} \\ A & \xrightarrow{p_A} & \text{Spec}(k), \end{array} $$

we obtain an isomorphism

$$ (2.8) \qquad Rp_{T*}Rp_{A*}(M_{\xi\chi}) \simeq Rp_{T!}Rp_{A*}(M_{\xi\chi}). $$

The projection formula gives an isomorphism $Rp_{A*}(M_{\xi\chi}) \simeq Rp_{A*}(M_\xi) \otimes \mathcal{L}_\chi$, and this object is perverse by the choice of $\xi$ and the fact that $\mathcal{L}_\chi$ is lisse. So from (2.8), we deduce that the cohomology with and without compact support of the perverse sheaf $Rp_{A*}(M_{\xi\chi})$ on T are isomorphic. By Artin's vanishing theorem, since T is affine, this implies that only the degree zero component may be non-zero. Hence, we have proved that $H_c^i(G, M_{\xi\chi}) = H^i(G, M_{\xi\chi}) = 0$ for $i \neq 0$, and that there is an isomorphism $H_c^0(G, M_{\xi\chi}) \simeq H^0(G, M_{\xi\chi})$.
In the case when $G = A \times U$, we apply the same argument, *mutatis mutandis*, simply replacing the use of Theorem 2.14 by that of Proposition 2.11.

**REMARK 2.32.** The proof of Theorem 2.6 relies in an essential way on the fact that we have, for the factors A and T (or U), the isomorphism between cohomology with and without compact support. Characters that only satisfy vanishing properties do not always satisfy the conclusion, as the following example shows.

Let $E$ be an elliptic curve over $k$ and let $p: E \times \mathbf{G}_m \to E$ and $q: E \times \mathbf{G}_m \to \mathbf{G}_m$ be the two projections. Let $i: C \to E \times \mathbf{G}_m$ be a closed one-dimensional irreducible subvariety of $E \times \mathbf{G}_m$, with dominant projections to both $E$ and $\mathbf{G}_m$. Let $M = i_*\overline{\mathbb{Q}}_\ell[1]$ be the perverse sheaf on $E \times \mathbf{G}_m$ which is the intermediate extension of the shifted constant sheaf on $C$.

We claim that both $p!M$ and $q!M$ are perverse. Indeed, we have $q!M = q|_C i_* \overline{\mathbb{Q}}_\ell[1]$, which is perverse since $q|_C: C \to \mathbf{G}_m$ is a finite morphism, and hence is $t$-exact by [8, Cor. 2.2.6]. Moreover, $p!M = p|_C i_* \overline{\mathbb{Q}}_\ell[1]$ and $p|_C$ can be factored as a finite morphism followed by an affine open immersion, both of which are $t$-exact by [8, Cor. 4.1.3]. It follows that the trivial characters on $\mathbf{G}_m$ and $E$ are in the generic sets for the relative vanishing Theorem 2.14 applied to $p$ and for Corollary 2.25 applied to $q$. However, $H_c^1((E \times \mathbf{G}_m)_{\bar{k}}, M) \simeq H_c^2(C_{\bar{k}}, \overline{\mathbb{Q}}_\ell) \neq 0$, and hence the trivial character is not in the generic set given by Theorem 2.1 applied to $E \times \mathbf{G}_m$ and $M$.
# Tannakian categories of perverse sheaves

## 3.1. Introduction

Throughout this chapter, $k$ denotes a finite field and $\bar{k}$ an algebraic closure of $k$. We denote by $\ell$ a prime number different from the characteristic of $k$. All complexes we consider are $\ell$-adic complexes.

Let $G$ be a connected commutative algebraic group over $k$.

Let $M$ be a perverse sheaf on $G$. We wish to define a “symmetry group” that governs the statistical behavior of the arithmetic Fourier transform

$$S(M, \chi) = \sum_{x \in G(k_n)} \chi(x) t_M(x; k_n)$$

for $\chi \in \hat{G}(k_n)$. The fundamental mechanism for this is that the symmetry group $G$ should come with a faithful linear representation $G \subset GL_r$ for some $r \ge 0$, and to almost all characters $\chi$ there should be assigned an element (or conjugacy class) $\text{Fr}_\chi \in G$ such that $S(M, \chi)$ is the trace of $\text{Fr}_\chi$.

The idea behind the construction of the group $G$ (following Katz [75]) is based on the fact that we have a “geometric” control on the algebra structure on the space of arithmetic Fourier transforms through the link with convolution: for two objects $M_1$ and $M_2$ on $G$, we have

$$S(M_1, \chi)S(M_2, \chi) = \sum_{x \in \hat{G}(k_n)} \chi(x) (t_{M_1} * t_{M_2})(x; k_n),$$

where

$$(t_{M_1} * t_{M_2})(x; k_n) = \sum_{y \in G(k_n)} t_{M_1}(y; k_n) t_{M_2}(y^{-1}x; k_n),$$

for $x \in G(k_n)$, is the convolution product in the classical sense of Fourier analysis on $G(k_n)$.

