### 3.5. Tannakian categories

By results of Gabriel [51] for abelian categories and Verdier (see the treatment in the book [107] of Neeman) for triangulated categories, we can define the quotient of an abelian or triangulated category by a Serre or thick subcategory. This allows us to make the following definition.

**DEFINITION 3.9 (Convolution categories).** The convolution category of $G$, denoted $\overline{D}(G)$, is the quotient category of $D(G)$ by $\text{Neg}_D(G)$; it is a triangulated category.

The *perverse convolution category* of $G$, denoted $\overline{\mathbf{P}}(G)$, is the quotient abelian category of $\mathbf{P}(G)$ by $\text{Neg}_{\mathbf{P}}(G)$.

Those two constructions are compatible, in the sense that the t-structure on $D(G)$ induces a t-structure on $\overline{D}(G)$ whose heart is the category $\overline{\mathbf{P}}(G)$ (see [50, Prop. 3.6.1]).

Since the functor $N \mapsto N^\vee$ preserves negligible objects, it induces a functor on $\overline{\mathbf{P}}(G)$ (resp. on $\overline{D}(G)$), which is still an involution.

**PROPOSITION 3.10.** With notation as above, the following properties hold:

(1) The convolution products $*!$ and $*_{*}$ induce bifunctors on $\overline{D}(G) \times \overline{D}(G)$.

(2) The canonical forget support morphisms $M*!N \to M*_{*}N$ induce isomorphisms in $\overline{D}(G)$, and define by passing to the quotient a convolution bifunctor denoted

$$*: \overline{D}(G) \times \overline{D}(G) \to \overline{D}(G).$$

(3) The subcategory $\overline{\mathbf{P}}(G)$ of $\overline{D}(G)$ is stable under the convolution $*$.

(4) The categories $\overline{D}(G)$ and $\overline{\mathbf{P}}(G)$, endowed with the bifunctor $*$, are symmetric $\overline{\mathbf{Q}}_{\ell}$-linear monoidal categories with unit object $\mathbf{1}$ the image of the skyscraper sheaf at the neutral element of $G$.

**PROOF.** The fact that $*!$ and $*_{*}$ induce functors on $\overline{D}(G) \times \overline{D}(G)$ follows from Lemma 3.8 (2). That they agree is Lemma 3.8 (1). The stability of $\overline{\mathbf{P}}(G)$ under $*$ is Lemma 3.8 (3). The fact that we obtain symmetric $\overline{\mathbf{Q}}_{\ell}$-linear monoidal categories is now clear. The last assertion follows from the canonical isomorphisms $\mathbf{1} *! M \simeq \mathbf{1} *_{*} M \simeq M$ which exist for any complex $M$. $\square$

It is also very useful that there exists a natural subcategory of $\mathbf{P}(G)$ that is equivalent to the perverse convolution category.

**DEFINITION 3.11.** The *internal convolution category* of $G$ is the full subcategory $\mathbf{P}_{\text{int}}(G)$ of the category $\mathbf{P}(G)$ whose objects are perverse sheaves that have no subobject or quotient in $\text{Neg}_{\mathbf{P}}(G)$.

**PROPOSITION 3.12.** The localization functor $\mathbf{P}(G) \to \overline{\mathbf{P}}(G)$ restricts to an equivalence of categories

$$\mathbf{P}_{\text{int}}(G) \to \overline{\mathbf{P}}(G),$$

hence the convolution product bifunctor $*$ on $\overline{\mathbf{P}}(G)$ induces a convolution bifunctor $*_{\text{int}}$ on $\mathbf{P}_{\text{int}}(G)$.

**PROOF.** The argument is the same as that of Gabber and Loeser [50, Déf.-Prop. 3.7.2]. $\square$

The convolution product on $\mathbf{P}_{\text{int}}(G)$ will sometimes be called the *internal* or *middle* convolution.

**REMARK 3.13.** One can give a more explicit form of the equivalence of categories above, and of the internal convolution.

First, Gabber and Loeser (loc. cit.) give an explicit quasi-inverse functor $M \mapsto M_{\text{int}}$ to the equivalence of categories $\mathbf{P}_{\text{int}}(G) \to \overline{\mathbf{P}}(G)$. Namely, let $M$ be an object of $\mathbf{P}(G)$. Let $M_t$ be the
largest subobject of M that belongs to $\mathrm{Neg}_P(G)$ and let $M^t$ be the smallest subobject of M such that $M/M^t$ belongs to $\mathrm{Neg}_P(G)$. Define $M_{\mathrm{int}} = M^t/(M^t \cap M_t)$. Then we have canonical isomorphisms

$$M_{\mathrm{int}} \simeq (M^t + M_t)/M_t,$$

and the assignment $M \mapsto M_{\mathrm{int}}$ is a functor which factors through $\bar{P}(G)$ and induces a quasi-inverse of the localization functor.

In particular, this implies that if M is a semisimple object of $\bar{P}(G)$, then $M_{\mathrm{int}}$ is the sum of all the simple constituents of M that are not in $\mathrm{Neg}_P(G)$.

Second, it follows from the argument in [50, Déf.-Prop. 3.7.3] that for M and N in $\mathbf{P}_{\mathrm{int}}(G)$, there are canonical isomorphisms

$$(3.1) \qquad M_{*\mathrm{int}} N \xrightarrow{\sim} \mathfrak{p}\mathcal{H}^0(M_{*!} N)_{\mathrm{int}} \xrightarrow{\sim} \mathfrak{p}\mathcal{H}^0(M_{*\mathrm{N}})_{\mathrm{int}}.$$

We continue using the notation of the remark.

**LEMMA 3.14.** Let M be an object of $\mathbf{P}(G)$. For all $\chi \in \mathcal{N}(M/M^t) \cap \mathcal{N}(M^t \cap M_t)$, the natural morphisms $M^t \xrightarrow{i} M$ and $M^t \xrightarrow{p} M^t/(M^t \cap M_t) = M_{\mathrm{int}}$ induce isomorphisms

$$H^*(G_{\bar{k}}, (M^t)_{\chi}) \xrightarrow{i_*} H^*(G_{\bar{k}}, M_{\chi}) \quad \text{and} \quad H^*(G_{\bar{k}}, (M^t)_{\chi}) \xrightarrow{p_*} H^*(G_{\bar{k}}, (M_{\mathrm{int}})_{\chi}).$$

In particular, the set of characters $\chi \in \hat{\mathcal{G}}$ such that the cohomology groups $H^*(G_{\bar{k}}, (M_{\mathrm{int}})_{\chi})$ and $H^*(G_{\bar{k}}, M_{\chi})$ are isomorphic is generic.

