simple perverse sheaf is $\mathcal{F}$[1]. Conversely, for a simple perverse sheaf M with support equal to X, of the form $j_*\mathcal{F}$[1] as in the theorem, the corresponding (irreducible) middle extension sheaf is $j_*\mathcal{F}$.

For simple perverse sheaves, the bounds on the dimension of the support of the cohomology sheaves have an “automatic improvement” from the bound given by the semi-perversity, except for $\mathcal{H}^{-\dim(X)}$.

**PROPOSITION A.13.** Let M be a simple perverse sheaf on X which is not punctual. Then for any $i \neq -\dim(\text{supp}(X))$, we have

$$ \dim \text{supp}(\mathcal{H}^i(M)) \leq -i - 1. $$

**PROOF.** This results from the classification of simple perverse sheaves and from the general description of the intermediate extension functor in [8, Prop. 2.1.11]. □

**EXAMPLE A.14.** In the case of a curve, this property can be seen from Example A.10, since in that case any simple perverse sheaf which is not punctual is supported on a dense open subset.

We thank S. Morel for communicating us a proof of the following lemma (see also [73, Sublemma 1.10.5]).

**LEMMA A.15.** Let $k$ be an algebraically closed field. Let $X$ be an irreducible projective variety of dimension $d$ over $k$, and let $M$ be a simple perverse sheaf on $X$ such that $H^{-d}(X, M)$ is non-zero. Then the support of $M$ is $X$ and there exists an open immersion $j: U \hookrightarrow X$ such that $M = j_*\bar{Q}_\ell[d]$.

**PROOF.** By the classification of simple perverse sheaves in Theorem A.11, there exists a locally closed immersion $j: U \hookrightarrow X$ and a simple lisse sheaf $\mathcal{F}$ on $U$ such that $M = j_*\mathcal{F}[\dim(U)]$. The cohomolog $H^i(X, M)$ vanishes unless $|i| \leq \dim(U)$ (see [8, 4.2.4]), so the assumption implies that $\dim(U) = d$. Besides, the formula for intermediate extensions from [8, 2.1.11] implies the vanishing $\mathcal{H}^i(M) = 0$ for $i < -d$. From the spectral sequence

$$ E_2^{p,q} = H^p(X, \mathcal{H}^q(M)) \implies H^{p+q}(X, M), $$

we then get an isomorphim $H^{-d}(X, M) \simeq H^0(X, \mathcal{H}^{-d}(M))$. The non-vanishing of this cohomology group implies that $\mathcal{H}^{-d}(M)$ has a global section. Hence, $\mathcal{F}$ has a global section and is therefore trivial. □

### A.3. Weights

In this section we assume that $k$ is a finite field of characteristic $p$, and denote by $\bar{k}$ an algebraic closure. We also fix an isomorphism $\iota: \bar{Q}_\ell \to \mathbb{C}$.

Let $q$ be a prime power and let $w \in \mathbb{Z}$ be an integer. An element $x \in \bar{Q}_\ell$ is said to be a $q$-Weil number of weight $w$ if it is algebraic over $\mathbb{Q}$, and if all the complex conjugates of $\iota(x)$ are complex numbers with modulus $q^{w/2}$.

Let $X$ be a separated scheme of finite type over $k$ and $\mathcal{F}$ a $\bar{Q}_\ell$-sheaf on $X$. Let $x$ be a closed point of $X$, with residue field $k(x)$. Viewing $k(x)$ as a subfield of the fixed algebraic closure $\bar{k}$ of $k$ defines a geometric point $\tilde{x}: \text{Spec}(\bar{k}) \to X$ supported at $x$. The geometric Frobenius automorphism, inverse of $y \mapsto y^{k(x)}$ in $\text{Gal}(\bar{k}/k)$, acts on the stalk $\mathcal{F}_{\tilde{x}}$ of $\mathcal{F}$ at $\tilde{x}$. We denote by $\text{Fr}_x$ this endomorphism of $\mathcal{F}_{\tilde{x}}$, which is well-defined up to conjugacy.

**DEFINITION A.16 ([28, 1.2], [8, 5.1.5]).** Let $X$ be a separated scheme of finite type over $k$, $\mathcal{F}$ a $\bar{Q}_\ell$-sheaf on $X$, and $M$ an object of $\text{D}_c^b(X)$.
(1) The sheaf $\mathcal{F}$ is punctually pure of weight $w$ if for every $x \in |X|$, the eigenvalues of $\text{Fr}_x$ are $|k(x)|$-Weil numbers of weight $w$.

(2) The sheaf $\mathcal{F}$ is mixed if it admits a finite filtration with successive quotients that are punctually pure. The weights of the non-zero quotients are called the punctual weights of $\mathcal{F}$.

(3) The complex M is mixed if all its cohomology sheaves are mixed. It is mixed of weights $\le w$ if for every $i \in \mathbb{Z}$, the sheaf $\mathcal{H}^i(M)$ is mixed with punctual weights $\le w + i$. It is mixed of weights $\ge w$ if its Verdier dual $\text{D}(M)$ is mixed of weights $\le -w$.

(4) The complex M is pure of weight $w$ if it is both mixed of weights $\le w$ and of weights $\ge w$.

### REMARK A.17.
Deligne also defines $\iota$-weights and $\iota$-pure or mixed sheaves and complexes for any fixed isomorphism $\iota$; the notion above means that the objects are $\iota$-pure for all $\iota$ (see [28, 1.2.6]).

We write $\text{D}_{\le w}(X)$ and $\text{D}_{\ge w}(X)$ for the full subcategories of $\text{D}_c^b(X)$ of objects mixed of weights $\le w$ and $\ge w$. Thanks to the shift in the definition, one has in particular $\text{D}_{\le w}[1] = \text{D}_{\le w+1}$.

### EXAMPLE A.18.
(1) Suppose that $X$ is smooth of pure dimension $d$, and that $M \in \text{D}_c^b(X)$ is such that all its cohomology sheaves are lisse on $X$. Then $M$ is pure of weight $w$ if and only if each sheaf $\mathcal{H}^i(M)$ is punctually pure of weight $w + i$.

(2) The characterization of (1) does not apply in general. For instance, let $X = \mathbf{A}^1$ be the affine line, and $j: \mathbf{G}_m \to X$ the open immersion. Let $M = (j_*\text{Kl}_2)[1](1/2)$ be the Kloosterman sheaf of rank 2 shifted to be in degree $-1$ and Tate-twisted to be of weight 0 (see (B.2)). Then $M$ is pure of weight 0. However, the cohomology sheaf $\mathcal{H}^{-1}(M) = j_*\text{Kl}_2(1/2)$ is not punctually pure of weight $-1$: indeed, the stalk of this sheaf at 0 has rank 1 with a Frobenius eigenvalue of weight $-2$.

(3) If $\mathcal{F}$ is a middle extension sheaf on $X$ (see Example A.12 for the definition), we say that $\mathcal{F}$ is pure of weight $w$ if the perverse sheaf $\text{M}[\text{dim}(X)](\text{dim}(X)/2)$ is pre of weight $w$. This is equivalent to the condition that the restriction of $\mathcal{F}$ to any dense open set where it is lisse is punctually pure of weight $w$.

Deligne's main theorem in [28, 3.3.1, 6.2.3], which directly implies the most general form of the Riemann Hypothesis over finite fields, is the following:

### THEOREM A.19 (Deligne).
Let $f: X \to Y$ be a separated morphism of schemes of finite type over $k$. Then the functor $\text{Rf}_!$ sends $\text{D}_{\le w}(X)$ to $\text{D}_{\le w}(Y)$.

Using duality, one gets the following list of compatibilities of the different functors on $\text{D}_c^b(X)$ (see [8, 5.1.14]):

(1) $\text{Rf}_!$ and $f^*$ preserve $\text{D}_{\le w}$;

(2) $\text{Rf}_*$ and $f'$ preserve $\text{D}_{\ge w}$;

(3) $\otimes$ sends $\text{D}_{\le w} \times \text{D}_{\le w'}$ to $\text{D}_{\le w+w'}$;

(4) $\text{RHom}$ sends $\text{D}_{\le w} \times \text{D}_{\ge w'}$ to $\text{D}_{\ge -w+w'}$;

(5) Verdier duality exchanges $\text{D}_{\le w}$ and $\text{D}_{\ge -w}$.

## A.4. Trace functions

We continue with the notation of the previous section, so that $X$ is an algebraic variety over a finite field $k$.
Let M be a complex in $D_c^b(X)$. For any integer $n \ge 1$ and $x \in X(k_n)$, the stalk $M_{\bar{x}}$ of M at a geometric point $\bar{x}$ above $x$ is a complex of finite-dimensional $\overline{\mathbb{Q}}_\ell$-vector spaces, with only finitely many non-zero cohomology spaces. The geometric Frobenius $\text{Fr}_{k_n}$ of $k_n$ (the inverse of the automorphisme $a \mapsto a^{|k_n|}$ of $k_n$) acts on $M_{\bar{x}}$, and this action is independent of the choice of $\bar{x}$ up to conjugacy. We denote

$$t_M(x; k_n) = \sum_{i \in \mathbb{Z}} (-1)^i \text{Tr}(\text{Fr}_{k_n} | \mathcal{H}^i(M)_{\bar{x}}),$$

which is also independent of $\bar{x}$ above $x$.

Whenever we have fixed the isomorphism $\iota_0: \overline{\mathbb{Q}}_\ell \to \mathbb{C}$ (as in the whole of the main text, see Section 3), we will view the trace function as a function $X(k_n) \to \mathbb{C}$ whenever convenient.

**DEFINITION A.20.** The trace function $t_M$ of M is the data of the whole family of functions $(t_M(\cdot; k_n))_{n \ge 1}$.

**REMARK A.21.** We will sometimes write simply $t_M(x)$ for $t_M(x; k)$, when $x \in X(k)$.

Viewing $X(k_n)$ as a subset of $X(\bar{k})$, we will also sometimes denote the stalk of M simply by $M_x$, instead of introducing explicitly a specific geometric point over $x$.

Let $f: X \to Y$ be a morphism of algebraic varieties over $k$. The following properties holds for objects $M_i$ and $M$ of $D_c^b(X)$ and $N$ of $D_c^b(Y)$:

$$ \begin{align*} t_{\overline{\mathbb{Q}}_\ell} &= 1 \quad (\overline{\mathbb{Q}}_\ell \text{ in degree } 0) \\ t_{M[k]} &= (-1)^k t_M, \quad t_{M(w)} = q^{-w/2} t_M \\ t_{M_2} &= t_{M_1} + t_{M_3} \quad \text{for any distinguished triangle } M_1 \to M_2 \to M_3 \to, \\ t_{M_1 \otimes M_2} &= t_{M_1} t_{M_2} \\ t_{f^*N} &= t_N \circ f, \quad \text{i.e.} \quad t_{f^*N}(x; k_n) = t_N(f(x); k_n) \text{ for all } n \ge 1 \text{ and } x \in X(k_n) \\ t_{R_f M}(y; k_n) &= \sum_{\substack{x \in X(k_n) \\ f(x)=y}} t_M(x; k_n). \end{align*} $$

The last of these properties is a form of the Grothendieck-Lefschetz trace formula (see [62, Exp. III, §4]). Applied to a complex M and to the structure morphism $X \to \text{Spec}(k)$, it takes the customary form

$$(A.5) \qquad \sum_{x \in X(k_n)} t_M(x; k_n) = \sum_{i \in \mathbb{Z}} (-1)^i \text{Tr}(\text{Fr}_{k_n} | \text{H}_c^i(X_{\bar{k}}, M)).$$

Suppose that M is a semisimple perverse sheaf which is pure of weight 0. Then by a result of Gabber (see [115, proof of Prop. 6.40]), the equality

$$(A.6) \qquad t_{\text{D}(M)}(x; k_n) = \overline{t_M(x; k_n)}$$

holds for all $n \ge 1$ and $x \in X(k_n)$.

We also recall a useful injectivity statement:

**PROPOSITION A.22.** Let $M_1$ and $M_2$ be objects of $D_c^b(X)$. The trace functions of $M_1$ and $M_2$ coincide, in the sense that

$$t_{M_1}(x; k_n) = t_{M_2}(x; k_n)$$

for all $n \ge 1$ and all $x \in X(k_n)$, if and only if the classes of $M_1$ and $M_2$ in the Grothendieck group $K(X)$ are equal. In particular, if M and N are semisimple perverse sheaves, then M and N are isomorphic.
Moreover, the classes of simple perverse sheaves form a basis of the Z-module K(X).

This is proved in [98, Th. 1.1.2].
<figure><img src="image_17.png" /></figure>
# The arithmetic Mellin transform over finite fields

We summarize here the most important results of Katz [75] concerning the arithmetic Mellin transform on $G_m$. These results are used in various places in the book.

## B.1. The category $\mathcal{P}$

Let $k$ be a finite field with algebraic closure $\bar{k}$ and with finite extensions $k_n/k$ for $n \ge 1$.

Katz defines a category $\mathcal{P}$ as the full subcategory of the category of perverse sheaves on $G_m$ over $\bar{k}$ whose objects are perverse sheaves $N$ such that, for any perverse sheaf $M$ on $G_m$, the objects $M *_! N$ and $M *_* N$ are both perverse (see [75, Ch. 2] and [71, 2.6.2]). Katz proved that a perverse sheaf $N$ is an object of $\mathcal{P}$ if and only if it admits no shifted Kummer sheaf $\mathcal{L}_\chi[1]$ as either subobject or quotient (this follows, e.g, from the combination of [71, Lemma 2.6.13, Lemma 2.6.14, Cor. 2.6.15]).

The category $\mathcal{P}_{\text{ari}}$ is defined as the full subcategory of perverse sheaves on $G_m$ over $k$ whose objects are those perverse sheaves $N$ such that the base change of $N$ to $\bar{k}$ is an object of $\mathcal{P}$ ([75, Ch. 4]).

Using the correct notion of exactness from the work of Gabber and Loeser, the categories $\mathcal{P}_{\text{ari}}$ and $\mathcal{P}$ are neutral tannakian categories with the middle convolution

$$M *_{\text{int}} N = \text{Im}(M *_! N \to M *_* N)$$

as tensor operation (see [50, p. 535]).

The tannakian dimension of an object of $\mathcal{P}$ is its Euler-Poincaré characteristic.