It is fundamental that by the proper base change theorem and the trace formula, we can view this function as a trace function, namely

$$(t_{M_1} * t_{M_2})(x; k_n) = t_{M_1 * ! M_2}(x; k_n),$$

where $M_1 * ! M_2$ is the convolution with compact support (Section 1.5).

This geometric interpretation suggests to use the convolution as “tensor operation” to define a *tannakian category*, which would be equivalent to the category of representations of the desired symmetry group.

In essence, this is what we will do. However, there are some significant issues to handle:

- The first one, already present in the work of Katz for $G_m$, has to do with the fact that convolution with compact support does not always preserve perverse sheaves (for instance, if $G$ has dimension $d$, then the convolution $\mathcal{L}_{\chi_1}[d] * ! \mathcal{L}_{\chi_2}[d]$ is not perverse) or duality (because duality transforms the convolution $*!$ into the convolution $*$, which is different in general).
We can solve this first problem using a suitable quotient category where the two geometric convolution products turn out to coincide (this idea goes back to Gabber and Loeser and was also used by Katz).

- A related issue is that weights do not always behave well under convolution, in the case of affine groups at least. Since weights dictate the size of the sums $S(M, \chi)$, this is a crucial issue for our intended applications. This is again related to the difference between the two geometric convolutions, each of which leads in practice to inequalities in one direction for the weights.

- Finally there is a major new difficulty in comparison with the work of Katz. The link between the abstract tannakian ideas and the arithmetic Fourier transform is that for a character $\chi \in \hat{G}(k_n)$, the formula

$$S(M, \chi) = \mathrm{Tr}(\mathrm{Fr}_{k_n} | H_c^0(G_{\bar{k}}, M_{\chi}))$$

should hold. This is in fact (by the generic vanishing theorem) only true in general for a generic set of $\chi$ where the contributions of $H_c^i$ in the trace formula vanish for $i \neq 0$. But we also want “higher-order” versions of this formula to hold, namely for instance

$$S(M, \chi)^2 = \mathrm{Tr}(\mathrm{Fr}_{k_n} | H_c^0(G_{\bar{k}}, M_{\chi} *! M_{\chi})),$$

and so on for further powers (intuitively, this is because understanding the limits of averages of such expressions is necessary to apply the Weyl equidistribution criterion, as we will do in the next chapter). This amounts roughly to requesting that $M \mapsto H_c^0(G_{\bar{k}}, M_{\chi})$ should be compatible with convolution and so should (roughly) the generic vanishing theorem.

Thus we need to distinguish various types of characters depending on their behavior with respect to operations of this type.

## 3.2. Categories of objects defined over finite fields

We denote by $D(G)$ and $P(G)$ the full subcategories of $D_c^b(G_{\bar{k}})$ and $Perv(G_{\bar{k}})$ respectively whose objects are defined over some finite extension of the base field $k$. These categories are abelian categories, stable by shifts and duality. Moreover, the perverse cohomology sheaves of an object of $D(G)$ belong to $P(G)$.

We recall from Section 1.5 the definition and properties of the two convolution bifunctors $(M, N) \mapsto M_* N$ and $(M, N) \mapsto M_*! N$ for objects M and N of $D_c^b(G)$ or $D_c^b(G_{\bar{k}})$. These are compatible with base change, so that the convolutions on $G_{\bar{k}}$ preserve the category $D(G)$. In addition, the functor $M \mapsto M^V$ also induces a functor on $D(G)$ and $P(G)$.

## 3.3. Weakly unramified characters

**DEFINITION 3.1** (Weakly unramified characters). Let M be an object of $P(G)$. A character $\chi \in \hat{G}$ is said to be weakly unramified for M if the following holds:

$$H^i(G_{\bar{k}}, M_{\chi}) = H_c^i(G_{\bar{k}}, M_{\chi}) = 0 \quad \text{for all } i \neq 0,$$

$$H_c^0(G_{\bar{k}}, M_{\chi}) = H^0(G_{\bar{k}}, M_{\chi}).$$

We denote by $\mathcal{X}_w(M)$ the set of weakly unramified characters for M.

**REMARK 3.2.** (1) The terminology is suggested by analogy with the case of the additive group, in which the characters for which generic vanishing holds correspond to points at which the Fourier transform is lisse. However, we will see that the generic vanishing condition is not in general strong enough to obtain the properties we seek (namely, that the assignment $M \mapsto H_c^0(G_{\bar{k}}, M_{\chi})$ defines a
fiber functor on a suitable tannakian category of perverse sheaves on $G_{\bar{k}}$), see Remark 3.26 and Example 9.15. We will introduce unramified characters in Definition 3.25, as well as the variant of Frobenius-unramified characters in Definition 3.36.