**PROOF.** The first statement follows immediately from the exact sequences

$$M^t \to M \to M/M^t, \quad M^t \cap M_t \to M^t \to M^t/(M^t \cap M_t) = M_{\mathrm{int}},$$

and the assumption on $\chi$. Since $\mathcal{N}(M/M^t)$ and $\mathcal{N}(M^t \cap M_t)$ are both generic by the definitions of $M^t$ and $M_t$, the final assertion follows. $\square$

Recall from Section 1.5 that for $M \in \mathbf{P}(G)$, the identity morphism $\mathrm{id}_M: M \to M$ defines evaluation and coevaluation maps

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

They correspond to maps in $\bar{P}(G)$ which we denote in the same way.

**PROPOSITION 3.15.** The monoidal category $\bar{P}(G)$ is rigid. That is, for each object M of $\bar{P}(G)$, the morphisms

$$M \simeq M * \mathbf{1} \xrightarrow{\mathrm{id}_M * \mathrm{coev}} M * M^{\vee} * M \xrightarrow{\mathrm{ev} * \mathrm{id}_M} \mathbf{1} * M \simeq M$$

$$M^{\vee} \simeq \mathbf{1} * M^{\vee} \xrightarrow{\mathrm{coev} * \mathrm{id}_M} M^{\vee} * M * M^{\vee} \xrightarrow{\mathrm{id}_M * \mathrm{ev}} M^{\vee} * \mathbf{1} \simeq M^{\vee}$$

are the identity on M and on $M^{\vee}$ respectively.

**PROOF.** The argument is the same as that of Krämer in [91, Th. 5.2]. $\square$

For any object M of $\bar{P}(G)$ (resp. of $\mathbf{P}_{\mathrm{int}}(G)$), we denote by $\langle M \rangle$ the subcategory of $\bar{P}(G)$ (resp. of $\mathbf{P}_{\mathrm{int}}(G)$) which is tensor-generated by M, i.e., the full subcategory whose objects are the subquotients of all convolution powers of $M \oplus M^{\vee}$.

Our next goal is to prove the following crucial result:

**THEOREM 3.16.** The categories $\bar{P}(G)$ and $\mathbf{P}_{\mathrm{int}}(G)$ are neutral tannakian categories.

In particular, for any object M of $\mathbf{P}_{\mathrm{int}}(G)$ or of $\mathbf{P}(G)$, the category $\langle M \rangle$ is a neutral tannakian category over $\bar{\mathcal{Q}}_{\ell}$.
Recall that this means that there exists a fiber functor, namely a faithful exact tensor functor from $\overline{\mathbf{P}}(G)$ to the category $\mathrm{Vect}_{\overline{\mathbb{Q}}_\ell}$ of finite dimensional $\overline{\mathbb{Q}}_\ell$-vector spaces.

We begin the proof with an auxiliary result. Recall that the trace $\mathrm{Tr}(f) \in \overline{\mathbb{Q}}_\ell = \mathrm{End}(\mathbf{1})$ of an endomorphism $f$ of $M \in \mathbf{P}_{\mathrm{int}}(G)$ is defined as the composition

$$ \mathbf{1} \xrightarrow{\mathrm{coev}} M *_{\mathrm{int}} M^{\vee} \xrightarrow{f *_{\mathrm{int}} \mathrm{id}_{M^{\vee}}} M *_{\mathrm{int}} M^{\vee} \xrightarrow{\mathrm{ev}} \mathbf{1}. $$

The dimension of $M \in \mathbf{P}_{\mathrm{int}}(G)$ is then intrinsically defined as $\mathrm{dim}(M) = \mathrm{Tr}(\mathrm{id}_M)$. It is, a priori, an element of $\overline{\mathbb{Q}}_\ell$.

**PROPOSITION 3.17.** Let $M$ be an object of $\mathbf{P}_{\mathrm{int}}(G)$. Then there is a generic set of characters $\chi \in \hat{G}$ such that the following equality holds:

$$ (3.2) \qquad \dim \mathrm{H}^0(G_{\bar{k}}, M_{\chi}) = \dim(M). $$

In particular, $\dim(M)$ is a non-negative integer, and there exists a generic set of characters $\chi$ such that the dimension of $\mathrm{H}^0(G_{\bar{k}}, M_{\chi})$ is independent of $\chi$.

**PROOF.** We need to determine the morphism

$$ \mathbf{1} \xrightarrow{\mathrm{coev}} M *_{\mathrm{int}} M^{\vee} \xrightarrow{\mathrm{ev}} \mathbf{1}. $$

Twisting by $\chi$ and taking cohomology, the sequence above induces a sequence

$$ \overline{\mathbb{Q}}_{\ell} \to \mathrm{H}^*(G_{\bar{k}}, (M *_{\mathrm{int}} M^{\vee})_{\chi}) \to \overline{\mathbb{Q}}_{\ell}. $$

Let $P = \mathfrak{p}\mathcal{H}^0(M *_{\mathrm{int}} M^{\vee})$. By Lemma 3.8 (3), the objects $\mathfrak{p}\mathcal{H}^i(M *_{\mathrm{int}} M^{\vee})$ are negligible for all $i \neq 0$, so that for $\chi$ in a generic set of characters, we have isomorphisms

$$ \mathrm{H}^*(G_{\bar{k}}, (M *_{\mathrm{int}} M^{\vee})_{\chi}) \simeq \mathrm{H}^*(G_{\bar{k}}, P_{\chi}) $$

by the spectral sequence computing the left-hand side in terms of perverse cohomology objects.

By Lemma 3.14, for $\chi$ in another generic set, we also have isomorphisms

$$ \mathrm{H}^*(G_{\bar{k}}, P_{\chi}) \simeq \mathrm{H}^*(G_{\bar{k}}, (P_{\mathrm{int}})_{\chi}). $$

By (3.1), the object $P_{\mathrm{int}}$ is isomorphic to $M *_{\mathrm{int}} M^{\vee}$, so these isomorphisms combine to imply that, for $\chi$ generic, we have an isomorphism

$$ \mathrm{H}^*(G_{\bar{k}}, (M *_{\mathrm{int}} M^{\vee})_{\chi}) \xrightarrow{\sim} \mathrm{H}^*(G_{\bar{k}}, (M *_{\mathrm{int}} M^{\vee})_{\chi}). $$

By the Künneth formula (1.5) from Lemma 1.15, there are canonical isomorphisms

$$ \mathrm{H}^*(G_{\bar{k}}, (M *_{\mathrm{int}} M^{\vee})_{\chi}) \simeq \mathrm{H}^*(G_{\bar{k}}, M_{\chi}) \otimes \mathrm{H}^*(G_{\bar{k}}, (M_{\chi})^{\vee}) $$

for all $\chi$. Finally, if the character $\chi$ is also in the generic set $\mathcal{X}_w(M)$, then $\mathrm{H}^*(G_{\bar{k}}, M_{\chi}) = \mathrm{H}^0(G_{\bar{k}}, M_{\chi})$ and $\mathrm{H}^*(G_{\bar{k}}, M_{\chi}^{\vee}) = \mathrm{H}^0(G_{\bar{k}}, M_{\chi})^{\vee}$ hold, and therefore there exists a generic set of characters for which the sequence above becomes

$$ \overline{\mathbb{Q}}_{\ell} \to \mathrm{End}(\mathrm{H}^0(G_{\bar{k}}, M_{\chi})) \to \overline{\mathbb{Q}}_{\ell}. $$