## B.2. Deligne's fiber functor and Frobenius conjugacy classes

One remarkable canonical fiber functor on the tannakian category $\mathcal{P}$ is given by a theorem of Deligne.

**THEOREM B.1 (Deligne).** Let $k$ be a finite field with algebraic closure $\bar{k}$. Let $j_0: G_m \to A^1$ be the open immersion. Then the functor

$$\omega_{\text{Del}}: M \mapsto \text{H}^0(\mathbf{A}_k^1, j_0!M)$$

is a fiber functor on the category $\mathcal{P}$.

This is [75, Th. 3.1 and Appendix].

Let $N$ be an object of $\mathcal{P}_{\text{ari}}$ which is arithmetically semisimple and pure of weight 0. Let $G_N^{\text{ari}}$ be the tannakian group of the tannakian subcategory of $\mathcal{P}_{\text{ari}}$ generated by $N$. Using Deligne's fiber functor and the tannakian formalism, Katz defines a Frobenius conjugacy class $\text{Fr}_{N,k_n}(\chi)$ in $G_N^{\text{ari}}$ for any $n \ge 1$ and any $\ell$-adic character $\chi$ of $k_n^\times$ by considering the fiber functor $\omega_\chi: M \mapsto \omega_{\text{Del}}(M_\chi)$ (see [75, Ch. 5]).
## B.3. Finite tannakian groups

**THEOREM B.2 (Katz).** Let $k$ be a finite field with algebraic closure $\bar{k}$. Let $N$ be a perverse sheaf in the category $\mathcal{P}_{\text{ari}}$. Assume that $N$ is arithmetically semisimple and pure of weight 0.

(1) If every Frobenius conjugacy class of $N$ is quasi-unipotent, then the object $N$ is punctual.

(2) If the geometric tannakian group of $N$, i.e., the tannakian group of the tannakian subcategory of $\mathcal{P}$ generated by $N \otimes \bar{k}$, is finite, then the object $N$ is punctual.

These statements are [75, Th 6.2 and Th. 6.4].

## B.4. Hypergeometric complexes and sheaves

Katz has also classified the perverse sheaves on $\mathbf{G}_m$ with tannakian dimension 0 and 1. Indeed, since the tannakian dimension is equal to the Euler-Poincaré characteristic in this case, the question is to classify simple perverse sheaves $M$ on $\mathbf{G}_m$ with $\chi(M) = 0$ or 1.

For Euler-Poincaré characteristic zero, we have:

**PROPOSITION B.3.** Let $k$ be an algebraically closed field of characteristic $p > 0$ with $p \neq l$. Let $M$ be a simple perverse sheaf on $\mathbf{G}_m$ with $\chi(\mathbf{G}_m, M) = 0$. Then there exists a tame character $\chi$ of $\mathbf{G}_m$ such that $M$ is isomorphic to $\mathcal{L}_{\chi}[1]$.

This is [70, Prop. 8.5.2].

Katz has furthermore shown that the objects with Euler-Poincaré characteristic 1 are exactly the hypergeometric complexes on $\mathbf{G}_m$, defined in [70, 8.2, 8.3].

We recall the definition and notation for hypergeometric complexes. Let $k$ be a field of positive characteristic. Fix a pair $(m, n)$ of non-negative integers and a non-trivial $l$-adic additive character $\psi$ of a finite subfield of $k$. Denote by $j: \mathbf{G}_m \to \mathbf{A}^1$ the open immersion. Let

$$ \chi = (\chi_1, \dots, \chi_n), \quad \varrho = (\varrho_1, \dots, \varrho_m) $$

be two tuples of tame $l$-adic continuous characters $\pi_1^t(\mathbf{G}_m) \to \overline{\mathbb{Q}}_l^\times$. Denote by $\bar{\psi}$ the inverse of $\psi$ and write

$$ \bar{\chi} = (\chi_1^{-1}, \dots, \chi_n^{-1}), \quad \bar{\varrho} = (\varrho_1^{-1}, \dots, \varrho_m^{-1}). $$

The hypergeometric complex $\text{Hyp}(!, \psi, \chi; \varrho)$ in $\text{D}_c^b(\mathbf{G}_m)$ is then defined inductively as follows:

(1) If $(m, n) = (0, 0)$ then $\text{Hyp}(!, \psi, \emptyset; \emptyset)$ is the skyscraper sheaf supported at 1.

(2) If $(m, n) = (1, 0)$ then $\text{Hyp}(!, \psi, \chi; \emptyset) = j^*(\mathcal{L}_{\bar{\psi}}) \otimes \mathcal{L}_{\chi}[1]$.

(3) If $(m, n) = (0, 1)$ then $\text{Hyp}(!, \psi, \emptyset, \varrho) = \text{inv}^*(j^*(\mathcal{L}_{\bar{\psi}}) \otimes \mathcal{L}_{\bar{\varrho}})[1]$.

(4) If $(m, n) = (m, 0)$ with $m \ge 2$ then $\text{Hyp}(!, \psi, \chi; \emptyset)$ is the convolution
$$ \text{Hyp}(!, \psi, \chi_1; \emptyset) *_! \cdots *_! \text{Hyp}(!, \psi, \chi_n; \emptyset). $$

(5) If $(m, n) = (0, n)$ with $n \ge 2$ then $\text{Hyp}(!, \psi, \emptyset; \varrho)$ is the convolution
$$ \text{Hyp}(!, \psi, \emptyset; \varrho_1) *_! \cdots *_! \text{Hyp}(!, \psi, \emptyset; \varrho_n). $$

(6) In the general case, we have

$$ \text{Hyp}(!, \psi, \chi; \varrho) = \text{Hyp}(!, \psi, \chi; \emptyset) *_! \text{Hyp}(!, \psi, \emptyset; \varrho). $$

For $\lambda \in k^\times$, define also

$$ \text{Hyp}_\lambda(!, \psi, \chi; \varrho) = [x \mapsto \lambda x]_* \text{Hyp}(!, \psi, \chi; \varrho). $$
It follows from these definitions that the general convolution formula

$$ \mathrm{Hyp}_{\lambda}(!, \psi, \chi; \varrho) *! \mathrm{Hyp}_{\mu}(!, \psi, \chi'; \varrho') = \mathrm{Hyp}_{\lambda\mu}(!, \psi, \chi, \chi'; \varrho, \varrho') $$

holds.

Let $K$ be an extension of $k$. We say that a complex $M$ on $G_m$ over $K$ is hypergeometric over $k$ if there exists $\lambda \in k^*$, an additive character $\psi$, and families of tame multiplicative characters $\chi$ and $\varrho$ over $k$ such that $M \otimes K$ is isomorphic to $\mathrm{Hyp}_{\lambda}(!, \psi, \chi; \varrho)$, where the characters involved are defined on $G_m$ over $K$ by composition with the canonical morphism $\pi_1^t((G_m)_K) \to \pi_1^t((G_m)_k)$.

Before stating some of the main results concerning hypergeometric sheaves, we need a further definition: the tuples $\chi$ and $\varrho$ are said to be disjoint if $(n, m) \neq (0, 0)$ and $\chi_i \neq \varrho_j$ for all $i$ and $j$.

**THEOREM B.4 (Katz).** Assume that $k$ is algebraically closed.

(1) If the tuples $\chi$ and $\varrho$ are disjoint, then $\mathrm{Hyp}_{\lambda}(!, \psi, \chi; \varrho)$ is a simple nonpunctual perverse sheaf of Euler characteristic 1 on $G_m$.

(2) Let $K$ be an extension of $k$ and $\bar{K}$ an algebraic closure of $K$. Let $M$ be a simple perverse sheaf on $G_m$ over $K$ with Euler-Poincaré characteristic equal to 1. Then the base change $M \otimes \bar{K}$ of $M$ to $\bar{K}$ is hypergeometric over $K$.

(3) Let $k$ be a finite field and $\bar{k}$ an algebraic closure of $k$. If the tuples $\chi$ and $\varrho$ are disjoint, then the tannakian group of the hypergeometric object $\mathrm{Hyp}_{\lambda}(!, \psi, \chi; \varrho)$ on $G_m$ over $\bar{k}$ is $GL_1$.

**PROOF.** The first statement follows from [70, Th. 8.4.2], and the third is explained, e.g., in [75, proof of Cor. 6.3].

The second statement is [70, Th. 8.5.3] if $K = \bar{K} = k$. Applied to $\bar{K}$ instead of an algebraic closure of $k$, this gives the result except that we only know a priori that $\lambda \in \bar{K}^*$. We need to check that in fact $\lambda \in K^*$. To do this, we check the steps of the proof of loc. cit., which is easily seen to provide this extra information.

Say that $M$ is of type $(m, n)$ if $m$ is the dimension of the tame part of $M$ at 0 and $n$ the dimension of the tame part at infinity. The strategy of the proof is to reduce by induction to the case $m > n$, then to $n = 0$ and finally to the case $m = n = 0$.

Each of these reduction steps follows a similar pattern. First, up to tensoring $M$ by $\mathcal{L}_{\Lambda}$ for some tame continuous character $\Lambda$ of $\pi_1^t(G_{m,k})$, one can assume that the trivial character occurs in the local monodromy at 0. From Kummer theory, we have an isomorphism $\pi_1^t(G_{m,k}) \simeq \hat{Z}(1)_{p'}$; since $M$ is defined over $k(\eta)$, the character $\Lambda$ must be of finite order and hence is a character of $k^*$. All the characters $\chi$ and $\varrho$ appear as such $\Lambda$.

After this tensoring step, one considers the Fourier transform $\mathrm{FT}_{\psi}(j_*M)$, and one checks that it is of type $(n, m - 1)$, and is still a geometrically simple perverse sheaf of Euler characteristic 1.

At the end of the induction, one is left either with a skyscraper sheaf, which must be supported on some $\lambda \in K^*$ since $M$ is geometrically simple over $K$, or with a perverse sheaf that is geometrically isomorphic to $\mathcal{L}_{\psi(\lambda x)}$ for some $\lambda \in \bar{K}^*$, and since this sheaf is defined over $K$, we must have $\lambda \in K$, as desired. $\square$

**REMARK B.5.** (1) In [70, Ch. 8], Katz has also determined the geometric monodromy group of almost all hypergeometric sheaves. We observe in passing that this computation has recently been used by Fresán and Jossen [44] to construct examples of $E$-functions that are not related to hypergeometric functions, answering a question raised by Siegel in his fundamental paper [121].

(2) Theorem B.4 is a key ingredient in the proof of the theorem of Gabber and Loeser that determines the group $\mathbf{H}_{\mathrm{int}}(G_{m,\bar{k}}^r)$ of isomorphism classes of objects on $G_{m,\bar{k}}^r$ with tannakian rank 1,
which is explained in Example 3.52. In fact, it is not very difficult to deduce from Theorem B.4, (3), that the group $\mathbf{H}_{\text{int}}(\mathbf{G}_{m,\bar{k}})$ is isomorphic to

$$ \bar{k}^{\times} \times \mathbf{Z}^{\Pi(\mathbf{G}_{m,\bar{k}}, \bar{\mathbf{Q}}_{\ell})}, $$

where we recall that $\Pi(\mathbf{G}_{m,\bar{k}}, \bar{\mathbf{Q}}_{\ell})$ denotes the set of continuous tame characters of $\mathbf{G}_{m,\bar{k}}$ (see Section 1.10).

An isomorphism $\Phi$ between these groups is determined as follows: given $\lambda \in \bar{k}^{\times}$ and a function $f \in \mathbf{Z}^{\Pi(\mathbf{G}_{k,\bar{k}}, \bar{\mathbf{Q}}_{\ell})}$, let $\chi$ be the tuple whose distinct elements are the characters $\chi$ such that $f(\chi) \ge 1$, each repeated with multiplicity $f(\chi)$, and let $\varrho$ be the tuple whose distinct elements are the characters $\chi$ such that $f(\chi) \le -1$, each repeated with multiplicity $-f(\chi)$. Then one has

$$ \Phi(\lambda, f) = \text{Hyp}_{\lambda}(!, \psi, \chi; \varrho). $$

Conversely, the function $f$ can be recovered from an element $M$ of $\mathbf{H}_{\text{int}}(\mathbf{G}_m)$ by looking at the tame characters appearing in the local monodromy at $0$ and $\infty$, and their multiplicities.

Let now $k$ be a finite field, with $\psi$ a non-trivial additive character of $k$. Let $\lambda \in k^{\times}$ and let

$$ M = \text{Hyp}_{\lambda}(!, \psi, \chi; \varrho) $$

for tuples $\chi$ and $\varrho$ of tame characters associated to multiplicative characters $k^{\times} \to \bar{\mathbf{Q}}_{\ell}^{\chi}$ (denoted in the same manner). The trace function of $M$ is then given by

$$ t_{\mathbf{M}}(x; k) = (-1)^{m-n} \sum_{\substack{(x_i) \in (k^{\times})^n, \ (y_j) \in (k^{\times})^m \\ x_1 \cdots x_n = \lambda^{-1} xy_1 \cdots y_m}} \psi\left(\sum_{i=1}^{n} x_i - \sum_{j=1}^{m} y_j\right) \prod_{i=1}^{n} \chi_i(x_i) \prod_{j=1}^{m} \varrho_j(y_j), $$

with the obvious analogue for finite extensions of $k$ (see [70, (8.2.7)]).

For a multiplicative character $\chi: k^{\times} \to \bar{\mathbf{Q}}_{\ell}^{\chi}$, let

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

denote the Gauss sums over $k$. Then the arithmetic Mellin transform of the hypergeometric complex $M$ is

$$ (B.1) \qquad \sum_{x \in k^{\times}} \chi(x) t_{\mathbf{M}}(x; k) = \chi(\lambda) \prod_{i=1}^{m} \tau(\psi, \chi \chi_i) \prod_{j=1}^{n} \tau(\bar{\psi}, \bar{\chi} \bar{\varrho}_j) $$

for $\chi: k^{\times} \to \bar{\mathbf{Q}}_{\ell}^{\chi}$ (a monomial in Gauss sums; see [70, (8.2.7), (8.2.8)]).