(2) Since $G$ is not proper in general, the condition on the cohomology groups with support is not implied by the one on the cohomology groups without support.

With this definition, we can reformulate the Stratified Generic Vanishing Theorem 2.1 as follows:

**THEOREM 3.3.** *The subset $\mathcal{X}_w(M) \subset \hat{G}$ of weakly unramified characters for an object $M$ of $\mathbf{P}(G)$ is generic.*

### 3.4. Negligible objects

In general, none of the two convolution bifunctors on the derived category preserves the sub-category of perverse sheaves. As first observed in the case of tori by Gabber and Loeser [50], there is however a suitable quotient of the category $\mathbf{P}(G)$ on which both convolution functors induce the same bifunctor.

**DEFINITION 3.4.** An object $M$ of $\mathbf{P}(G)$ is said to be negligible if the set of characters $\chi \in \hat{G}$ satisfying $H^0(G_{\bar{k}}, M_\chi) = 0$ is generic. An object $N$ of $\mathbf{D}(G)$ is said to be negligible if all its perverse cohomology objects $\mathfrak{p}\mathcal{H}^i(N)$ are negligible.

We denote by $\text{Neg}_{\mathbf{P}}(G)$ and $\text{Neg}_{\mathbf{D}}(G)$ the full subcategories of $\mathbf{P}(G)$ and $\mathbf{D}(G)$ respectively consisting of negligible objects.

We denote by $K_{\text{neg}}(G)$ the subgroup of the Grothendieck group $K(G)$ generated by classes of negligible perverse sheaves, or equivalently by classes of negligible objects.

Given an object $M$ of $\mathbf{P}(G)$, set

$$ \mathcal{N}(M) = \{\chi \in \hat{G} \mid \mathrm{H}^i(G_{\bar{k}}, M_\chi) = \mathrm{H}_c^i(G_{\bar{k}}, M_\chi) = 0 \text{ for all } i\}. $$

Using Theorem 2.1, we see that $M$ is negligible if and only if $\mathcal{N}(M)$ is a generic subset of $\hat{G}$. For $M \in \text{Neg}_{\mathbf{D}}(G)$, we set

$$ \mathcal{N}(M) = \bigcup_{i} \mathcal{N}(\mathfrak{p}\mathcal{H}^i(M)). $$

It follows from the definition that, for each negligible perverse sheaf $M$ (resp. object of $\text{Neg}_{\mathbf{D}}(G)$), the perverse sheaf $M^\vee$ is also negligible (resp. the complex $M^\vee$ is negligible).

**EXAMPLE 3.5.** Any character sheaf $\mathcal{L}_\chi$ on $G$ is negligible. More generally, let $f: G \to H$ be a surjective morphism of algebraic groups such that the dimension $d$ of the kernel $\ker(f)$ is positive. Let $\eta \in \hat{G}$ and let $N$ be any object of $\mathcal{D}_c^b(H)$. We claim that the object $M = (f^*N)_\eta$ is negligible.

Indeed, let $i \in \mathbf{Z}$. We can factor $f = f_1 \circ f_2$, where $f_2$ is smooth of relative dimension $d$ and $f_1$ is radicial. Then $f_2^*[d]$ is t-exact (see [8, § 4.2.4]), and so is tensoring by $\mathcal{L}_\eta$ (Lemma 1.17), so there is a canonical isomorphism

$$ \mathfrak{p}\mathcal{H}^i((f^*N)_\eta) \simeq f_2^*(\mathfrak{p}\mathcal{H}^{i-d}(f_1^*(N)))_\eta. $$

For $\chi \in \hat{G}$, the projection formula leads to canonical isomorphisms

$$ \mathrm{H}^*(G_{\bar{k}}, M_\chi) \simeq \mathrm{H}^*(G_{\bar{k}}, f_2^*(\mathfrak{p}\mathcal{H}^{i-d}(f_1^*(N)))) \otimes \mathcal{L}_{\eta\chi}) \simeq \mathrm{H}^*(H_{\bar{k}}, \mathfrak{p}\mathcal{H}^{i-d}(f_1^*(N))) \otimes \mathrm{Rf}_{2!}\mathcal{L}_{\eta\chi}). $$