Since the evaluation and coevaluation maps are sent to evaluation and covevaluation maps in vector spaces (see the proof of [91, Th. 5.2]), this composition is the multiplication by the dimension of $\mathrm{H}^0(G_{\bar{k}}, M_{\chi})$, which is therefore equal to the dimension of $M$ in $\mathbf{P}_{\mathrm{int}}(G)$. $\square$
**PROOF OF THEOREM 3.16.** By Proposition 3.15, the equivalent categories $\overline{\mathbf{P}}(\mathcal{G})$ and $\mathbf{P}_{\text{int}}(\mathcal{G})$ are $\overline{\mathbf{Q}}_{\ell}$-linear rigid tensor symmetric categories. Since the unit **1** is (the image of) a skyscraper sheaf, we have $\text{End}(\mathbf{1}) \simeq \overline{\mathbf{Q}}_{\ell}$.

**Proposition 3.17** and **Theorem 3.3** imply that the dimension $\dim(\mathcal{M})$ of every object $\mathcal{M}$ of $\overline{\mathbf{P}}(\mathcal{G})$ is a non-negative integer. By a theorem of Deligne [30, Th. 7.1], it follows that the category $\overline{\mathbf{P}}(\mathcal{G})$ is a tannakian category. A further theorem of Deligne (see the proof by Coulembier in [23, Th.6.4.1]) implies that it is indeed neutral (i.e., there exists a fiber functor defined over $\overline{\mathbf{Q}}_{\ell}$). $\square$

**REMARK 3.18.** (1) In Example 9.15, we will give examples to show that there may exist weakly unramified characters for which formula (3.2) does not hold.

(2) In this book, we will exclusively consider from now on the categories $\langle\mathcal{M}\rangle$ generated by a single object. A simpler proof that these are neutral tannakian categories is then provided by combining [30, Th. 7.1] with [30, Cor. 6.20].

**COROLLARY 3.19.** Let $\mathcal{M}$ be an object of $\mathbf{P}_{\text{int}}(\mathcal{G})$. There exists an affine algebraic group $\mathcal{G}$ over $\overline{\mathbf{Q}}_{\ell}$ such that the category $\langle\mathcal{M}\rangle$ is equivalent to the category $\text{Rep}_{\overline{\mathbf{Q}}_{\ell}}(\mathcal{G})$ of finite-dimensional $\overline{\mathbf{Q}}_{\ell}$-representations of $\mathcal{G}$. If $\mathcal{M}$ is semisimple, then the group $\mathcal{G}$ is reductive and the category $\langle\mathcal{M}\rangle$ is semisimple.

**PROOF.** The first part follows from the tannakian reconstruction theorem [32, Th. 2.11]. If $\mathcal{M}$ is semisimple then since the category of representations of $\mathcal{G}$ is equivalent to the category $\langle\mathcal{M}\rangle$ generated by the semisimple object $\mathcal{M}$, it follows, e.g., from [106, Th. 22.42] that the group $\mathcal{G}$ is reductive, and that every object $N \in \langle\mathcal{M}\rangle$ is semisimple. $\square$

**DEFINITION 3.20.** For any object $\mathcal{M}$ of $\mathbf{P}_{\text{int}}(\mathcal{G})$ or of $\overline{\mathbf{P}}(\mathcal{G})$, we denote by $\mathcal{G}_{\mathcal{M}}^{\text{geo}}$ the affine algebraic group over $\overline{\mathbf{Q}}_{\ell}$ given by the corollary, and we say that it is the geometric tannakian group of the object $\mathcal{M}$.

**EXAMPLE 3.21.** (1) Let $\mathcal{G} = \mathcal{G}_m$. A perverse sheaf $N$ on $\mathcal{G}_m$ is negligible if and only if it is a successive extension of shifted Kummer sheaves $\mathcal{L}_{\chi}[1]$ for some characters $\chi$, and it follows that the category $\mathbf{P}_{\text{int}}(\mathcal{G}_m)$ is the same as the category $\mathcal{P}$ of Katz [75, Ch. 2] (see also Section B.1).

(2) Let $\mathcal{G} = \mathcal{G}_a$. Fix an additive character $\psi$ of $k$. By the proper base change theorem, a perverse sheaf $N$ on $\mathcal{G}_a$ is negligible if and only if its Fourier transform $\text{FT}_{\psi}(\mathcal{N})$ is punctual, which means that $N$ is a finite direct sum of Artin-Schreier sheaves $\mathcal{L}_{\psi(yx)}[1]$ for some $y \in \mathcal{G}_a$. This implies that the category $\mathbf{P}_{\text{int}}(\mathcal{G}_a)$ coincides with the category of perverse sheaves on $\mathcal{G}_a$ with “property $\mathcal{P}$”, as defined by Katz [71, (2.6.2)] (this follows by combining Cor. 2.6.14, Cor. 2.6.15 and Lemma 2.6.13 of [71]; see Remark 2.10.4 in loc. cit.).

**3.6. Euler-Poincaré characteristic and Grothendieck groups**

Proposition 3.17 has some other useful corollaries which we state now.

**PROPOSITION 3.22.** Let $\mathcal{M}$ be an object of $\mathcal{D}_c^{\mathbf{b}}(\mathcal{G})$.

(1) There exists a generic set $\mathcal{X} \subset \hat{\mathcal{G}}$ such that the Euler-Poincaré characteristic $\chi(\mathcal{G}_{\bar{k}}, \mathcal{M}_{\chi})$ is independent of $\chi \in \mathcal{X}$.

(2) If $\mathcal{M}$ is negligible, then $\chi(\mathcal{G}_{\bar{k}}, \mathcal{M}_{\chi}) = 0$ for all $\chi$ in a generic set of characters. The converse holds if $\mathcal{M}$ is a perverse sheaf.