In particular, if $n \ge 1$ and $\chi_i = 1$ for all $i$, and if $\varrho$ is empty and $\lambda = 1$, we obtain the unnormalized hyper-Kloosterman sums

$$ (-1)^n \sum_{\substack{x_1, \dots, x_n \in k^{\times} \\ x_1 \cdots x_n = x}} \psi(x_1 + \cdots + x_n). $$

The corresponding hypergeometric complex

$$ (B.2) \qquad \mathcal{H}_{\ell, \psi} = \text{Hyp}(!, \psi, (1, \dots, 1); \emptyset) $$

is called a Kloosterman complex; it is of the form $\text{Kl}_n[1]$ for a lisse sheaf $\text{Kl}_n$ of rank $n$ on $\mathbf{G}_m$, called the Kloosterman sheaf of rank $n$ (see [70, Rem. 8.4.3]).
The product formula for epsilon factors

We recall in this Appendix the formula of Laumon [98] for the epsilon factor of an object of $D_c^b(X)$ on a curve X, and recall the main parts of the formalism of local epsilon factors. We also include the general Euler-Poincaré characteristic formula.

C.1. The product formula

The results in this section are quoted directly from [98, § 3].

Let $k$ be a finite field of characteristic $p$, with $k_n/k$ the extension of $k$ of degree $n$ in an algebraic closure $\bar{k}$ of $k$.

Let $X$ be a smooth projective curve over $k$. We denote by $[X]$ the set of closed points of $X$. For a complex $M$ in $D_c^b(X)$, the $L$-function of $M$ is defined by the product

$$L(M, T) = \prod_{x \in [X]} \det(1 - T^{\deg(x)} \mathrm{Fr}_{k_{\deg(x)}} | M_x)^{-1}.$$

It satisfies the relation

$$L(M, T) = \det(1 - T \mathrm{Fr}_k | H^*(X_{\bar{k}}, M))^{-1}$$

and the functional equation

$$L(M, T) = \varepsilon(M)T^{a(M)}L(D(M), T^{-1}),$$

where

$$a(M) = -\chi(X_{\bar{k}}, M), \quad \varepsilon(M) = \det(-\mathrm{Fr}_k | H^*(X_{\bar{k}}, M))^{-1}.$$

Laumon's product formula, which had been conjectured by Deligne, is an expression for $\varepsilon(M)$ in terms of local epsilon factors.

Consider a fixed non-trivial $\ell$-adic additive character $\psi$ of $\mathbf{F}_p$, and denote $\psi_k = \psi \circ \mathrm{Tr}_{k/\mathbf{F}_p}$. Furthermore, consider a fixed non-zero meromorphic 1-form $\omega$ on $X$.

THEOREM C.1 (Laumon). Suppose that $X$ is connected. Let $g$ be the common genus of all the connected components of $X_{\bar{k}}$, and $n \ge 1$ the number of these connected components.

Let $M$ be an object of $D_c^b(X)$ of generic rank $r(M)$. There exist specific local constants $\varepsilon_x(M)$, depending on the choice of $\omega$, such that

$$(C.1) \qquad \varepsilon(M) = |k|^c \prod_{x \in [X]} \varepsilon_x(M)$$

where $c = n(1 - g)r(M)$.

This is [98, Th. 3.2.1.1], defining (in the notation of loc. cit.) the local factors by

$$(C.2) \qquad \varepsilon_x(M) = \varepsilon(X_{(x)}, M|X_{(x)}, \omega|X_{(x)}).$$
## C.2. Local epsilon factors

We summarize here the basic identities and formal properties of the local epsilon factors $\epsilon_x(M)$ in Laumon's Theorem C.1. The existence and uniqueness of these local factors, subject to certain conditions, are given precisely by Laumon in [98, Th 3.1.5.4]; they were defined earlier by Deligne [26].

The local epsilon factors are attached to a triple $(T, M, \omega)$, where $T$ is a strictly henselian local ring of equal characteristic with residue field containing $k$, $M$ is an object of $D_c^b(T)$ and $\omega$ is a non-zero meromorphic 1-form on $T$.

The notation $\epsilon(X_{(x)}, M|X_{(x)}, \omega|X_{(x)})$ in (C.2) refers to these factors with the subscript $(x)$ referring to strict localization at $x$.

We now recall the local exponents $a(T, M, \omega)$ and $a(T, M)$, which require additional notation (see [98, 3.1.5]):

(1) We denote by $v$ the valuation of $T$, extended to 1-forms by $v(adb) = v(a)$ if $v(b) = 1$.

(2) We denote by $t$ the closed point of $T$ and by $\eta$ the generic point.

(3) We denote by $\bar{t}$ (resp. $\bar{\eta}$) a geometric generic point of $T$ above $t$ (resp. above $\eta$).

(4) We denote by $k_t$ the residue field of $T$ at $t$.

(5) For an object $M$ of $D_c^b(T)$, we denote by $r(M_{\bar{\eta}})$ (resp. $r(M_{\bar{t}})$) the generic rank of $M$ (resp. the rank of the stalk at the closed point) and by $s(M_{\bar{\eta}})$ the Swan conductor; all of these are defined for an étale sheaf first and extended by additivity, see [98, § 2.2.1].

With these notation, the local conductor exponents are defined by the formulas

$$ (C.3) \qquad a(T, M) = r(M_{\bar{\eta}}) + s(M_{\bar{\eta}}) - r(M_{\bar{t}}), $$

$$ (C.4) \qquad a(T, M, \omega) = a(T, M) + r(M_{\bar{\eta}})v(\omega). $$

(see [98, (3.1.5.1), (3.1.5.2)]).

In the global case, we will denote

$$ a_x(M, \omega) = a(X_{(x)}, M|X_{(x)}, \omega|X_{(x)}). $$

Furthermore, for a lisse $\overline{\mathbb{Q}}_\ell$-sheaf $\mathcal{F}$ on the generic point $\eta$ of $T$, one defines

$$ (C.5) \qquad \varepsilon_0(T, \mathcal{F}, \omega) = \varepsilon(T, j_!\mathcal{F}, \omega), $$

where $j: \{\eta\} \to T$ is the open immersion (see [98, 3.1.5.6, p. 187]).

For a short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$, we have

$$ (C.6) \qquad \varepsilon_0(T, \mathcal{F}, \omega) = \varepsilon_0(T, \mathcal{F}', \omega)\varepsilon_0(T, \mathcal{F}'', \omega). $$

The local epsilon factors satisfy (among other things) the following properties (see, respectively, formula (3.1.5.6), formula (3.1.5.5) and section 3.5.3.1 in [98]):

(1) For any lisse $\overline{\mathbb{Q}}_\ell$-sheaf $\mathcal{F}$ of rank $r$ on $T$, the formula

$$ (C.7) \qquad \varepsilon(T, M \otimes \mathcal{F}, \omega) = \det(Fr | \mathcal{F})^{a(T,M,\omega)} \varepsilon(T, M, \omega)^r $$

holds, where $Fr$ denotes the geometric Frobenius automorphism at the closed point $t$ of $T$.

(2) For a non-zero rational function $a$ on $T$, the formula

$$ (C.8) \qquad \varepsilon(T, M, a\omega) = \chi(a)|k_t|^{r(M_{\bar{\eta}})v(a)}\varepsilon(T, M, \omega) $$
holds, where $\chi$ is the character of the completion of the residue field at $\eta$ associated, by local class field theory,<fnref n="1" /> to the lisse sheaf $\det(M)|_{\eta}$ on $\eta$, viewed as a character of the local Galois group.

(3) For a non-trivial multiplicative character $\chi$ of the residue field $k_t$ and the corresponding lisse Kummer sheaf $\mathcal{L}_\chi$ on $\{\eta\}$, and for a uniformizer $\pi$ at $x$, we have

$$ (C.9) \qquad \varepsilon_0(\mathrm{T}, \mathcal{L}_\chi, d\pi) = \chi(-1) \sum_{a \in k_t^\times} \chi(a) \psi(\mathrm{Tr}_{k_t/\mathbb{F}_p}(a)). $$

We also have the elementary shift formula

$$ (C.10) \qquad \varepsilon(\mathrm{T}, \mathrm{M}[1], \omega) = \varepsilon(\mathrm{T}, \mathrm{M}, \omega)^{-1}. $$

### C.3. The Euler-Poincaré characteristic formula

We keep the notation of Section C.1. In particular, $X$ is a smooth projective curve over a finite field $k$ with algebraic closure $\bar{k}$. We assume that $X$ is geometrically connected, and denote by $g$ the genus of $X$.

Let $M$ be a complex in $\mathrm{D}_c^b(X)$. For any point $x \in X(\bar{k})$, the Swan conductor $\mathrm{swan}_x(M)$ is defined by additivity from the case of a $\overline{\mathbf{Q}}_\ell$-sheaf (in which case, it is defined for instance in [98, (2.1.2.5)] or [69, Ch. 1]). Similarly, the drop $\mathrm{drop}_x(M)$ is defined by additivity from the drop

$$ \mathrm{drop}_x(\mathcal{F}) = \mathrm{rank}(\mathcal{F}) - \mathrm{dim}(\mathcal{F}_x) $$

of a $\overline{\mathbf{Q}}_\ell$-sheaf $\mathcal{F}$.

**THEOREM C.2 (Grothendieck-Ogg-Shafarevich).** Let $U \subset X$ be an open dense subset. Let $M$ be a complex in $\mathrm{D}_c^b(U)$, let $V$ be an open dense subset of $U$ on which $M$ is lisse of generic rank $r(M)$.

We have

$$ \chi(\mathrm{U}_{\bar{k}}, \mathrm{M}) = \chi(\mathrm{U}_{\bar{k}}, \overline{\mathbf{Q}}_{\ell})r(\mathrm{M}) - \sum_{x \in \mathrm{X}(\bar{k})} \mathrm{swan}_x(\mathrm{M}) - \sum_{x \in \mathrm{U}(\bar{k})} \mathrm{drop}_x(\mathrm{M}), $$

where $\chi(\mathrm{U}_{\bar{k}}, \overline{\mathbf{Q}}_{\ell}) = (2 - 2g) - |(\mathrm{X} - \mathrm{U})|$.

This statement follows from [98, Th. 2.2.1.2], which corresponds to $X = U$ (up to changes in notation) by applying this result to $j_*M$, where $j: U \to X$ is the open immersion, and using the additivity of the Euler-Poincaré characteristic, in the sense that

$$ \chi(\mathrm{X}_{\bar{k}}, j_*\mathrm{M}) = \chi(\mathrm{U}_{\bar{k}}, \mathrm{M}) + \chi((\mathrm{X} - \mathrm{U})_{\bar{k}}, i^*j_*\mathrm{M}) $$

with $i$ the closed immersion of $X - U$ in $X$.

For the case of a $\overline{\mathbf{Q}}_\ell$-sheaf, the statement is also given for instance in [75, Ch. 14].

We consider some special cases that appear in this book.

(1) If $U = X$ and $M = \mathcal{F}[1]$ for some $\overline{\mathbf{Q}}_\ell$-sheaf $\mathcal{F}$ of generic rank $r$ on $X$, then the formula becomes

$$ (C.11) \qquad \chi(\mathrm{X}_{\bar{k}}, \mathrm{M}) = (2g - 2)r + \sum_{x \in \mathrm{X}(\bar{k})} (\mathrm{swan}_x(\mathcal{F}) + \mathrm{drop}_x(\mathcal{F})). $$

(2) If $U = \mathbf{G}_m \subset X = \mathbf{P}^1$ and $M = \mathcal{F}[1]$ for some $\overline{\mathbf{Q}}_\ell$-sheaf $\mathcal{F}$ of generic rank $r$ on $\mathbf{G}_m$, then

$$ (C.12) \qquad \chi((\mathbf{G}_m)_{\bar{k}}, \mathrm{M}) = \mathrm{swan}_0(\mathcal{F}) + \mathrm{swan}_\infty(\mathcal{F}) + \sum_{x \in \bar{k}^\times} (\mathrm{swan}_x(\mathcal{F}) + \mathrm{drop}_x(\mathcal{F})). $$

<footnote n="1">¹ Normalized as explained in [98, (3.1.4)].</footnote>
<figure><img src="image_18.png" /></figure>
Deligne's letter to Kazhdan

We reproduce below the content of Deligne's letter to Kazhdan, in which the ℓ-adic Fourier transform was defined for the first time (the typography is not faithfully reproduced).

29-11-76

Dear Каждан,

This is perhaps a partial answer to an old letter of yours. I thought to the matter again because
of some estimations of trigonometrical sums Hooley asked me about. As I am in a hurry to continue
writing up Weil II, I will leave many open ends and soon turn to French.

Theme: many functions correspond to sheaves, and operations on functions to operations on sheaves. What about harmonic analysis on $\mathbb{G}_a$?

(a) If X is a scheme $\mathbb{F}_q$, we will consider

α) objects of the derived category $D^b(X, \overline{\mathbb{Q}}_\ell)$
↓ by $\sum (-1)^i \mathbb{H}^i$

β) virtual ℓ-adic sheaves [this means either: elements of the Grothendieck group of the abelian category of constructible sheaves — or if possible and useful, objects of some Picard category having this $K^0$ as set of isomorphism classes of objects]
↓ by Tr($F_x^*$, $\mathcal{F}_{\bar{x}}$) (this map is injective)

γ) “functions”: a system of functions on the X($\mathbb{F}_{q^n}$)

(b) Here are corresponding operations:

On functions: $+$, $\cdot$, $\sum$ \qquad On $\alpha, \beta$:\ $\oplus$, $\otimes$, $\pi_!$

Convolution of functions: if $G$ is a group, and $K, L \in D^b(X, \overline{\mathbb{Q}}_\ell)$, one considers the product

$$
\pi: G \times G \to G, \quad \text{and} \quad K \boxtimes L = \text{pr}_1^*K \otimes \text{pr}_2^*L, \text{ and} \\
K * L = \text{R}\pi_!(K \boxtimes L)
$$

Kernel: given $Z \to X \times Y$ and $K \in D^b(Z, \overline{\mathbb{Q}}_\ell)$, this defines an operation $D^b(X) \to D^b(Y)$
$L_X \mapsto \text{R}_{2!}(K \otimes \text{R}_1^* L_X)$.

(c) Now I want to consider Fourier transform.

Let us choose $\psi: \mathbb{F}_p \to \overline{\mathbb{Q}}_\ell^*$. If $f$ is a function on $X$, we get a sheaf $\mathcal{F}(\psi f)$.<fnref n="1" /> Fourier transform, on $\mathbb{G}_a$, is given by the kernel $\mathcal{F}(\psi(xy))$ on $\mathbb{G}_a \times \mathbb{G}_a$.