The complex $\mathrm{Rf}_{2!}\mathcal{L}_{\eta\chi}$ is zero if the restriction of $\eta\chi$ to the subgroup $\ker(f_2)^\circ$ is not the trivial character (see Lemma 1.16). Since this condition defines a generic set of characters $\chi$, we deduce that $\mathfrak{p}\mathcal{H}^i(M)$ is negligible, and the result follows.
REMARK 3.6. Intuitively, to say that M is negligible means that the arithmetic Fourier transform of M (see Section 1.8) satisfies $S(M, \chi) = 0$ for $\chi$ in a generic subset of $\hat{G}$. To illustrate this concrete aspect, we show how it explains the previous example. Thus consider $M = (f^*N)_\eta$, with notation as above for some $\eta \in \hat{G}(k)$. Let $\chi \in \hat{G}(k_n)$; the corresponding value of the Fourier transform is

$$ S(M, \chi) = \sum_{x \in G(k_n)} \chi(x) t_M(x; k_n) = \sum_{x \in G(k_n)} \chi(x) (\eta \circ N_{k_n/k})(x) t_N(f(x); k_n) \\ = \sum_{y \in H(k_n)} t_N(y; k_n) \sum_{\substack{x \in G(k_n) \\ f(x)=y}} \chi(\eta \circ N_{k_n/k})(x), $$

and the inner sum is either empty or a sum of a character over the $k_n$-points of a coset of the kernel of $f$, which vanishes unless $\chi = (\eta \circ N_{k_n/k})^{-1}$ on the kernel of $f$.

In some cases, one can show that, conversely, all simple negligible perverse sheaves are of the form $(f^*N)_\eta$ for some quotient morphism $f$ with kernel of dimension at least 1. This is for instance the case for abelian varieties, by a result of Weissauer [127, Lemma 6, Th. 3] (see also Remark 5.15) and we will prove later that this is also the case for $G_a \times G_m$ (see Section 9.4).

This structural property is however not always true. For instance, if $G$ is a unipotent group of dimension at least 2 (e.g., $G = G_a^d$ with $d \ge 2$), with Serre dual $G^\vee$, then we can take any object $N \in \mathrm{D}_c^b(G^\vee)$ whose support $S$ has codimension at least 1, and the inverse Fourier transform $M$ of $N$ will be a negligible object on $G$. If $S$ is not a translate of a subgroup of $G$, then the object $M$ is not obtained by pullback from any quotient of $G$. (In the terminology of [39, § 4], in the case of $G_a^d$, such objects are said to have A-number equal to 0, and they play a delicate role in certain analytic applications.)

We recall that a full subcategory $S$ of an abelian category $C$ is said to be a *Serre subcategory* if it is not empty, stable by extension and by subobject and quotient. A strictly full triangulated subcategory $S$ of a triangulated category $C$ is said to be *thick* if, for any morphism $f: X \to Y$ in $C$ which factors through an object of $S$, and which appears in a distinguished triangle

$$ X \xrightarrow{f} Y \to Z $$

with $Z$ object of $S$, the objects $X$ and $Y$ are in $S$.

LEMMA 3.7. The category $\mathrm{Neg}_P(G)$ is a Serre subcategory of $\mathbf{P}(G)$, and $\mathrm{Neg}_D(G)$ is a thick triangulated subcategory of $\mathbf{D}(G)$.

PROOF. Fix an exact sequence $X \to Y \to Z$ in $\mathbf{P}(G)$ such that $X$ and $Z$ are objects of $\mathrm{Neg}_P(G)$. By Theorem 3.3, there is a generic set of characters $\chi \in \hat{G}$ that are weakly unramified for $X$, $Y$, and $Z$. From the long exact sequence in cohomology, we find that for any such $\chi$, the vanishing $H^i(G_{\bar{k}}, Y_\chi) = H_c^i(G_{\bar{k}}, Y_\chi) = 0$ holds for all $i$, and hence $Y$ is negligible. The first statement follows easily. An argument of Gabber–Loeser (see [50, Prop. 3.6.1(i)]) then implies that $\mathrm{Neg}_D(G)$ is a thick triangulated subcategory of $\mathbf{D}(G)$. $\square$

LEMMA 3.8. For all objects $M$ and $N$ of $\mathbf{D}(G)$, the following properties hold:

(1) The cone of the canonical morphism $M *_! N \to M *_* N$ lies in $\mathrm{Neg}_D(G)$.

(2) If $M$ belongs to $\mathrm{Neg}_D(G)$, then so do $M *_! N$ and $M *_* N$ for each object $N$.

(3) If $M$ and $N$ belong to $\mathbf{P}(G)$, then $\mathfrak{P}\mathcal{H}^i(M *_! N)$ and $\mathfrak{P}\mathcal{H}^i(M *_* N)$ lie in $\mathrm{Neg}_P(G)$ for each non-zero integer $i$.

We omit the proof, which is the same as that of [91, Lem. 4.3].