(3) If $\mathcal{G}$ is a semiabelian variety, then the Euler-Poincaré characteristic $\chi(\mathcal{G}_{\bar{k}}, \mathcal{M}_{\chi})$ is independent of $\chi \in \hat{\mathcal{G}}$ and it is non-negative if $\mathcal{M}$ is a perverse sheaf.
PROOF. The decomposition

$$M = \sum_{i \in \mathbb{Z}} (-1)^i \mathfrak{P}\mathcal{H}^i(M)$$

in the Grothendieck group K(G), together with Lemma 1.17, implies that

$$\chi(G_{\bar{k}}, M_{\chi}) = \sum_{i \in \mathbb{Z}} (-1)^i \chi(G_{\bar{k}}, \mathfrak{P}\mathcal{H}^i(M)_{\chi})$$

for all $\chi \in \hat{G}$. Thus the first statement is an immediate consequence of Proposition 3.17, combined with the generic vanishing theorem, applied to each perverse cohomology sheaf.

If N is a negligible perverse sheaf, then by definition we get $H^*(G_{\bar{k}}, N_{\chi}) = 0$ for a generic set of characters, hence also $\chi(G_{\bar{k}}, N_{\chi}) = 0$ for a generic set of characters. The previous formula shows that this is also true for any complex M.

Conversely, assume that M is a perverse sheaf and $\chi(G_{\bar{k}}, M_{\chi}) = 0$ for all $\chi$ in a generic set. Combined with the generic vanishing theorem, this implies that $H^*(G_{\bar{k}}, M_{\chi}) = 0$ for $\chi$ generic, hence M is negligible.

If G is a semiabelian variety, then the Euler-Poincaré characteristic $\chi(G_{\bar{k}}, M_{\chi})$ is independent of $\chi$ by a result of Deligne (see [63]), because all the $\chi \in \hat{G}$ are tame. In this case, the tannakian dimension of a perverse sheaf on G is therefore the same as its Euler-Poincaré characteristic. $\square$

COROLLARY 3.23. A perverse sheaf M in $\mathbf{P}(G)$ is negligible if and only if its class in the Grothendieck group K(G) belongs to the subgroup $K_{\text{neg}}(G)$ generated by classes of negligible perverse sheaves.

PROOF. It suffices to prove that a perverse sheaf M is negligible if the class of M in K(G) can be expressed as a finite sum

$$M = \sum_{i \in \mathbb{I}} \varepsilon_i M_i$$

in K(G), where $M_i$ is a negligible perverse sheaf for all $i \in \mathbb{I}$ and $\varepsilon_i \in \{-1, 1\}$. Such a formula implies the equality

$$\chi(G_{\bar{k}}, M_{\chi}) = \sum_{i \in \mathbb{I}} \varepsilon_i \chi(G_{\bar{k}}, M_{i, \chi})$$

for all $\chi \in \hat{G}$. For a generic set of characters we have $\chi(G_{\bar{k}}, M_{i, \chi}) = 0$ for all $i \in \mathbb{I}$, since $M_i$ is negligible by assumption, hence $\chi(G_{\bar{k}}, M_{\chi}) = 0$ for a generic set of characters; thus M is negligible by Proposition 3.22, (2). $\square$

COROLLARY 3.24. Suppose that G is a semiabelian variety. Let M be a negligible perverse sheaf on G. The Euler-Poincaré characteristic of M is 0 and the set of characters $\chi \in \hat{G}$ such that the space $H^0(G_{\bar{k}}, M_{\chi})$ is non-zero is contained in a finite union of tacs.

PROOF. The fact that $\chi(M) = 0$ has been stated in Proposition 3.22. By Theorem 2.19, there exists a finite family $(S_f)$ of tacs of G such that $H^i(G_{\bar{k}}, M_{\chi}) = 0$ for all $i \neq 0$ and $\chi$ not belonging to the union $\mathcal{S}$ of these tacs. For any $\chi$ not in $\mathcal{S}$, we then deduce by loc. cit. that

$$\dim H^0(G_{\bar{k}}, M_{\chi}) = \chi(M_{\chi}) = \chi(M) = 0.$$

$\square$
### 3.7. Arithmetic fiber functors

We now address the question of constructing arithmetic fiber functors that will be used to define conjugacy classes of elements in the tannakian groups.

**DEFINITION 3.25** (Unramified characters). Let $M$ be an object of $\mathbf{P}_{\text{int}}(G)$. A weakly unramified character $\chi \in \hat{G}$ for $M$ is said to be *unramified for* $M$ if the functor

$$N \mapsto \omega_{\chi}(N) = H^0(G_{\bar{k}}, N_{\chi})$$

is a fiber functor on the category $\langle M \rangle \subset \mathbf{P}_{\text{int}}(G)$. We denote by

$$\mathcal{X}(M) \subset \mathcal{X}_w(M) \subset \hat{G}$$

the set of unramified characters for $M$. We say that the perverse sheaf $M$ is *generically unramified* if the subset $\mathcal{X}(M) \subset \hat{G}$ is generic.

**REMARK 3.26.** In Example 9.15, we will give examples to show that there may exist weakly unramified characters which are not unramified. An example is given by the sheaf $M = \mathcal{L}_{\eta(f)}(1/2)[1]$ on $\mathbf{G}_m \times \mathbf{G}_a$, where $f$ is a polynomial of degree $d$ such that $f(0) \neq 0$ and $\eta$ is a multiplicative character such that $\eta^d$ is non-trivial.

We shall prove that every character $(\chi, a)$ is weakly unramified for $M$, that $\dim(H^0(\mathbf{G}_m \times \mathbf{G}_a, M_{(\chi, a)}))$ is $d+1$ if $a \neq 0$ but $d$ if $a = 0$, implying that $(\chi, 0)$ is not unramified for $M$.

We expect that all semisimple objects of $\mathbf{P}_{\text{int}}(G)$ are generically unramified. We can currently only prove this property for the three fundamental types of algebraic groups.

**THEOREM 3.27.** If $G$ is a torus, an abelian variety or a unipotent group, then any semisimple object of $\mathbf{P}_{\text{int}}(G)$ is generically unramified.

For tori or abelian varieties, we need a general technical criterion ensuring that an object $M$ is generically unramified.

**LEMMA 3.28.** Let $M$ be a semisimple object of $\mathbf{P}_{\text{int}}(G)$. Set $L = M \oplus M^{\vee}$. For each $m \geq 2$, let $C_m$ be the cone of the canonical morphism $L^{*1} \to L^{*m}$. All characters $\chi$ in

$$(3.3) \qquad \mathcal{X}_w(M) \cap \bigcap_{m \geq 2} \mathcal{N}(C_m)$$

are unramified for $M$.

**PROOF.** Let $\chi$ be a character in the set (3.3). By Proposition 1.30, every object $N$ of $\langle M \rangle$ is a direct sum of direct factors of $m$-fold convolution products $L^{*m}_{\text{int}}$ for some integers $m$. By the definition of (3.3) and Lemma 1.15, we have canonical isomorphisms

$$H^*(G_{\bar{k}}, L_{\chi}^{*m}) \simeq H^*(G_{\bar{k}}, L_{\chi}^{*m}) \simeq H^*(G_{\bar{k}}, L_{\chi})^{\otimes m}$$

for any $m$.