Definition: $\mathbb{F}(K) = \text{Rpr}_{2!}(\mathcal{F}(\psi(xy)) \otimes \text{Rpr}_{1}^{*}K)$

Proposition 1: $\underline{\mathbb{F}(K * L)} = \underline{\mathbb{F}(K) \otimes \mathbb{F}(L)}$ (from $\mathcal{F}(\psi(x(y' + y'')) = \mathcal{F}(\psi(xy'))$ $\otimes$

<footnote n="1">¹ Pull back by $f$ of the sheaf on $\mathbb{G}_a$, rank 1, defined by $\psi$ and Artin-Schreier $T^p - T = X$.</footnote>
**Proposition 2:** $\overline{\mathbb{F}}(K) = K^{\vee}(-1)[-2]: \vee$ is for "image by $x \mapsto -x$", $(-1)$ for a Tate twist, and $[-]$ for décalage.

Kernels compose like expected: we have to compute $\mathrm{R}\pi_!\mathcal{F}(\psi(x+z)y)$ for $\pi: \mathbb{G}_a \times \mathbb{G}_a \times \mathbb{G}_a \xrightarrow{(13)} \mathbb{G}_a \times \mathbb{G}_a$, one gets

$$ \begin{cases} \mathbb{Q}_{\ell}(-1) & \text{on the diagonal, in degree 2} \\ 0 & \text{elsewhere} \end{cases} $$

hence the result.

It is convenient in such computations to forget writing $\psi$ and writing $\int \cdots dy$ for a $\mathrm{R}\pi_!$.

**Remark:** this defines, via prop 1, an isomorphism $\overline{\mathbb{F}}(K \otimes L)(-1)[-2] = \overline{\mathbb{F}}(K) * \overline{\mathbb{F}}(L)$.

For Plancherel formula, one suffer somewhat of not having complex conjugation. Let $\overline{\mathbb{F}}$ be $\mathbb{F}$ defined using $\psi(-x)$. Then

a) inner product: $\langle K, L \rangle = \mathrm{R}\Gamma(K \otimes L)$

b) Proposition $\langle \overline{\mathbb{F}}K, \overline{\mathbb{F}}L \rangle = \langle K, L \rangle(-1)[-2]$.

This boils down to the usual

$$ \int \psi((x' - x'')y)K(x)L(x'')dx'dx''dy = \int_{\substack{\uparrow \\ \text{by } \int_y}} \delta^{(-1)[-2]}(x' - x'')K(x')L(x'')dx'dx''. $$

Everything done above can be generalized to any abelian connected unipotent group $U$. The dual $U^*$ is to be taken in Serre's sense (it is natural only up to inseparable isogenies, but this does not matter. For $n$ large enough, one has a pairing

$$ U \times U^* \xrightarrow{\cdot} W_n $$

(better: the pairing is in the cowitt vectors $W_{-\infty} = \lim_{\underset{\sim}{\rightarrow}} W_n$). Given $\psi: \mathbb{Q}_p/\mathbb{Z}_p = W_{-\infty}(\overline{\mathbb{F}}_p) \to \overline{\mathbb{Q}}_\ell^*$, and using the sheaf given by the Lang covering of $W_{-\infty}/\overline{\mathbb{F}}_p$ and $\psi$, everything can be repeated, with $(-1)[-2]$ replaced by $(-d)[-2d]$ where $d$ is the dimension.

This requires to be careful if one wants to consider $\mathbb{Q}_p$ as a (ind pro quasi) unipotent algebraic group $/\overline{\mathbb{F}}_p$.

④ Where $\overline{\mathbb{F}}$ is, there should also be an action of the metaplectic group! (here symplectic). Let me work for $\mathbb{G}_a$, and for $p \neq 2$. The most precise way of speaking I see is working over $\overline{\mathbb{F}}_p$, with kernels. [It gives more than actions of $\mathrm{SL}(2, k)$ on $\mathrm{D}^b(\mathbb{G}_a, \mathbb{Q}_\ell)$, $k/\overline{\mathbb{F}}_q$.]

**Wanted:** $P \in \mathrm{D}^b(\mathrm{SL}(2) \times \mathbb{G}_a \times \mathbb{G}_a)$, viewed as a family of kernels on $\mathbb{G}_a \times \mathbb{G}_a$ parametrized by $\mathrm{SL}(2)$. Plus "$P_{g'} \cdot P_{g''} = P_{g' \cdot g''}$"

We know what is wanted for generators:

$$ U^{-} \quad \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \quad \mapsto \quad ( \otimes \mathcal{F} \left( \psi \left( \frac{ax^2}{2} \right) \right) ) \quad ( \text{noyau sur la diagonale} ) $$

$$ H \quad \begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix} \quad \mapsto \quad (x \to \lambda x)_*(\quad) \quad (\text{noyau sur } y = \lambda x) $$

$$ a \neq 0 \ U_0^+ \quad \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \quad \mapsto \quad \left( \int_x \mathcal{F} \psi \left( \frac{a^{-1}x^2}{2} \right) \right)^{-1} \mathcal{F} \psi \left( \frac{a^{-1}x^2}{2} \right) * \quad (\text{noyau: faisceau loc } c^\ell, \text{ de rg } 1, \\ \text{en degré } -1) $$
An explanation: $\Gamma\mathcal{F}\psi(a^{-1}\frac{x^2}{2})$ is of dimension 1, and degree 1, and I take the dual one dimensional vector space [in $a$: a sheaf], in degree $-1$.

En français:

Raisonnons un peu a priori. Comme “fonctions”, on sait ce que sont les noyaux cherchés. On cherche des faisceaux leur donnant naissance. Sur $U^- \times H \times U_0^+ \times U^- \times G_a \times G_a$, composant les générateurs, on trouve un faisceau localement constant de rang 1, placé en degré $-1$, qui convient. En chaque point de $U^- \times H \times U_0^+ \times U^-$, comme fonction de $x, y$, il est de la forme $\psi f$, pour $f$ une fonction qui, en $x, y$ (sur $G_a \times G_a$) est quadratique homogène. Regardons la surjection

$$U^{-} \times H \times U_{0}^{+} \times U^{-} \rightarrow G - B^{-} \qquad (G = \text{SL}(2), B^{-} = (\begin{smallmatrix} * & 0 \\ * & * \end{smallmatrix}))$$

Puisque comme “fonctions” ce que nous cherchons existe, le faisceau obtenu est constant sur les fibres de (cette application $\times G_a \times G_a$).

**Obtenu:** un faisceau de rang 1, en degré $-1$, localement constant, sur $(G - B^{-}) \times G_a \times G_a$.

Pour compléter ce tableau, il est bon de comprendre en quel sens, pour $a \to 0$, on a

$$(\int \mathcal{F}\psi(a^{-1}\frac{x^2}{2})dx)^{-1} \cdot \mathcal{F}\psi(a^{-1}\frac{x^2}{2}) \longrightarrow \delta \text{ (faisceau } \mathbb{Q}_{\ell} \text{ en } x=0)$$

[où]

$$\int \mathcal{F}\psi(a^{-1}\frac{x^2}{2})dx$$

est un faisceau de rang 1 (degré $-1$) sur la droite de $a$; ce faisceau se trivialise sur le revêtement de la droite de $a$ donné par $\sqrt{a}$, car

$$\int \mathcal{F}\psi(a^{-2}\frac{x^2}{2})dx = \int \mathcal{F}\psi(\frac{(a^{-1}x)^2}{2})dx = \int \mathcal{F}\psi(\frac{x^2}{2})dx \quad \text{par ch}^{\text{nt}} \text{ de variable}$$

Il correspond à une somme de Gauss; sur $\int \dots$, $|\text{Frobenius}| = q^{1/2}$.

Traçons le plan $a, x$ ; le faisceau considéré est défini pour $a \neq 0$; il se ramifie (sauvagement) le long de $a = 0$, et la ramification est équisingulière pour $x \neq 0$. Si $j$ est l'inclusion de $a \neq 0$ dans le plan, on a

$$\left\{ \begin{array}{ll} j_*(\text{faisceau}) = j_!(\text{faisceau}) & \text{nul pour } a = 0 \\ \text{R}^1 j_*(\text{faisceau}) & \text{concentré en } (0,0), \text{ où c'est } \delta \\ \text{R}^2 j_*(\text{faisceau}) & \text{concentré en } (0,0) \end{array} \right.$$

Ceci se vérifie assez facilement en éclatant 2 fois (0,0), la 2ème fois en éclatant (courbe exceptionnelle) $\cap$ (transformé pur de l'axe des $x$) : on utilise

<figure><img src="image_19.png" /></figure>

On contrôle en projetant sur la droite des $a$ : si $p_a$ est cette projection, on a

$$R p_{a*} R j_{*} = R j_{*} R p_{a*}$$
[où] $Rp_{a*}$ donne $\mathbb{Q}_\ell$ sur la droite $a, -\{0\}$, et $Rj_*$ sur cette droite ($a \neq 0$) $\mapsto a = 0$, donc

$$ \begin{cases} \deg. 0 & \mathbb{Q}_\ell \\ \deg. 1 & \mathbb{Q}_\ell(-1) \text{ en } 0. \end{cases} $$

Ceci nous dit ce que nous devons faire pour construire P:

(a) sur $U^+$, le noyau s'obtient à partir de

$$ \tau_{\le 0} (Rj_* ((\int \mathcal{F}\psi(a^{-1}\frac{x^2}{2})dx)^{-1} \mathcal{F}(\psi(a^{-1}\frac{x^2}{2}))) \text{ sur } U^+ \times \mathbb{G}_a, \text{ comme convolution.} $$

(b) sur $G \times \mathbb{G}_a \times \mathbb{G}_a$, on prend le noyau déjà construit sur $(G - B^-) \times \mathbb{G}_a \times \mathbb{G}_a$, et pour $j$ l'inclusion dans $G \times \mathbb{G}_a \times \mathbb{G}_a$, on lui applique $\tau_{\le 0}Rj_*$.

Je me suis convaincu que la formule $P_g \cdot P_{g'} = P_{gg'}$ vaut au sens le plus fort possible:

a) sur $G \times G \times \mathbb{G}_a \times \mathbb{G}_a$, on prend $P_{g''}(y, z)P_{g'}(x, y)$.

b) on intègre par rapport à $y$: $(P \cdot P)_{g',g''} = \int dy \cdots$ sur $G \times G \times \mathbb{G}_a \times \mathbb{G}_a$

c) si $\pi$ est $G \times G \to G: g', g'' \mapsto g'g''$, on a un isomorphisme

$$ (P \cdot P) = \pi^*P $$

d) on a une compatibilité pour un composé triple [en c), on a unicité à une constante près, et on normalise par ce qui se passe à l'origine].

Bien sûr, tout ceci devrait valoir pour un espace vectoriel $V$, et $\text{Sp}(V \oplus V^*)$. Il est facile de se convaincre qu'on a en tout cas un noyau $P_g(v, v')$ qui est un faisceau virtuel, et que sur la cellule des $g \in \text{Sp}$ où $gV^* \cap V^* = 0$, il est donné de façon naturelle par un faisceau de rang 1, localement constant, en degré $-n$. J'espère que le noyau lui-même s'en déduit par une suite d'opérations $\tau \le j_*$, avec un résultat localement constant de rang 1 sur un sous-espace, en degré $-k$, sur la strate $\text{dim}(V^*/V^* \cap gV^*) = k...$ (qu'on ait un noyau ainsi stratifié doit pouvoir se vérifier par Fourier).

**Question:** Le foncteur $K \mapsto (x \mapsto -x)_* \text{RHom}(K, \mathbb{Q}_\ell)$ commute-t-il à l'action de $\text{SL}(2)$ ?

**Question bis:** pour $P_g$ le noyau, et $K$ sur $\mathbb{G}_a$, a-t-on

$$ \mathrm{Rpr}_{2!}(P_g \otimes \mathrm{pr}_1^* K) \xrightarrow{\sim} \mathrm{Rpr}_{2*}(P_g \otimes \mathrm{pr}_1^* K) ? $$

**Question ter:** y commute-t-il virtuellement – au moins virtuellement sur $\mathbb{F}$ ?

Bien à toi,

P. Deligne
# APPENDIX E

## Intuition for analytic number theorists

The goal of this informal appendix is to provide readers who have a background in analytic number theory with some intuition and feeling for objects such as $\ell$-adic complexes, perverse sheaves, or tannakian categories, all of which are essential tools in this book.

The focus here concerns trace functions of more than one variable. On the other hand, the theory of trace functions in one variable is more accessible, as the algebraic objects can be presented more concretely using Galois theory of function fields. Some familiarity with this point of view will certainly also be very helpful in developing intuition. A very concise introduction can be found in the Pisa survey of Fouvry, Kowalski and Michel [40], and a more detailed treatment is contained in the lectures of Michel at the 2016 Arizona Winter School [42].

We fix a finite field $k$, and denote by $k_n$ the extension of $k$ of degree $n$ inside a fixed algebraic closure $\bar{k}$. For simplicity of notation, we will mostly speak about trace functions on the affine space $\mathbf{A}^m$ for some integer $m \ge 0$. However, it will be implicit that most of what we discuss can be done for any algebraic variety $Y$ over $k$ (and this is needed, for instance because we often naturally wish to restrict a trace function to a subvariety, where some particular property holds), for instance for powers of the multiplicative group $\mathbf{G}_m$ (i.e., $Y$ such that $Y(k_n) = (k_n^\times)^d$ for some $d \ge 0$). The reader should keep in mind that for such a subvariety, of dimension $d \le m$, the size of the finite set $Y(k_n)$ of points of $Y$ with coordinates in $k_n$ is approximately $|k_n|^d$ when $n$ is large.

Throughout, we fix a non-trivial additive character $\psi: k \to \mathbf{C}^\times$ and, for $n \ge 1$, we define

$$ \psi_n: k_n \to \mathbf{C}^\times \\ x \mapsto \psi(\mathrm{Tr}_{k_n/k}(x)). $$

We finally note that we will completely ignore (here) the distinction between $\overline{\mathbf{Q}}_\ell$ and $\mathbf{C}$.

### E.1. Trace functions

The concrete origin for the use of methods of algebraic geometry and étale cohomology in analytic number theory lies in trace functions, and especially in exponential sums. Properly speaking, a trace function on $\mathbf{A}^m$ is the data of a family $(t_n)_{n \ge 1}$ of functions $k_n^m \to \mathbf{C}$, and it is associated to some algebraic object $M$, which we call a “coefficient object”. This object is not uniquely determined by $(t_n)$, but we will not worry about this matter in this appendix.