By (3.3) again, we have $H^*(G_{\bar{k}}, L_{\chi}) = H^0(G_{\bar{k}}, L_{\chi})$, and hence $\omega_{\chi}(L^{*m}_{\text{int}}) = \omega_{\chi}(L)^{\otimes m}$. This proves that the functor $\omega_{\chi}$ is compatible with the tensor product; other compatibilities are elementary, and the functor $\omega_{\chi}$ is exact on $\langle M \rangle$, hence the result (see [32, Prop. 1.19]). $\square$

**PROOF OF THEOREM 3.27 FOR ABELIAN VARIETIES.** If $G$ is an abelian variety, then both convolution functors are canonically isomorphic; hence, all objects $C_m$ vanish and the set (3.3) is the same as $\mathcal{X}_w(M) = \mathcal{X}(M)$, which is generic. $\square$
REMARK 3.29. There is a more precise result if G is an abelian variety. Indeed, we have recalled that $\mathcal{X}_w(M) = \mathcal{X}(M)$ for any semisimple object of $\mathbf{P}_{\text{int}}(G)$, and by the strong form of the Stratified Generic Vanishing Theorem (Theorem 2.3), it follows that the set of ramified characters is contained in a finite union of tacs of G.

PROOF OF THEOREM 3.27 FOR TORI. We use the notation of the previous lemma. For a torus G, a result of Gabber and Loeser [50, Prop. 3.9.3 (iv)] implies that there is an inclusion $\mathcal{N}(C_2) \subset \mathcal{N}(C_m)$ for all integers $m \ge 2$. So the set

$$ \mathcal{X}_w(M) \cap \bigcap_{m \ge 2} \mathcal{N}(C_m) = \mathcal{X}_w(M) \cap \mathcal{N}(C_2) $$

is generic, by the generic vanishing theorem and the definition of negligible objects. $\square$

Finally we consider unipotent groups.

PROOF OF THEOREM 3.27 FOR G UNIPOTENT. We denote by $G^\vee$ a form of the Serre dual of G, and we fix an additive character $\psi$ to compute the Fourier transform $\text{FT}_\psi$ on G (see Section 2.2).

Let M be a semisimple object of $\mathbf{P}_{\text{int}}(G)$. We claim that there exists a dense open set $V \subset G^\vee$ such that for all objects N and N' of $\langle M \rangle$, the restriction of $\text{FT}_\psi(N)$ to V is lisse and there exists a canonical isomorphism

$$ (3.4) \qquad \text{FT}_\psi(N *_{\text{int}} N')|V \to (\text{FT}_\psi(N) \otimes \text{FT}_\psi(N'))|V. $$

Indeed, if this claim holds, then it is elementary that for any $a \in V(\bar{k})$, the corresponding character $\psi_a \in \hat{G}$ is unramified for M.

The claim above follows in turn from a more general statement: for all objects $M_1$ and $M_2$ of $\mathbf{P}_{\text{int}}(G)$, and for any open dense subset $W \subset G^\vee$ such that the Fourier transforms $\text{FT}_\psi(M_1)$ and $\text{FT}_\psi(M_2)$ are lisse on W, there exists a canonical isomorphism

$$ \text{FT}_\psi(M_1 *_{\text{int}} M_2)|W \to (\text{FT}_\psi(M_1) \otimes \text{FT}_\psi(M_2))|W. $$

Indeed, the isomorphism shows in particular that the Fourier transform of $M_1 *_{\text{int}} M_2$ is also lisse on W; since the same is true of the dual $\text{D}(M_1)$, it follows that the Fourier transform of any object of $\langle M_1 \rangle$ is lisse on W, leading to the previous claim (with $V = W$).

We now prove the general statement above. Let $M = \mathfrak{p}\mathcal{H}^0(M_1*M_2)$. By definition of $M_1*M_{\text{int}}M_2$, we have $M_1*M_{\text{int}}M_2 = M_{\text{int}}$ (see Remark 3.13).

Let $\mathfrak{p}\tau_{\le 0}$ and $\mathfrak{p}\tau_{\ge 0}$ be the perverse truncation functors. We have canonical morphisms

$$ (3.5) \qquad \mathfrak{p}\tau_{\le 0}(M_1 *! M_2) \to M_1 *! M_2 $$

and

$$ (3.6) \qquad \mathfrak{p}\tau_{\le 0}(M_1 *! M_2) \to \mathfrak{p}\tau_{\ge 0}(\mathfrak{p}\tau_{\le 0}(M_1 *! M_2)) = \mathfrak{p}\mathcal{H}^0(M_1 *! M_2) = M. $$

By Lemma 3.8, the mapping cones of both morphisms are negligible. By the vanishing theorem for unipotent groups (Proposition 2.9), there is a dense open subset $W'$ of W such that the induced morphisms

$$ (3.7) \qquad \text{FT}_\psi(\mathfrak{p}\tau_{\le 0}(M_1 *! M_2))|W' \to \text{FT}_\psi(M_1 *! M_2)|W' $$

and

$$ (3.8) \qquad \text{FT}_\psi(\mathfrak{p}\tau_{\le 0}(M_1 *! M_2))|W' \to \text{FT}_\psi(M)|W' $$
are isomorphisms. Inverting (3.7) and composing with (3.8), we obtain a canonical isomorphism

$$ (3.9) \qquad \mathrm{FT}_{\psi}(M_1 *! M_2)|W' \to \mathrm{FT}_{\psi}(M)|W'. $$

Let $M^t$ be the smallest subobject of $M$ such that $M/M^t$ is negligible. We then have a canonical injection $M^t \to M$ with negligible cokernel and a canonical surjection $M^t \to M_{\mathrm{int}}$ with negligible kernel, by Remark 3.13. By the vanishing theorem for unipotent groups (Proposition 2.9), up to replacing $W'$ by a smaller dense open subset, we can assume that the canonical morphisms

$$ (3.10) \qquad \mathrm{FT}_{\psi}(M^t)|W' \to \mathrm{FT}_{\psi}(M)|W' $$

and

$$ (3.11) \qquad \mathrm{FT}_{\psi}(M^t)|W' \to \mathrm{FT}_{\psi}(M_{\mathrm{int}})|W' $$

are isomorphisms. Inverting (3.10) and composing with (3.11), we get a canonical isomorphism

$$ (3.12) \qquad \mathrm{FT}_{\psi}(M)|W' \to \mathrm{FT}_{\psi}(M_{\mathrm{int}})|W' = \mathrm{FT}_{\psi}(M_1 *_{\mathrm{int}} M_2)|W'. $$

Composing (3.9) and (3.12), we get a canonical isomorphism

$$ (3.13) \qquad \mathrm{FT}_{\psi}(M_1 *! M_2)|W' \simeq \mathrm{FT}_{\psi}(M_1 *_{\mathrm{int}} M_2)|W'. $$

Denote by $j: W' \to W$ the open immersion. By the definition of the category $\mathbf{P}_{\mathrm{int}}(G)$, the Fourier transform $\mathrm{FT}_{\psi}(M_1 *_{\mathrm{int}} M_2)$ (which is a perverse sheaf up to shift) has no shifted perverse component supported in $G^\vee - W'$ (such a component would be negligible), and therefore we have a canonical isomorphism

$$ (3.14) \qquad j_* j^*(\mathrm{FT}_{\psi}(M_1 *_{\mathrm{int}} M_2)|W) \simeq \mathrm{FT}_{\psi}(M_1 *_{\mathrm{int}} M_2)|W $$

by the properties of the intermediate extension functor $j_*$ (see Proposition A.9).

By Lemma 1.15, there is a canonical isomorphism $\mathrm{FT}_{\psi}(M_1 *! M_2) \simeq \mathrm{FT}_{\psi}(M_1) \otimes \mathrm{FT}_{\psi}(M_2)$. Since $\mathrm{FT}_{\psi}(M_1)$ and $\mathrm{FT}_{\psi}(M_2)$ are lisse on $W$, we have also a canonical isomorphism

$$ j_* j^* ((\mathrm{FT}_{\psi}(M_1) \otimes \mathrm{FT}_{\psi}(M_2))|W) \simeq (\mathrm{FT}_{\psi}(M_1) \otimes \mathrm{FT}_{\psi}(M_2))|W, $$
hence a canonical isomorphism

$$ (3.15) \qquad j_* j^* (\mathrm{FT}_{\psi}(M_1 *! M_2)|W) \simeq (\mathrm{FT}_{\psi}(M_1) \otimes \mathrm{FT}_{\psi}(M_2)|W). $$

We now apply the functor $j_*$ to the isomorphism (3.13), and use (3.14) and (3.15) to obtain the desired canonical isomorphism (3.4); this concludes the proof of the claim. $\square$

### 3.8. The arithmetic tannakian group

In this section, we consider the situation over the finite field $k$. Base change $M \mapsto M_{\bar{k}}$ gives a functor $\mathbf{Perv}(G) \to \mathbf{P}(G)$. For a perverse sheaf $M$ on $G$, we define the set of unramified characters for $M$ as $\mathcal{X}(M) = \mathcal{X}(M_{\bar{k}})$.

We denote by $\mathrm{Neg}_{\mathbf{P}}^{\mathrm{ari}}(G)$ (resp. $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G)$) the full subcategory of $\mathbf{Perv}(G)$ whose objects are the perverse sheaves $M$ such that $M_{\bar{k}}$ is an object of $\mathrm{Neg}_{\mathbf{P}}(G)$ (resp. of $\mathbf{P}_{\mathrm{int}}(G)$). As in the geometric case, we find that $\mathrm{Neg}_{\mathbf{P}}^{\mathrm{ari}}(G)$ is a Serre subcategory of $\mathbf{Perv}(G)$ and that the localization functor induces an equivalence from $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G)$ to the quotient abelian category $\bar{\mathbf{P}}^{\mathrm{ari}}(G) = \mathbf{Perv}(G)/\mathrm{Neg}_{\mathbf{P}}^{\mathrm{ari}}(G)$.

Also similarly to the geometric case, the two convolution bifunctors on $\mathbf{Perv}(G)$ induce equivalent bifunctors on $\bar{\mathbf{P}}^{\mathrm{ari}}(G)$ (compare with Proposition 3.10). The categories $\bar{\mathbf{P}}^{\mathrm{ari}}(G)$ and $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G)$ are then rigid symmetric $\bar{\mathbf{Q}}_\ell$-linear tensor categories, with unit object **1** still the skyscraper sheaf at the unit of $G$, which again satisfies $\mathrm{End}(\mathbf{1}) \simeq \bar{\mathbf{Q}}_\ell$.
Let $M$ be a perverse sheaf on $G$. To distinguish between the arithmetic and geometric situations, we denote from now on by $\langle M \rangle^{\text{ari}}$ (resp. $\langle M \rangle^{\text{geo}}$) the subcategory of $\mathbf{P}_{\text{int}}^{\text{ari}}(G) \simeq \bar{\mathbf{P}}^{\text{ari}}(G)$ (resp. of $\mathbf{P}_{\text{int}}(G) \simeq \bar{\mathbf{P}}(G)$) that is tensor-generated by (the image of) $M$ (resp. by $M_{\bar{k}}$). Base change $N \mapsto N_{\bar{k}}$ gives a functor from $\langle M \rangle^{\text{ari}}$ to $\langle M \rangle^{\text{geo}}$.

**THEOREM 3.30.** Let $M$ be an object of $\mathbf{Perv}(G)$. The categories $\langle M \rangle^{\text{ari}}$ and $\langle M \rangle^{\text{geo}}$ are neutral $\overline{\mathbb{Q}}_{\ell}$-linear tannakian categories. There exist algebraic groups $\mathbf{G}_{M}^{\text{geo}}$ and $\mathbf{G}_{M}^{\text{ari}}$ over $\overline{\mathbb{Q}}_{\ell}$ such that $\langle M \rangle^{\text{ari}}$ is equivalent to the category $\text{Rep}_{\overline{\mathbb{Q}}_{\ell}}(\mathbf{G}_{M}^{\text{ari}})$ and $\langle M \rangle^{\text{geo}}$ is equivalent to the category $\text{Rep}_{\overline{\mathbb{Q}}_{\ell}}(\mathbf{G}_{M}^{\text{geo}})$.

Moreover, if $r$ is the tannakian dimension of $M$, then the objects $M$ and $M_{\bar{k}}$ of $\langle M \rangle^{\text{ari}}$ and $\langle M \rangle^{\text{geo}}$, respectively, correspond to faithful representations of $\mathbf{G}_{M}^{\text{ari}}$ and $\mathbf{G}_{M}^{\text{geo}}$ in $\mathbf{GL}_r(\overline{\mathbb{Q}}_{\ell})$.

**PROOF.** The case of $\langle M \rangle^{\text{geo}}$ is dealt with by Theorem 3.16 and Corollary 3.19. The case of $\langle M \rangle^{\text{ari}}$ follows by the same argument because Proposition 3.17 also applies to $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$.

The last assertion is a tautological consequence of the formalism. $\square$

**DEFINITION 3.31.** In the context of Theorem 3.30, we call $\mathbf{G}_{M}^{\text{ari}}$ the arithmetic tannakian group of $M$, and $\mathbf{G}_{M}^{\text{geo}}$ its geometric tannakian group.