The first examples of trace functions arise from polynomials $f \in k[X_1, \dots, X_m]$ by means of

$$ (E.1) \qquad t_n(x_1, \dots, x_m) = \psi_n(f(x_1, \dots, x_m)); $$

the corresponding coefficient object is denoted by $\mathcal{L}_{\psi(f)}$. Many other examples are then obtained by applying various operations, which are known to preserve the set of trace functions (these are operations on the coefficient objects, which are reflected in a specific operation at the level of trace functions). These operations include the following, where we indicate the algebraic notation for the corresponding coefficient objects:

- The constant function 1 is associated to the coefficient object $M = \overline{\mathbf{Q}}_\ell$.
- The sum of the trace functions associated to $M_1$ and $M_2$ is associated to $M_1 \oplus M_2$.

- If $(t_n)$ is a trace function associated to M, then $((-1)^k t_n)$ is a trace function for each integer $k \in \mathbb{Z}$, associated to a coefficient denoted by $M[k]$ and called a “shift” of M.

- If $(t_n)$ is a trace function associated to M, then $(|k_n|^r t_n)$ is a trace function for each integer $r \in \mathbb{Z}$, associated to a coefficient denoted by $M(-r)$ and called a “(Tate) twist” of M.

- The product of the trace functions associated to $M_1$ and $M_2$ is associated to $M_1 \otimes M_2$.

- If $f = (f_1, \dots, f_d): \mathbf{A}^m \to \mathbf{A}^d$ is a tuple of polynomials in $k[X_1, \dots, X_m]$, and $s = (s_n)$ is a trace function on $\mathbf{A}^d$ associated to a coefficient N, then

$$t_n(x_1, \dots, x_m) = s_n(f(x_1, \dots, x_m))$$

defines a trace function $(t_n)$ on $\mathbf{A}^m$, which we also denote by $s \circ f$. The corresponding coefficient is $f^*N$.

- If $f = (f_1, \dots, f_d): \mathbf{A}^m \to \mathbf{A}^d$ is a tuple of polynomials in $k[X_1, \dots, X_m]$, and $t = (t_n)$ is a trace function on $\mathbf{A}^m$, associated to a coefficient object M, then

$$(E.2) \qquad s_n(y_1, \dots, y_d) = \sum_{\substack{x \in k_n^m \\ f(x)=y}} t_n(x)$$

defines a trace function on $\mathbf{A}^d$; the associated coefficient object is denoted by $Rf_!M$.

**EXAMPLE E.1 (Fourier transform).** This formalism is already sufficient to explain Deligne's Fourier transform. Let $m \ge 1$ be an integer, and consider the projections

$$p_1, p_2: \mathbf{A}^{2m} \to \mathbf{A}^m$$

given by

$$p_1(x_1, \dots, x_m, y_1, \dots, y_m) = (x_1, \dots, x_m), \quad p_2(x_1, \dots, x_m, y_1, \dots, y_m) = (y_1, \dots, y_m).$$

We write

$$X \cdot Y = X_1 Y_1 + \dots + X_m Y_m$$

for variables $X_i$ and $Y_j$. This is a polynomial with coefficients in $k$, so the functions

$$F_n(x, y) = \psi_n(x_1 y_1 + \dots + x_m y_m)$$

define a trace function $F = (F_n)$ on $\mathbf{A}^{2m}$, associated to the coefficient object $\mathcal{L}_{\psi(X,Y)}$.

Let $t = (t_n)$ be a trace function on $\mathbf{A}^m$ with coordinates $(x_1, \dots, x_m)$. Then the discrete Fourier transforms $(\hat{t}_n)$, which are defined for $n \ge 1$ and $y \in k_n^m$ by

$$\hat{t}_n(y) = \sum_{x \in k_n^m} t_n(x) F_n(x, y) = \sum_{x \in k_n^m} t_n(x) \psi_n(x \cdot y),$$

also define a trace function $\hat{t} = (\hat{t}_n)$. Indeed, for any $y$, the set of all $x \in k_n^m$ can be identified with the set of $(x, y) \in k_n^{2m}$ such that $p_2(x, y) = y$, and we have $t_n(x) = t_n(p_1(x, y))$, so that if $t$ is associated to the coefficient object M, then the formalism above shows that $\hat{t}$ is associated to

$$\hat{M} = \mathrm{R} p_2! (p_1^* M \otimes \mathcal{L}_{\psi(X, Y)}).$$
## E.2. Weights and purity: lisse sheaves

The formalism of trace functions is useful in analytic number theory because of Deligne's Riemann hypothesis over finite fields. This also leads to some understanding of the important qualitative differences between various types of trace functions—corresponding to classes of coefficients which may (for instance) be lisse sheaves, constructible sheaves, complexes of constructible sheaves, or perverse sheaves. We will try in this and the following sections to provide the readers with some intuition of the concrete meaning of these notions.

The key concept (due to Deligne) is that of a coefficient M which is punctually pure, or pure, of some weight $w \in \mathbb{Z}$. The main conceptual difficulty is that the meaning of this property for the corresponding trace function is not straightforward in general.

The simplest case (from which the others will be derived) is that of M which is a single “lisse sheaf”. In that case, the concrete meaning<fnref n="1" /> of M being punctually pure of weight w, in terms of the trace function $t = (t_n)$, is that there exist

- an integer $r \ge 0$, the rank of M,

- for each $n \ge 1$ and $x \in k_n^m$, a unitary matrix $\Theta_M(x; k_n) \in \mathbf{U}_r(\mathbb{C})$, well-defined up to conjugacy,

so that the following equality holds:

$$t_n(x) = |k_n|^{w/2} \mathrm{Tr}(\Theta_M(x; k_n)).$$

In particular, note that this implies the estimate

$$|t_n(x)| \le r |k_n|^{w/2}$$

for all $n$ and $x \in k_n^m$.

In the remainder of this appendix, we will sometimes say that a lisse sheaf, or its trace function, is “pure” instead of the more correct “punctually pure”.

**REMARK E.2.** The matrix $\Theta_M(x; k_n)$ is not arbitrary in $\mathbf{U}_r(\mathbb{C})$. For instance, its eigenvalues (which of course determine the trace) are Weil numbers of weight 0, i.e., algebraic numbers in $\mathbb{C}$ for which all Galois conjugates have modulus 1. Moreover, if $n'$ is a multiple of $n$, then $x \in k_n^m$ can also be viewed as an element of $k_{n'}^m$ through the inclusion $k_n \subset k_{n'}$, and the formula

$$\Theta_M(x; k_{n'}) = \Theta_M(x; k_n)^{n'/n}$$

holds (i.e., the eigenvalues of the matrix $\Theta_M(x; k_{n'})$ are those of $\Theta_M(x; k_n)$ raised to the power $n'/n$).

As one can expect, the trace functions defined by the formulas (E.1), associated to $\mathcal{L}_{\psi(f)}$, are of this type, with $r = 1$, $w = 0$, and the matrix $\Theta(x; k_n)$ reduced to the single complex number of modulus one $\psi_n(f(x))$. Moreover, it is also intuitively clear (and true) that some of the operations discussed above will respect the special class of trace functions associated to pure lisse sheaves.

For instance:

- If $t$ and $t'$ are trace functions associated to objects M and N which are both lisse sheaves pure of (the same) weight $w$, then $t + t'$ is also pure of weight $w$; we have

$$\Theta_{M \oplus N}(x; k_n) = \Theta_M(x; k_n) \oplus \Theta_N(x; k_n).$$

<footnote n="1">¹ But not exactly the precise definition.</footnote>
- If $t$ and $t'$ are trace functions associated to objects M and N which are both lisse sheaves pure of weights $w$ and $w'$, respectively, then $tt'$ is also pure of weight $w + w'$. In other words, $M \otimes N$ is still a lisse sheaf, pure of that weight; in fact, we have

$$ \Theta_{M \otimes N}(X; k_n) = \Theta_M(x; k_n) \otimes \Theta_N(x; k_n). $$

- If $f = (f_1, \dots, f_d): \mathbf{A}^m \to \mathbf{A}^d$ is a tuple of polynomials in $k[X_1, \dots, X_m]$, and $s$ is a trace function on $\mathbf{A}^d$ associated to a lisse sheaf of weight $w$, then $s \circ f$ is also pure of weight $w$. In other words, $f^*N$ is still a lisse sheaf, pure of weight $w$; in fact, we have

$$ \Theta_{f^*N}(x; k_n) = \Theta_N(f(x); k_n). $$

But elementary examples show that the crucially important operation of “summing over the fiber” (see (E.2)) does not always send a single lisse sheaf to a lisse sheaf, and may also not map a trace function which is pure of some weight to another one.

**EXAMPLE E.3.** (1) Let $m = d = 1$ and $f \in k[X]$ a polynomial of degree 2, viewed as a map from $\mathbf{A}^1$ to itself. We consider the trace function $(t_n)$ with $t_n(x) = \psi_n(x)$, associated to the lisse sheaf $\mathcal{L}_{\psi(X)}$ (of weight 0), and the trace function $(s_n)$ defined by

$$ s_n(x) = \sum_{\substack{y \in k_n \\ f(y)=x}} t_n(y) = \sum_{\substack{y \in k_n \\ f(y)=x}} \psi_n(y), $$

for $n \ge 1$ and $x \in k_n$, which is associated to the coefficient object $Rf!\mathcal{L}_{\psi(X)}$. For most $x$, the value of $s_n(x)$ is either 0 (if $f(y) = x$ has no solutions in $k_n$) or a sum of two roots of unity, but for the single point $x_0 = f(y_0)$, where $y_0$ is the unique zero of the derivative of $f$, the value $s_n(x_0)$ is a single root of unity (note that $y_0$, and hence $x_0$, belongs to $k$, so it also belongs to $k_n$ for all $n$, but the value of $s_n(x_0)$ does vary with $n$).

(2) We consider $m = 2$ and the trace function $(t_n)$ defined by $t_n(x, y) = \psi_n(xy^2)$ for $(x, y) \in k_n^2$. It is associated to the coefficient object $\mathcal{L}_{\psi(XY^2)}$, which is pure of weight 0. Let $d = 1$ and $f = X$. Then $Rf!\mathcal{L}_{\psi(XY^2)}$ has the trace function $(s_n)$ such that

$$ s_n(x) = \sum_{y \in k_n} \psi_n(xy^2) = \begin{cases} \text{a quadratic Gauss sum} & \text{if } x \neq 0, \\ |k_n| & \text{if } x = 0. \end{cases} $$

Neither of these examples of trace functions are associated to a single punctually pure lisse sheaf. However, it turns out that the underlying reason is not the same. In Example (1), the issue is that $(s_n)$ is associated to a single constructible sheaf which is “not lisse” at the point $x_0$. In Example (2), the issue is that $(s_n)$ is associated to a “complex” of constructible sheaves, i.e., not to a single sheaf.

### E.3. Weights and purity: constructible sheaves and complexes

In fact, the most general source of trace functions are (bounded) mixed complexes of constructible sheaves. We now try to outline the concrete interpretation of these more general conditions.

The first step goes from a single lisse sheaf to a single constructible sheaf. Such a sheaf is (punctually) pure of weight $w$ if there is a “stratification”

$$ \emptyset = X_0 \subset X_1 \subset \dots \subset X_q = \mathbf{A}^m $$

of $\mathbf{A}^m$, where $X_i$ is a closed subvariety of $X_{i+1}$, so that the restriction of M to each of the pieces $X_{i+1} - X_i$ is a single lisse sheaf, punctually pure of weight $w$, and of some rank $r_i \ge 0$ (which in general depends on $i$).
Concretely, for a given $x \in k_n^m$, there exists a unique $i$ such that $x \in X_{i+1} - X_i$, and then there exists a unitary matrix $\Theta_M(x; k_n)$ of size $r_i$ such that

$$t_n(x) = |k_n|^{w/2} \mathrm{Tr}(\Theta_M(x; k_n)).$$

EXAMPLE E.4. Example (1) above is of this kind, with the stratification

$$\emptyset \subset \{x_0\} \subset \mathbf{A}^1,$$

and with $r_0 = 1$ and $r_1 = 2$. On $\{x_0\}$, the unique eigenvalue is $s_n(x_0) = \psi_n(y_0)$, viewing $x_0$ as belonging to $k_n$. On $\mathbf{A}^1 - \{x_0\}$, the two eigenvalues are either opposite (hence the trace is zero) if $x \notin f(k_n)$, or are given by $\psi_n(y)$, for $y$ ranging over the two roots of the quadratic equation $f(y) = x$.

More generally, Deligne defined a mixed constructible sheaf of weights $\le w$ by the condition that there is a filtration with associated punctually pure quotients $M_j$, each of some weight $w_j \le w$. Concretely, this implies that the trace function $t = (t_n)$ is given by

$$t_n(x) = \sum_{j \in \mathbb{Z}} t_{n,j}(x)$$

for some finite set $\mathbb{J}$, where each family $(t_{n,j})_{n \ge 1}$ is the trace function of a constructible sheaf which is pure of weight $w_j \le w$.

Finally, the most general type of trace functions arises from objects M that are complexes of constructible sheaves. Such a complex gives in particular rise to a sequence $(\mathcal{H}^i(M))_{i \in \mathbb{Z}}$ of constructible sheaves, with $\mathcal{H}^i(M) = 0$ for all but finitely many $i$, in such a way that

$$t_n(x) = \sum_{i \in \mathbb{Z}} (-1)^i t_{n,i}(x)$$

for all $n \ge 1$ and $x \in k_n$, where $(t_{n,i})_{n \ge 1}$ is the system of trace functions for the constructible sheaf $\mathcal{H}^i(M)$. (These sheaves are called the cohomology sheaves of the complex M.)

EXAMPLE E.5. Example (2) above is obtained from a complex of constructible sheaves M, where there are two non-zero pieces, namely $\mathcal{H}^1(M)$ and $\mathcal{H}^2(M)$.

The sheaf $\mathcal{H}^1(M)$ is constructible for the stratification

$$\emptyset \subset \{0\} \subset \mathbf{A}^1,$$

with the piece on $\{0\}$ of rank 0, and the piece on $\mathbf{A}^1 - \{0\}$ of rank 1, pure of weight 1, with the corresponding unique eigenvalue equal to the quadratic Gauss sum

$$\sum_{y \in k_n} \psi_n(xy^2)$$

for $x \in k_n - \{0\}$.