**PROPOSITION 3.32.** Let $M$ be an object of $\mathbf{Perv}(G)$. The functor of base change to $\bar{k}$ is a tensor functor from $\langle M \rangle^{\text{ari}}$ to $\langle M \rangle^{\text{geo}}$ that induces a morphism $\varphi: \mathbf{G}_{M}^{\text{geo}} \to \mathbf{G}_{M}^{\text{ari}}$. This morphism is a closed immersion.

**PROOF.** The first assertion is immediate, and it implies by the tannakian formalism the existence of the homomorphism $\varphi$. According to [32, Prop. 2.21 (b)], this morphism $\varphi$ is a closed immersion if and only if every object of $\langle M \rangle^{\text{geo}}$ is isomorphic to a subquotient of an object in the essential image of the base-change functor.

Let $N$ be such an object of $\langle M \rangle^{\text{geo}}$, viewed as an object of $\mathbf{P}_{\text{int}}(G)$. By definition of the category $\mathbf{P}(G)$, there exists a finite extension $k_n$ of $k$ in $\bar{k}$ such that $N$ is the base change to $\bar{k}$ of a perverse sheaf $N_1$ on $G_{k_n}$. Then $N$ is a subquotient of the base change of the perverse sheaf $f_{n*}N_1$ to $\mathbf{G}_{\bar{k}}$, where $f_n: \text{Spec}(k_n) \to \text{Spec}(k)$ is the canonical morphism, hence the result. $\square$

From now on, we will identify the geometric tannakian group of a perverse sheaf $M$ on $G$ with its image in the arithmetic tannakian group.

We recall the convention from Section 1.12 concerning properties over $k$ and $\bar{k}$. Let $M$ be a perverse sheaf on $G$. We view $\langle M \rangle^{\text{ari}}$ as a subcategory of $\mathbf{P}_{\text{int}}(G)$, so that the weights of an object $N \in \langle M \rangle^{\text{ari}}$ are well-defined.

**THEOREM 3.33.** Let $M$ be a perverse sheaf on $G$. Assume that $M$ is arithmetically semisimple and pure of weight zero. Let $r$ be the tannakian dimension of $M$.

(1) The groups $\mathbf{G}_{M}^{\text{ari}}$ and $\mathbf{G}_{M}^{\text{geo}}$ are reductive subgroups of $\mathbf{GL}_r$.

(2) Every object $N$ of $\langle M \rangle^{\text{ari}}$ is arithmetically semisimple and pure of weight zero, and every object $N$ of $\langle M \rangle^{\text{geo}}$ is semisimple.

**PROOF.** Since any pure perverse sheaf on $G$ is geometrically semisimple by [8, Th. 5.3.8], the assertions for $\langle M \rangle^{\text{geo}}$ follow. The same proof is also valid for $\langle M \rangle^{\text{ari}}$, since $M$ is arithmetically semisimple, so that the group $\mathbf{G}_{M}^{\text{ari}}$ is also reductive, and all objects of $\langle M \rangle^{\text{ari}}$ are arithmetically semisimple.
We now prove the purity statement. Since M is pure of weight zero, it follows from the description of $M_{\text{int}}$ in Remark 3.13 that the corresponding object of $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$ is also pure of weight zero, and similarly for its dual

For any perverse sheaves $N_1$ and $N_2$ on $G$ that are pure of weight zero, the convolution $N_1 *_{\text{int}} N_2$ is also pure of weight zero. Indeed, by Deligne's Riemann Hypothesis [28, 3.3.1], the object $N_1 *! N_2$ is mixed of weights $\le 0$. Hence, the quotient $N_1 *_{\text{int}} N_2$ of $N_1 *! N_2$ is also mixed of weights $\le 0$ by [8, Prop. 5.3.1]. Thanks to Lemma 1.15, the same applies to the Verdier dual $\text{D}(N_1 *_{\text{int}} N_2)$, which implies the claim.

Hence, the property of being pure of weight zero is preserved by convolution, duality and taking subobjects. Thus we conclude that every object $N$ of $\langle M \rangle^{\text{ari}}$ is pure of weight zero. $\square$

We now show that the tannakian groups coincide with those of Katz for the multiplicative group using the category $\mathcal{P}$ (see [75, Ch. 2] and Section B.1), and with monodromy groups of the Fourier transform for unipotent groups.

**PROPOSITION 3.34.** Let $M$ be a perverse sheaf on $G$. Assume that $M$ is arithmetically semisimple and pure of weight zero.

(1) If $G = \mathbf{G}_m$, then the arithmetic and geometric tannakian groups of $G$ coincide with those defined by Katz using the category $\mathcal{P}$.

(2) If $G$ is unipotent of dimension $d$, and $\psi$ is a fixed additive character used to define its Fourier transform, then there exists a dense open subset $U$ of the Serre dual $G^{\vee}$ such that $(\text{FT}_{\psi} M_{\text{int}})|_{U}$ is isomorphic to a lisse sheaf $\mathcal{F}$ on $U$, pure of weight $d$, placed in degree 0. The arithmetic and geometric tannakian groups of $M$ coincide with the arithmetic and geometric monodromy groups of the lisse sheaf $\mathcal{F}$.

**PROOF.** In the case of $\mathbf{G}_m$, the statement follows directly from Example 3.21 (1) (see also Section B.1 for the definition of $\mathcal{P}$).

Suppose then that $G$ is unipotent. To prove the first assertion of (2), we may assume that $M$ is simple and non-negligible. Its Fourier transform is then a simple $d$-shifted perverse sheaf on the Serre dual $G^{\vee}$, pure of weight $d$, and with support equal to $G^{\vee}$ (since the object $M$ would be negligible if the support were smaller). Thus it is a single lisse sheaf, pure of weight $d$, on an open dense subset of $G^{\vee}$.

For the second part of (2), we note that by (the proof of) Theorem 3.27 for unipotent groups, the convolution product on $\langle M \rangle^{\text{ari}}$ can be identified with the tensor product on the subcategory generated by $\mathcal{F}$ of the category of lisse sheaves on $U$. The result then follows. $\square$

## 3.9. Frobenius conjugacy classes

We keep working over the finite field $k$ and use the same notation as in the previous subsection. For any finite extension $k_n$ of $k$, we denote by $\text{Fr}_{k_n}$ the geometric Frobenius automorphism of $k_n$.

For an object $M$ of $\text{D}_c^b(X)$, an integer $n \ge 1$ and a character $\chi \in \hat{G}(k_n)$, we denote by $\text{Fr}_{M,k_n}(\chi)$ the automorphism of the $\overline{\mathbb{Q}}_\ell$-vector space $\text{H}_c^0(G_{\bar{k}}, M_\chi)$ induced by the action of $\text{Fr}_{k_n}$. Recall from A.3 the notions of weights and purity.