The sheaf $\mathcal{H}^2(M)$ is also constructible, for the same stratification (but this is not a general feature), with the lisse sheaf of rank 0 on $\mathbf{A}^1 - \{0\}$, and a piece of rank 1 of weight 2 at $\{0\}$, with eigenvalue $|k_n|$.

However, for a complex M, the definition of what it means that M is pure of weight $w$ is much more subtle than for a single sheaf. In particular, it does not mean that each piece $\mathcal{H}^i(M)$ is itself a punctually pure sheaf of weight $w$. More precisely, one defines first the mixed complexes of weights $\le w$, which are those such that $\mathcal{H}^i(M)$ is a mixed constructible sheaf of weights $\le w + i$ for any $i \in \mathbb{Z}$. There is then furthermore defined another complex $\mathrm{D}(M)$, called the Verdier dual of M, and M is said to be pure of weight $w$ if M is mixed of weights $\le w$ and $\mathrm{D}(M)$ is mixed of weights $\le -w$.
### REMARK E.6. (1)
For a single lisse sheaf $M$ which is punctually pure of weight 0, the corresponding complex has $\mathcal{H}^0(M) = M$ and $\mathcal{H}^i(M) = 0$ for all $i \neq 0$. One can prove that the Verdier dual is a complex $D(M)$ such that $\mathcal{H}^{-2m}(D(M))$ is a lisse sheaf which is pure of weight $-2m$ and all the other cohomology sheaves vanish, so that the two definitions of purity coincide for lisse sheaves. In fact, the trace function of $D(M)$ is in this case the complex conjugate of the trace function of $M$.

(2) In practice, if an analytic number theorist is interested in a single trace function (e.g., one that represents a concrete family of exponential sums which one is interested in estimating) and one is not applying further operations like $Rf_!$, then one can quite often reduce to the case of a single lisse sheaf. This is for example the case for the hyper-Kloosterman sums in two variables

$$ \mathrm{Kl}_3(x; k_n) = \frac{1}{|k_n|} \sum_{\substack{a,b,c \in k_n^\times \\ abc=x}} \psi_n(a+b+c), $$

or the famous sums

$$ \mathrm{FI}(x, y; k_n) = \sum_{z \in k_n^\times} \mathrm{Kl}_3(xz; k_n) \mathrm{Kl}_3(yz; k_n) \psi_n(z) $$

which arose in the work of Friedlander and Iwaniec on the ternary divisor function [45], and reappeared in the work of Zhang [128].

Indeed, if the exponential sum is mixed, this will often be clear from the definition, or from a preliminary analysis, and one can “isolate” the part of most interest (of highest weight usually), which will be associated to a punctually pure constructible sheaf. Then by restricting the set of definition according to a suitable stratification, one will ensure that one handles a lisse sheaf.

For $m = 1$, this second step means avoiding finitely many values of $x$ where the sheaf has unusual behavior; for $m \ge 2$, this means avoiding those that satisfy some non-trivial polynomial equation $g(x_1, \dots, x_m) = 0$. These special parameters can then be handled separately—giving rise to a kind of inductive process which reflects exactly the algebraic stratification of the corresponding coefficient $M$.

One good explanation for the focus on mixed objects with bounded weights can be found (a posteriori) from the statement of Deligne's most general form of the Riemann hypothesis. In our context, it can be stated as follows:

### THEOREM E.7 (Deligne)
Let $(t_n)$ be a trace function on $\mathbf{A}^m$ associated to a complex $M$ which is mixed of weights $\le w$. Let $f = (f_1, \dots, f_d)$ be a tuple of polynomials in $k[X_1, \dots, X_m]$. The complex $Rf!M$ is mixed of weights $\le w$, and so its trace functions

$$ s_n(y) = \sum_{\substack{x \in k_n^m \\ f(x)=y}} t_n(x) $$

are also mixed of weights $\le w$.

### REMARK E.8.
On the other hand, even if $M$ is a single lisse sheaf, punctually pure of weight $w$, it is not always the case that $Rf!M$ is pure.

A benefit of introducing these more general definitions is that all operations now respect the property of being mixed for any trace function, with a good understanding of how the weights may change:

- The lisse sheaf $M = \overline{\mathbb{Q}}_\ell$ is pure of weight 0.

- If $M_1$ and $M_2$ have weights $\le w_1$ and $\le w_2$, respectively, then $M_1 \oplus M_2$ has weights $\le \max(w_1, w_2)$ and $M_1 \otimes M_2$ has weights $\le w_1 + w_2$.
- If M has weights $\le w$, then for any $k \in \mathbb{Z}$, the shifted complex $M[k]$ has weights $\le w - k$.

- If M has weights $\le w$, then for any $r \in \mathbb{Z}$, the twisted complex $M(r)$ has weights $\le w - 2r$.

- If $f = (f_1, \dots, f_d): \mathbf{A}^m \to \mathbf{A}^d$ is a tuple of polynomials in $k[X_1, \dots, X_m]$, and $s = (s_n)$ is a trace function on $\mathbf{A}^d$ associated to a mixed complex N of weights $\le w$, then $f^*N$ has weights $\le w$.

- If $f = (f_1, \dots, f_d): \mathbf{A}^m \to \mathbf{A}^d$ is a tuple of polynomials in $k[X_1, \dots, X_m]$, and if M has weights $\le w$, then $Rf!M$ has weights $\le w$ (this is again Deligne's Theorem E.7).

All objects that occur in practice in analytic number theory<fnref n="2" /> are mixed complexes. This means that any trace function $(t_n)$ has a decomposition

$$t_n = \sum_{a \le w \le b} t_{n,w}$$

for some $a$ and $b$ (independent of $n$), where $(t_{n,w})_{n \ge 1}$ is a trace function associated to a complex which is pure of weight $w$.

## E.4. Perverse sheaves

There remains the task of attempting to explain a further fundamental subclass of trace functions (hence of complexes), those associated to *perverse sheaves*. This is a distinguished class of complexes with remarkable geometric and arithmetic properties. For analytic purposes, the most important of these is maybe that the *simple* perverse sheaves provide a *canonical basis* of the abelian group of trace functions, and that if we restrict to pure perverse sheaves, then this is in a natural sense a *quasi-orthogonal basis* for the trace functions of pure complexes of weight 0. We will now explain these properties.

The rigorous definition of perverse sheaves is of a similar nature to that of pure complexes: it is the combination for both the complex M and its Verdier dual D(M) of a relatively simple condition, called *semiperversity*.<fnref n="3" /> The condition of semiperversity concerns the size of the support of the cohomology sheaves $\mathcal{H}^i(M)$ (which are intuitively the points $x$ where $\mathcal{H}^i(M)$ does not vanish; in the stratification in terms of lisse sheaves, this is where these sheaves have non-zero rank): for any $i \in \mathbb{Z}$, the support of $\mathcal{H}^i(M)$ should be of dimension at most $-i$. (In particular, if $i \ge 1$, then the support should be empty, so $\mathcal{H}^i(M)$ should be zero then.)

Remarkably, this condition can be recovered intuitively from basic analytic intuition (which highlights that it is extremely natural).

Thus consider a trace function $t = (t_n)$ associated to a complex M on $\mathbf{A}^m$ and assume that it is mixed of weights $\le 0$. From the analytic point of view, we are often in the situation where the mean-square of the values of the trace function $t_n$ are bounded (after some normalization maybe), and bounded away from zero, i.e., for $n$ large enough, we have

$$(E.3) \qquad \sum_{x \in k_n^m} |t_n(x)|^2 \asymp 1.$$

For $i \in \mathbb{Z}$, the cohomology sheaf $\mathcal{H}^i(M)$ should be “essentially” pure of weight $i$ (rigorously, we only know that it is mixed of weights $\le i$). So the contribution to the sum above of the $x$ in the support $S_i$ of $\mathcal{H}^i(M)$ should be expected to be of order of magnitude

$$|k_n|^{2\cdot i/2} \times |S_i(k_n)| \approx |k_n|^{i+d_i}$$

<footnote n="2">² And indeed more generally in algebraic geometry.</footnote>

<footnote n="3">³ The complication is that the Verdier dual is often difficult to compute.</footnote>
if $S_i$ has dimension $d_i$. Hence the estimate (E.3) only has a chance to hold if $i + d_i \le 0$ for all $i$, and this is precisely the semiperversity condition.

EXAMPLE E.9. Consider a family of exponential sums of type

$$ \frac{1}{|k_n|^m} \sum_{y \in k_n^m} \psi_n(f(y) + x_1 y_1 \cdots + x_m y_m) $$

with parameters $(x_1, \dots, x_m) \in k_n^m$ (these functions of $x$ are the trace functions of a complex M which is a normalized form of Deligne's Fourier transform of the lisse sheaf $\mathcal{L}_{\psi(f)}$).

We expect "generic" square-root cancellation, so as $n$ varies, for "most" choices of $x \in k_n^m$, this sum should be of size about $|k_n|^{-m/2}$. Since $\mathcal{H}^i(M)$ is of weight $\le i$, and hence contributes terms of size typically expected to be $|k_n|^{i/2}$, this expectation corresponds to the fact that $\mathcal{H}^i(M)$ should be "generically" zero unless $i = m$, while $\mathcal{H}^{-m}(M)$ contributes a fixed number of complex numbers of modulus $\le |k_n|^{-m/2}$.

But for special values of $x$, those satisfying some non-trivial polynomial equation $g(x) = 0$, one may obtain a larger sum than square-root cancellation. Experience teaches that usually this size only jumps by one factor $|k_n|^{1/2}$ (so the sum is about $|k_n|^{-m/2+1/2}$) if only this one condition is imposed; if it is bigger (say of size $|k_n|^{-m/2+1}$), this should mean that a second (independent) equation $h(x) = 0$ holds, and so on.

This "stratification" of bounds getting steadily worse only on smaller subsets corresponds to cohomology sheaves $\mathcal{H}^i(M)$ (contributing terms of size $|k_n|^{i/2}$) vanishing outside of subvarieties of dimension at most $-i$.

In the extreme case, the exponential sum is of size 1 (i.e., there is no cancellation at all) at worse for finitely many values of the parameters, corresponding to $\mathcal{H}^0(M)$ being supported on finitely many points.

This particular example is at the root of the results of Katz, Laumon and Fouvry on stratification for additive exponential sums [38, 78, 39]. It should suggest to analytic readers that semiperversity is a relatively easy condition to check, and that it should be natural and ubiquitous in analytic number theory.

The following statement provides a concrete illustration of the advantages of perverse sheaves.

THEOREM E.10. The $\mathbb{Z}$-module of trace functions on $\mathbb{A}^m$ over $k$ is generated by the trace functions of perverse sheaves, and the trace functions of simple perverse sheaves form a basis.

The first statement is in fact very explicit. Indeed, if $t = (t_n)$ is an arbitrary trace function, associated to a complex M, one can define (in addition to its "usual" cohomology sheaves $\mathcal{H}^i(M)$) its *perverse cohomology sheaves* $\mathcal{P}\mathcal{H}^i(M)$, which are perverse sheaves, zero for $|i| > m$, such that their trace functions $(\mathcal{P}t_{i,n})_{n \ge 1}$ satisfy the equation

$$ t_n = \sum_{i \in \mathbb{Z}} (-1)^i \mathcal{P}t_{i,n} $$

for all $n \ge 1$. Furthermore, a complex M is mixed of weights $\le w$ if and only if each $\mathcal{P}\mathcal{H}^i(M)$ is also mixed of weights $\le w + i$ (similarly to the cohomology sheaves; see [8, Th. 5.4.1]).

REMARK E.11. To say that a complex M is perverse is to say that its perverse cohomology sheaves are $M = \mathcal{P}\mathcal{H}^0(M)$ and $\mathcal{P}\mathcal{H}^i(M) = 0$ for all $i \ne 0$.

Up to the terminology and notation, the second statement of Theorem E.10 is proved by Laumon in [98, Th. 1.1.2] (it was already mentioned by Deligne in his letter to Kazhdan; see Appendix D).
To understand it, one must explain what are the simple perverse sheaves which are mentioned there. We will content ourselves with stating the quasi-orthonormality property which holds for a simple perverse sheaf that is pure of weight 0. It is another consequence of Deligne's Riemann Hypothesis, proved by Katz, that if $t = (t_n)$ is the trace function of a perverse sheaf M, then

$$ (E.4) \qquad \limsup_{n \to +\infty} \sum_{x \in k_n^m} |t_n(x)|^2 = 1 $$

if and only if M is simple.

**REMARK E.12.** One of the fundamental results of Beilinson, Bernstein, Deligne and Gabber [8, Cor. 5.3.4] is that a simple perverse sheaf which is mixed, as a complex, is in fact *pure* of some weight; since non-mixed complexes do not appear in practice, this means that simple perverse sheaves in analytic number theory are always pure of some weight, and the quasi-orthonormality characterization can be extended to all simple perverse sheaves, up to normalization.

**EXAMPLE E.13.** We can illustrate how useful this quasi-orthonormality statement can be to guess or understand some properties of perverse sheaves by noting that it strongly suggests a non-trivial property of simple perverse sheaves. Namely, let M be a simple perverse sheaf, pure of weight 0, and generically non-zero (i.e., the support of M is all of $\mathbf{A}^m$). If we repeat the argument leading to the guess of the semiperversity condition, we see that we expect that the contribution to

$$ \sum_{x \in k_n^m} |t_n(x)|^2 $$

of each non-zero cohomology sheaf $\mathcal{H}^i(M)$ should be of size

$$ \alpha_i |k_n|^{i+d_i} $$

for some integer $\alpha_i \ge 1$, and comparison with (E.4) indicates that $i+d_i$ will be < 0 except for one single value of $i$. Moreover, one knows that the cohomology sheaf $\mathcal{H}^{-m}(M)$ is generically non-zero, so this value must be $i = -m$, so that we expect that

$$ d_i \le -i - 1 \quad \text{for} \quad i \ne -m, $$

which is stronger than the condition $d_i \le -i$ derived from semiperversity only. This is indeed true (it is the improved support condition of Proposition A.13).

## E.5. Tannakian categories

The results of this book also rely in an essential way on another tool that is most likely unfamiliar to analytic number theorists: the formalism of tannakian categories. In very rough terms, this refers to a method to construct or define a *group* (which in our case will be the “symmetry group” that governs the equidistribution properties of a trace function), by recovering it from the way it acts on finite-dimensional K-vector spaces, for some algebraically closed field K of characteristic zero (which can be considered to be $\mathbb{C}$). That this is possible is indicated by the following result:

**THEOREM E.14 (Tannaka).** Let $\mathbf{G}$ be a compact group. Assume that for every finite-dimensional complex vector space $V$ on which the group $\mathbf{G}$ acts linearly, via a continuous homomorphism $\rho: \mathbf{G} \to \mathbf{GL}(V)$, we are given an invertible linear transformation $\alpha(\rho): V \to V$, and suppose that these data satisfy the following conditions:

“Whenever $\mathbf{G}$ acts by $\rho$ on $V$ and by $\pi$ on $W$, we have

$$ \alpha(1) = \mathrm{Id}_{\mathbb{C}}, \qquad \alpha(\rho \otimes \pi) = \alpha(\rho) \otimes \alpha(\pi); $$
whenever $\mathbf{G}$ acts by $\rho$ on $V$, we have

$$ \alpha(\bar{\rho}) = \overline{\alpha(\rho)}, $$

where $\bar{\rho}$ is the same action as $\rho$ but viewed as a representation on the conjugate vector space; and
whenever we have a linear map $u: V \to W$ such that

$$ u(\rho(g)v) = \pi(g)u(v) $$

for all $g \in \mathbf{G}$ and $v \in V$, then we have

$$ u \circ \alpha(\rho) = \alpha(\pi) \circ u $$

as linear maps from $V$ to $W$.”

Then there exists a unique element $g \in \mathbf{G}$ such that $\alpha(\rho) = \rho(g)$ for all actions $\rho$ of $\mathbf{G}$.

More generally, note that the “set” of all data of all $\alpha(\rho)$ of the type considered in this theorem can naturally be used to form a group (with $(\alpha\beta)(\rho) = \alpha(\rho) \circ \beta(\rho)$), and then the result identifies the group $\mathbf{G}$ with these data.

In a converse direction, the main theorem of the theory of tannakian categories establishes a list of conditions on a suitable category which guarantees that it is “equivalent” to the category of representations of a group $\mathbf{G}$ (although the context is that of algebraic groups, such as $\mathbf{GL}_n(\mathbb{C})$, instead of compact groups). A key property to apply the “reconstruction theorem” is that one must be able to associate to each object $M$ a finite-dimensional vector space $\omega(M)$ (corresponding to the abstract space on which the group acts), and one needs to have defined a bilinear operation on these objects, say $M \star N$, in such a way that $\omega(M \star N) = \omega(M) \otimes \omega(N)$. Such an “assignment” $\omega$ is called a *fiber functor*; it is not unique, and its construction may be a delicate matter.

In the applications in this book (following the idea of Katz in [75]), the objects that will correspond in this abstract way to the actions of $\mathbf{G}$ on vector spaces are certain perverse sheaves, and the operation $\star$ is a form of algebraic convolution which respects the corresponding usual convolution operation on trace functions.

For the classical form of Tannaka duality for compact groups, we refer to the presentation by Joyal and Street [68, § 1]. For an accessible treatment of tannakian categories, emphasizing the natural evolution from Galois theory, we refer to the book [124] of Szamuely.

## E.6. Frequently asked questions

We conclude by trying to answer some natural questions that an analytically-minded reader of little faith may raise:

- Is it possible to describe trace functions (or the underlying algebraic objects) “by generators and relations”, by listing a number of basic examples and a list of operations preserving trace functions, so that all trace functions are obtained from these basic data in finitely many steps?

It is true that in many applications to analytic number theory, the sheaves or trace functions which occur are constructed precisely in such a way (e.g., starting from an additive character, replacing the variable by a polynomials, taking the Fourier transform, etc).

However, it seems extremely unlikely that one could provide a satisfactory and rigorous version of such an idea, for instance because it is known that there are $q$ geometrically irreducible middle extension sheaves of rank 2 on the projective line over $\mathbb{F}_q$ with 4 singular points and principal tame local monodromy at each point (see for instance [31, Prop. 7.1]; the proof of this fact relies on automorphic methods). All these sheaves have bounded complexity as $q$ varies. However, only six such sheaves are explicitly known (they are
associated to certain elliptic curves over $F_q(t)$ with four singular fibers), as shown by Beauville [4].

Since operations on sheaves tend to increase the complexity in general (although in a controllable manner), it seems very difficult to imagine how one could construct the “other” $q-6$ sheaves in a straightforward way.

- *Why are perverse sheaves essential to the results of this book? Why can one not (even in the simplest cases, such as exponential sums parameterized by multiplicative characters) work around the requirement to use such objects in a way similar to the previous papers of Fouvry, Kowalski and Michel?*

The simplest reason for this (not the only one) is that the use of tannakian methods (which is the only way we know to produce the symmetry group for arithmetic Fourier transforms) depends on applying many times a number of operations which will have uncontrollable effect on the type of complex we work with, even when starting with a single lisse sheaf.

More technically, the same tannakian idea requires the construction of an *abelian* category (which will “be” the category of representations of the symmetry group); general complexes do not form an abelian category, whereas perverse sheaves form one—certainly the best known abelian category beyond that of lisse sheaves.

- *Conversely, if perverse sheaves are so natural and have such remarkable properties, and suffice to describe all trace functions, why not dispense with general complexes then?*

Here the issue is that, although perverse sheaves and their trace functions are individually wonderful things, they are not *in toto* stable by all the operations that one might want to apply. In particular, if $M, M_1, M_2$ and $N$ are perverse sheaves, then it is *not true* in general that $M_1 \otimes M_2$, or $f^*N$, or $Rf!M$, are perverse sheaves (on their respective affine spaces). (A significant and highly non-trivial exception, however, is that if $M$ is perverse on $\mathbf{A}^m$, then its Fourier transform in the sense of Deligne is still perverse.) In the case of our applications, the problem appears in the definition of the algebraic convolution that is used to apply the tannakian formalism—a priori, even for $M$ and $N$ perverse, their algebraic convolution is simply a complex of constructible sheaves.

- *Why is there no normalization by the size of the sum in a formula like (E.4)?*

It is a useful property of perverse sheaves, although surprising at first sight, that the definition itself implies a normalization for these sums. If $M$ is a perverse sheaf with support $\mathbf{A}^m$ which is pure of weight 0, then the local eigenvalues at a “generic” point $x$ of $k_n^m$ are of weight $-m$, i.e., they are typically of size $|k|^{-m/2}$. So the sum (E.4) is naturally expected to be of bounded size, *without normalizing*.
<figure><img src="image_20.png" /></figure>
Index

27 lines of a smooth cubic surface, 174

A-number, 58

α-symmetric Sidon set, 127

L̂-function, 97

i-symmetric Sidon morphism, 127

q-Weil number of weight w, 184

r-Sidon set, 127

“forget support” morphism, 22

Albanese variety of the Fano variety, 171

Albanese morphism, 171

algebraic group, 12

algebraic variety, 12

almost simple group, 30

alteration, 39

alteration topology, 46

arithmetic fiber functor, 64

arithmetic Fourier transform, 27

arithmetic tannakian group, 67

Artin L-function, 169

Artin–Schreier sheaf, 62

automorphic representation, 179

Cesàro mean, 81

character codimension, 28

character sheaf, 24

Chebotarev density theorem, 86, 174

classical group, 126

coefficient object, 201

coevaluation map, 23

compatible system, 117

complexity, 19

convolution, 22

convolution category, 59

convolution with compact support, 22

Deligne's equidistribution theorem, 5, 77

Deligne's fiber functor, 74, 178, 189

diagonal embedding, 138

Diophantine group theory, 10

diophantine irreducibility criterion, 123

discrete Plancherel formula, 178

disjoint tuples of characters, 191

dual homomorphism, 25

embedding dimension, 12

equidistribution on average, 77

evaluation map, 23

exceptional group E₆, 134

faithful representation, 32

Fano variety of lines, 171

Fermat curve and threefold, 176

finite Heisenberg groups, 136

Fourier L-function, 97

Fourier sheaf, 141

Fourier inversion formula, 27

Fourier transform, 36

Fourier–Mellin transform on semi-abelian variety, 30

Frobenius reciprocity, 101

Frobenius automorphism of M associated to χ, 69

Frobenius conjugacy class of M associated to χ, 69

Frobenius–Schur indicator, 123

Frobenius-unramified character, 70

functional equation, 99, 193

Gabber's torus trick, 133

Gauss sum, 13, 192

generalized jacobian, 131, 140

generic Fourier invertibility, 115

generic set of characters, 28

generic vanishing theorem, 6

generically unramified, 64

generically unramified perverse sheaf, 64

geometric Frobenius automorphism, 24, 68

geometric tannakian group, 62, 67

gonality, 130

Goursat–Kolchin–Ribet criterion, 148

half-spin representation, 135

hyper-Kloosterman sums, 5, 192

hyperelliptic curve, 130

hypergeometric sheaf, 190

hypergeometric complex, 190

hyperplane sections, 174

intermediate jacobian, 171

internal convolution, 59

internal convolution category, 59

jacobian of a curve, 130
Kloosterman sums, 5, 138

Kloosterman–Salié sums, 138

Kummer sheaf, 62, 189

Lang torsor, 24

Langlands correspondence, 166, 179

Larsen's alternative, 126

Legendre elliptic curve, 162

local epsilon factor, 145, 193

local monodromy representation, 161

middle extension sheaf, 74, 183

mixed complex of weights $\le w$, 185

moments of a representation, 124

monodromy group, 5

monoidal category, 59

negligible object, 57

neutral tannakian category, 60

norm map, 23

permutation representation, 174

perverse amplitude, 13

perverse convolution category, 59

Peter–Weyl theorem, 69, 80

Picard group, 131

Picard variety of the Fano variety, 171

Plancherel formula, 27

primitive characters, 97

punctual weight, 185

punctually pure sheaf, 185

pure complex of weight $w$, 185

quasi-orthogonality, 21

quasi-projective scheme, 12

quasi-projective variety, 12

quasi-unipotent object, 100

reductive algebraic group, 12

representation

of symplectic type, 123

of orthogonal type, 123

Riemann Hypothesis, 21

rigid monoidal category, 60

Salié sums, 138

Sato–Tate law, 5

Sato–Tate measure, 5, 84, 138

Schur's Lemma, 81, 123

semicontinuity of monodromy, 176

Serre dual, 35

Serre subcategory, 58

set containing most characters, 29

sheaf tame and constructible along $\mathcal{X}$, 13

Sidon morphism, 127

Sidon set, 127

Sidon subvariety, 127

skyscraper sheaf, 23

smooth cubic threefold, 171

square-root cancellation, 208

standard representation, 86

stratification, 13

stratified vanishing theorem, 6

summation method, 82

Swan conductor, 194, 195

symmetry group, 209

tac, 29

tame monodromy representation, 74

tannakian reconstruction theorem, 62

tannakian determinant, 145

theta divisor, 172

thick subcategory, 58

unipotent Jordan blocks, 74

unit object, 59

unitary Frobenius conjugacy class associated to $\chi$, 69

unitary Frobenius conjugacy class of $M$ associated to $\chi$, 69

unramified character, 64

virtually central subgroup, 127

von Mangoldt function, 160

weakly generic set of characters, 29

weakly unramified character, 56

Weil number, 13

Weyl criterion, 11, 87

Weyl dimension formula, 135

Weyl group, 135, 174
Glossary

<table>
  <tr>
    <td>1</td>
    <td>skyscraper sheaf, unit for convolution, 22</td>
  </tr>
  <tr>
    <td>A(X)</td>
    <td>Albanese variety of F(X), 161</td>
  </tr>
  <tr>
    <td>D(M)</td>
    <td>Verdier dual, 14</td>
  </tr>
  <tr>
    <td>D(G)</td>
    <td>subcategory of D<sup>b</sup><sub>c</sub>(G<sub>k</sub>) of objects defined over a finite field, 54</td>
  </tr>
  <tr>
    <td>D&#773;(G)</td>
    <td>convolution category, 57</td>
  </tr>
  <tr>
    <td>D<sup>b</sup><sub>c</sub>(X) = D<sup>b</sup><sub>c</sub>(X, Q&#773;<sub>l</sub>)</td>
    <td>category of bounded constructible complexes of Q&#773;<sub>l</sub>-sheaves on X, 14</td>
  </tr>
  <tr>
    <td>&Delta;</td>
    <td>diagonal embedding G<sub>m</sub> &rarr; G<sub>m</sub> &times; G<sub>a</sub>, 130</td>
  </tr>
  <tr>
    <td>FM!</td>
    <td>Fourier-Mellin transform with compact support, 30</td>
  </tr>
  <tr>
    <td>FM*</td>
    <td>Fourier-Mellin transform, 30</td>
  </tr>
  <tr>
    <td>Fr<sub>M,k<sub>n</sub></sub>(&chi;)</td>
    <td>Frobenius action on H<sup>0</sup><sub>c</sub>(G<sub>k</sub>, &chi;), 66</td>
  </tr>
  <tr>
    <td>Fr<sub>M</sub>(&chi;)</td>
    <td>Fr<sub>M,k</sub>(&chi;), 66</td>
  </tr>
  <tr>
    <td>Fr<sub>k<sub>n</sub></sub></td>
    <td>geometric Frobenius automorphism of k<sub>n</sub>, 66</td>
  </tr>
  <tr>
    <td>Fr<sub>M,k<sub>n</sub></sub>(&chi;)</td>
    <td>Frobenius conjugacy class in G<sup>ari</sup><sub>M</sub>, 66</td>
  </tr>
  <tr>
    <td>Fr<sub>M</sub>(&chi;)</td>
    <td>Fr<sub>M,k</sub>(&chi;), 66</td>
  </tr>
  <tr>
    <td>FT<sub>&psi;</sub></td>
    <td>Fourier transform on a unipotent group, 36</td>
  </tr>
  <tr>
    <td>F(X)</td>
    <td>Fano variety of lines, 161</td>
  </tr>
  <tr>
    <td>[G&#770;]</td>
    <td>primitive characters modulo Galois action, 91</td>
  </tr>
  <tr>
    <td>G&#770;</td>
    <td>disjoint union of G&#770;(k<sub>n</sub>), 23</td>
  </tr>
  <tr>
    <td>G&#770;(k<sub>n</sub>)</td>
    <td>characters of G(k<sub>n</sub>), 23</td>
  </tr>
  <tr>
    <td>G&#770;<sup>*</sup></td>
    <td>primitive elements of G&#770;, 91</td>
  </tr>
  <tr>
    <td>G<sup>(&ell;)</sup></td>
    <td>&ell;-adic characters, 111</td>
  </tr>
  <tr>
    <td>G<sup>ari</sup><sub>M</sub></td>
    <td>arithmetic tannakian group of M, 64</td>
  </tr>
  <tr>
    <td>G<sup>geo</sup><sub>M</sub></td>
    <td>geometric tannakian group of M, 60</td>
  </tr>
  <tr>
    <td>H<sub>int</sub>(T)</td>
    <td>hypergeometric group of Gabber and Loeser, 180</td>
  </tr>
  <tr>
    <td>Hyp(!, &psi;, &chi;; &varrho;)</td>
    <td>hypergeometric sheaf, 178</td>
  </tr>
  <tr>
    <td>J<sub>S</sub></td>
    <td>generalized jacobian, 123</td>
  </tr>
  <tr>
    <td>K(X) = K(X, Q&#773;<sub>l</sub>)</td>
    <td>Grothendieck group, 14</td>
  </tr>
  <tr>
    <td>K<sup>&#8317;</sup></td>
    <td>space of conjugacy classes in K, 75</td>
  </tr>
  <tr>
    <td>K<sub>neg</sub>(G)</td>
    <td>subgroup of K(G) generated by negligible objects, 55</td>
  </tr>
  <tr>
    <td>&Lambda;<sub>M</sub></td>
    <td>von Mangoldt function of M, 148</td>
  </tr>
  <tr>
    <td>L&#770;(M, T)</td>
    <td>L&#770;-function of M, 91</td>
  </tr>
  <tr>
    <td>L<sup>&lambda;</sup><sub>&chi;,a</sub></td>
    <td>sheaf &lambda;<sub>*</sub>(L<sub>&chi;</sub> &otimes; j<sup>*</sup>L<sub>&psi;(ay)</sub>), 130</td>
  </tr>
  <tr>
    <td>L<sub>G</sub></td>
    <td>&Omega;<sub>G</sub>-sheaf associated to can<sub>G</sub>, 30</td>
  </tr>
  <tr>
    <td>L<sub>U,U<sup>&varnothing;</sup>,&psi;</sub></td>
    <td>lisse sheaf on U &times; U<sup>&varnothing;</sup>, 36</td>
  </tr>
  <tr>
    <td>L<sub>U,U<sup>&varnothing;</sup></sub></td>
    <td>bi-extension, 35</td>
  </tr>
  <tr>
    <td>L<sub>&chi;</sub></td>
    <td>character sheaf, 24</td>
  </tr>
  <tr>
    <td>M<sub>*N</sub></td>
    <td>convolution, 22</td>
  </tr>
  <tr>
    <td>M<sub>*!N</sub></td>
    <td>convolution with compact support, 22</td>
  </tr>
  <tr>
    <td>M<sub>*int</sub>N</td>
    <td>internal convolution, 57</td>
  </tr>
  <tr>
    <td>M<sup>&varnothing;</sup></td>
    <td>tannakian dual of M, 22</td>
  </tr>
  <tr>
    <td>M<sub>2m</sub>(G)</td>
    <td>M<sub>2m</sub>(G, Std), 116</td>
  </tr>
  <tr>
    <td>M<sub>2m</sub>(G, V)</td>
    <td>2m-th absolute moment of the representation V of G, 116</td>
  </tr>
</table>
<table>
  <tr>
    <td>M<sub>χ</sub></td>
    <td>twist of M by L<sub>χ</sub>, 25</td>
  </tr>
  <tr>
    <td>M<sub>int</sub></td>
    <td>quasi-inverse of P<sub>int</sub>(G) → P&#773;(G), 57</td>
  </tr>
  <tr>
    <td>Neg<sub>D</sub>(G)</td>
    <td>negligible complexes of D(G), 55</td>
  </tr>
  <tr>
    <td>Neg<sub>P</sub>(G)</td>
    <td>negligible objects of P(G), 55</td>
  </tr>
  <tr>
    <td>Neg<sub>P</sub><sup>ari</sup>(G)</td>
    <td>arithmetic negligible objects, 64</td>
  </tr>
  <tr>
    <td>P<sub>int</sub><sup>ari</sup>(G)</td>
    <td>arithmetic internal convolution category, 64</td>
  </tr>
  <tr>
    <td>N(M)</td>
    <td>characters χ with H<sup>*</sup>(G<sub>k</sub>, M<sub>χ</sub>) = H<sup>*</sup><sub>c</sub>(G<sub>k</sub>, M<sub>χ</sub>) = 0, 55</td>
  </tr>
  <tr>
    <td>N<sub>k<sub>n</sub>/k</sub></td>
    <td>norm map, 23</td>
  </tr>
  <tr>
    <td>O(B)</td>
    <td>orthogonal group of B, 118</td>
  </tr>
  <tr>
    <td>Ω<sub>G</sub></td>
    <td>completed group algebra of π<sub>1</sub><sup>t</sup>(G<sub>k</sub>)<sub>ℓ</sub>, 29</td>
  </tr>
  <tr>
    <td>Perv(X) = Perv(X, Q&#773;<sub>ℓ</sub>)</td>
    <td>category of perverse sheaves, 14</td>
  </tr>
  <tr>
    <td>Π(G)</td>
    <td>disjoint union of Π(G)<sub>ℓ</sub>, 28</td>
  </tr>
  <tr>
    <td>Π(G)<sub>ℓ</sub></td>
    <td>Q&#773;<sub>ℓ</sub>-scheme whose Q&#773;<sub>ℓ</sub>-points are Π(G, Q&#773;<sub>ℓ</sub>)<sub>ℓ</sub>, 28</td>
  </tr>
  <tr>
    <td>Π(G, Q&#773;<sub>ℓ</sub>)</td>
    <td>continuous tame ℓ-adic characters, 28</td>
  </tr>
  <tr>
    <td>Π(G, Q&#773;<sub>ℓ</sub>)<sub>ℓ'</sub></td>
    <td>torsion characters of order prime to ℓ, 28</td>
  </tr>
  <tr>
    <td>Π(G, Q&#773;<sub>ℓ</sub>)<sub>ℓ</sub></td>
    <td>characters factoring through the pro-ℓ-quotient, 28</td>
  </tr>
  <tr>
    <td>P(G)</td>
    <td>subcategory of Perv(G<sub>k</sub>) of objects defined over a finite field, 54</td>
  </tr>
  <tr>
    <td>P&#773;(G)</td>
    <td>perverse convolution category, 57</td>
  </tr>
  <tr>
    <td>P&#773;<sup>ari</sup>(G)</td>
    <td>arithmetic convolution category, 64</td>
  </tr>
  <tr>
    <td>P</td>
    <td>category of perverse sheaves on G<sub>m</sub>, 60</td>
  </tr>
  <tr>
    <td>π<sub>1</sub><sup>t</sup>(G<sub>k</sub>)</td>
    <td>tame étale fundamental group, 28</td>
  </tr>
  <tr>
    <td>Rep<sub>Q&#773;<sub>ℓ</sub></sub>(G)</td>
    <td>category of representations of G, 59</td>
  </tr>
  <tr>
    <td>Std</td>
    <td>standard or tautological representation, 68</td>
  </tr>
  <tr>
    <td>S(M, χ)</td>
    <td>arithmetic Fourier transform, 26</td>
  </tr>
  <tr>
    <td>SO(B)</td>
    <td>special orthogonal group of B, 118</td>
  </tr>
  <tr>
    <td>Sp(B)</td>
    <td>symplectic group of B, 118</td>
  </tr>
  <tr>
    <td>Θ(X)</td>
    <td>theta divisor, 162</td>
  </tr>
  <tr>
    <td>Θ<sub>M,k<sub>n</sub></sub>(χ)</td>
    <td>unitary conjugacy class for Fr<sub>M,k<sub>n</sub></sub>(χ), 66</td>
  </tr>
  <tr>
    <td>Θ<sub>M</sub>(χ)</td>
    <td>Θ<sub>M,k</sub>(χ), 66</td>
  </tr>
  <tr>
    <td>Θ<sub>M,k<sub>n</sub></sub>(χ)</td>
    <td>unitary Frobenius conjugacy class in G<sub>M</sub><sup>ari</sup>, 66</td>
  </tr>
  <tr>
    <td>Θ<sub>M</sub>(χ)</td>
    <td>Θ<sub>M,k</sub>(χ), 66</td>
  </tr>
  <tr>
    <td>Unip(n)</td>
    <td>unipotent Jordan block of size n, 153</td>
  </tr>
  <tr>
    <td>U<sup>*</sup></td>
    <td>Serre dual, as group scheme, 35</td>
  </tr>
  <tr>
    <td>U<sup>√</sup></td>
    <td>algebraic group model of the Serre dual, 35</td>
  </tr>
  <tr>
    <td>[X]</td>
    <td>set of closed points of X, 181</td>
  </tr>
  <tr>
    <td>X&#773;<sub>F</sub>(N)</td>
    <td>X&#773;<sub>F</sub>(ρ), 67</td>
  </tr>
  <tr>
    <td>X&#773;<sub>F</sub>(N)<sub>M</sub></td>
    <td>Frobenius-unramified characters for N ∈ ⟨M⟩, 67</td>
  </tr>
  <tr>
    <td>X&#773;<sub>F</sub>(ρ)</td>
    <td>Frobenius-unramified characters for ρ, 67</td>
  </tr>
  <tr>
    <td>X(M)</td>
    <td>set of unramified characters, 61</td>
  </tr>
  <tr>
    <td>X<sub>w</sub>(M)</td>
    <td>set of weakly unramified characters for M, 54</td>
  </tr>
  <tr>
    <td>[s]</td>
    <td>the set {1, ..., s}, 30</td>
  </tr>
  <tr>
    <td>FS(ρ)</td>
    <td>Frobenius-Schur indicator of a representation, 116</td>
  </tr>
  <tr>
    <td>⟨M⟩</td>
    <td>subcategory tensor-generated by M, 58</td>
  </tr>
  <tr>
    <td>⟨M⟩</td>
    <td>tannakian category generated by M, 14</td>
  </tr>
  <tr>
    <td>⟨M⟩<sup>ari</sup></td>
    <td>subcategory of P<sub>int</sub><sup>ari</sup>(G) tensor-generated by M, 64</td>
  </tr>
  <tr>
    <td>⟨M⟩<sup>geo</sup></td>
    <td>subcategory of P<sub>int</sub>(G) tensor-generated by M<sub>k</sub>, 64</td>
  </tr>
  <tr>
    <td>a(T, M)</td>
    <td>local exponent, 182</td>
  </tr>
  <tr>
    <td>a(T, M, ω)</td>
    <td>local exponent, 182</td>
  </tr>
  <tr>
    <td>α<sup>deg</sup></td>
    <td>geometrically trivial lisse sheaf, 14</td>
  </tr>
  <tr>
    <td>can<sub>G</sub></td>
    <td>tautological character, 30</td>
  </tr>
  <tr>
    <td>ccodim(S)</td>
    <td>character codimension, 27</td>
  </tr>
  <tr>
    <td>⟨χ, a⟩</td>
    <td>character on G<sub>m</sub> × G<sub>a</sub>, 129</td>
  </tr>
</table>
<table>
  <tr>
    <td>coev</td>
    <td>coevaluation map, 58</td>
  </tr>
  <tr>
    <td>c(M)</td>
    <td>complexity on projective space, 19</td>
  </tr>
  <tr>
    <td>c<sub>u</sub>(M)</td>
    <td>complexity on quasi-projective variety, 19</td>
  </tr>
  <tr>
    <td>deg(χ)</td>
    <td>degree of a primitive character, 91</td>
  </tr>
  <tr>
    <td>drop<sub>x</sub>(M)</td>
    <td>drop of a complex at x, 183</td>
  </tr>
  <tr>
    <td>e</td>
    <td>neutral element of a group, 22</td>
  </tr>
  <tr>
    <td>ev</td>
    <td>evaluation map, 58</td>
  </tr>
  <tr>
    <td>e(z)</td>
    <td>exp(2iπz), 12</td>
  </tr>
  <tr>
    <td>ε<sub>0</sub>(T, 𝒯, ω)</td>
    <td>local epsilon factor, 182</td>
  </tr>
  <tr>
    <td>f̂</td>
    <td>dual homomorphism, 24</td>
  </tr>
  <tr>
    <td>f̂<sub>n</sub></td>
    <td>dual homomorphism, 24</td>
  </tr>
  <tr>
    <td>f ≍ g</td>
    <td>asymptotic notation, 12</td>
  </tr>
  <tr>
    <td>f ≪ g</td>
    <td>asymptotic notation, 12</td>
  </tr>
  <tr>
    <td>f ∼ g</td>
    <td>asymptotic notation, 12</td>
  </tr>
  <tr>
    <td>inv</td>
    <td>inverse map on a group, 22</td>
  </tr>
  <tr>
    <td>m</td>
    <td>group law, 22</td>
  </tr>
  <tr>
    <td>μ<sup>‡</sup></td>
    <td>image of Haar measure of K on K<sup>‡</sup>, 83</td>
  </tr>
  <tr>
    <td>ν<sub>cp</sub></td>
    <td>image of Haar measure on K on the space of conjugacy classes in U<sub>r</sub>(C)<sup>‡</sup>, 75</td>
  </tr>
  <tr>
    <td>ω<sub>χ</sub></td>
    <td>fiber functor defined by χ, 61</td>
  </tr>
  <tr>
    <td>ω<sub>Del</sub></td>
    <td>Deligne's fiber functor, 177</td>
  </tr>
  <tr>
    <td>ψ<sub>x</sub></td>
    <td>character associated to x ∈ U<sup>∨</sup>, 35</td>
  </tr>
  <tr>
    <td>swan<sub>x</sub>(M)</td>
    <td>Swan conductor of a complex at x, 183</td>
  </tr>
  <tr>
    <td>t<sub>M</sub>(x) = t<sub>M</sub>(x; k)</td>
    <td>trace function of M on X(k), 14</td>
  </tr>
  <tr>
    <td>t<sub>M</sub>(x; k<sub>n</sub>)</td>
    <td>trace function of M on X(k<sub>n</sub>), 14</td>
  </tr>
  <tr>
    <td>τ(ξ, ψ)</td>
    <td>Gauss sum, 14</td>
  </tr>
</table>
<figure><img src="image_21.png" /></figure>
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