Let $r$ be the dimension of this space. If the automorphism $\text{Fr}_{M,k_n}(\chi)$ is pure of weight zero, for instance if $M$ is pure of weight 0 and $\chi$ is weakly unramified for $M$, then there is a unique conjugacy class $\Theta_{M,k_n}(\chi)$ in the complex unitary group $\mathbf{U}_r(\mathbf{C})$ containing the semisimple part of $\iota_0(\text{Fr}_{M,k_n}(\chi))$.
We call $\mathrm{Fr}_{M,k_n}(\chi)$ the *Frobenius automorphism of M associated to $\chi$ over $k_n$* and $\Theta_{M,k_n}(\chi)$ the *unitary Frobenius conjugacy class of M associated to $\chi$ over $k_n$*.

Suppose now that M is an arithmetically semisimple perverse sheaf on G.

Let $n \ge 1$ and let $\chi \in \hat{G}(k_n)$ be an *unramified character* for M, so that the functor $\omega_\chi: N \mapsto H^0(G_{\bar{k}}, N_\chi)$ is a fiber functor on the tannakian category $\langle M \rangle^{\mathrm{ari}}$. For any object N of $\langle M \rangle^{\mathrm{ari}}$, the Frobenius automorphism $\mathrm{Fr}_{k_n}$ now induces an automorphism of $\omega_\chi(N)$, and thus defines an automorphism of the fiber functor $\omega_\chi$. By the tannakian formalism, this corresponds to a unique conjugacy class in $G_M^{\mathrm{ari}}(\bar{Q}_\ell)$. We denote by $\mathrm{Fr}_{M,k_n}(\chi)$ the corresponding conjugacy class of $G_M^{\mathrm{ari}}(C)$, and call it the *Frobenius conjugacy class of M associated to $\chi$ over $k_n$*.

Suppose furthermore that M is pure of weight zero. Let $K_M$ be a maximal compact subgroup of the reductive group $G_M^{\mathrm{ari}}(C)$. Since all objects of $\langle M \rangle^{\mathrm{ari}}$ are pure of weight zero (by Theorem 3.33), the eigenvalues of any element of the conjugacy class $\mathrm{Fr}_{M,k_n}(\chi)$ are complex numbers with modulus 1, so that the semisimple part of this conjugacy class is a unitary matrix. One can then deduce from the Peter–Weyl Theorem that the $G_M^{\mathrm{ari}}(C)$-conjugacy class of the semisimple part of $\mathrm{Fr}_{M,k_n}(\chi)$ intersects $K_M$ in a unique conjugacy class, which is denoted $\Theta_{M,k_n}(\chi)$, and is called the *unitary Frobenius conjugacy class of M associated to $\chi$*.

(See, e.g., [79, 9.2.4] for this argument.) For an unramified character $\chi$, the space $\omega_\chi(M)$ has dimension $r$, the tannakian dimension of M, and the conjugacy class of $\mathrm{Fr}_{M,k_n}(\chi)$ in the automorphism group of $H_c^0(G_{\bar{k}}, M_\chi)$ coincides with that of $\mathrm{Fr}_{M,k_n}(\chi)$, and similarly for $\Theta_{M,k_n}(\chi)$.

When $k_n = k$, we will sometimes use simply the notation $\mathrm{Fr}_M(\chi)$, $\Theta_M(\chi)$, $\mathrm{Fr}_M(\chi)$, $\Theta_M(\chi)$.

We have the following important consequences of the formalism.

**LEMMA 3.35.** Let M be an arithmetically semisimple perverse sheaf on G that is pure of weight zero and of tannakian dimension $r \ge 0$.

(1) Let $\chi \in \mathcal{X}_w(M)(k)$ be a weakly unramified character for M. For any integer $n \ge 1$, we have

$$ \mathrm{Tr}(\Theta_{M,k_n}(\chi)) = \mathrm{Tr}(\Theta_M(\chi)^n) = \sum_{x \in G(k_n)} \chi(N_{k_n/k}(x))t_M(x; k_n), $$

where $t_M$ is the trace function of M and the trace on the left is that on $GL_r$.

(2) Let $\chi \in \mathcal{X}(M)(k)$ be an unramified character. Let $\varrho$ be an algebraic $\bar{Q}_\ell$-representation of $G_M^{\mathrm{ari}}$ and denote by $\varrho(M)$ the corresponding object of $\langle M \rangle^{\mathrm{ari}}$. The character $\chi$ is unramified for $\varrho(M)$ and

$$ \mathrm{Tr}(\varrho(\mathrm{Fr}_M(\chi))) = \mathrm{Tr}(\mathrm{Fr}_k | H_c^0(G_{\bar{k}}, \varrho(M)_\chi)). $$

**PROOF.** (1) By definition, we have

$$ \mathrm{Tr}(\Theta_M(\chi)^n) = \mathrm{Tr}(\mathrm{Fr}_M(\chi)^n) = \mathrm{Tr}(\mathrm{Fr}_k^n | H_c^0(G_{\bar{k}}, M_\chi)). $$

Since $\chi$ is weakly unramified, we have $H_c^i(G_{\bar{k}}, M_\chi) = 0$ for all $i \ne 0$ and $H^0(G_{\bar{k}}, M_\chi) = H_c^0(G_{\bar{k}}, M_\chi)$, so that we can write

$$ \mathrm{Tr}(\Theta_M(\chi)^n) = \sum_{i \in \mathbb{Z}} (-1)^i \mathrm{Tr}(\mathrm{Fr}_k^n | H_c^i(G_{\bar{k}}, M_\chi)) = \sum_{x \in G(k)} \chi(N_{k_n/k}(x))t_M(x; k_n), $$

by the trace formula.

(2) The fact that $\chi$ is unramified for $\varrho(M)$ follows from the definition and Proposition 1.30, and the formula follows then from the definition of the Frobenius conjugacy class of $\chi$ for $\varrho(M)$. $\square$
### 3.10. Frobenius-unramified characters

Because weakly unramified characters do not always give rise to fiber functors, and moreover we do not always know if there exist sufficiently many (if any) unramified characters, we introduce an intermediate notion.

**DEFINITION 3.36** (Frobenius-unramified characters). Let $M$ be an object of $\mathbf{Perv}(G)$ which is arithmetically semisimple and pure of weight zero, of tannakian dimension $r$. Let $\varrho$ be a representation of $\mathbf{GL}_r$ and let $N$ be the object of $\langle M \rangle^{\text{ari}}$ corresponding to the restriction of $\varrho$ to $\mathbf{G}_M^{\text{ari}}$. Let $n \ge 1$ and let $\chi \in \mathcal{X}_w(M)(k_n)$ be a weakly unramified character for $M$. We say that $\chi$ is Frobenius-unramified for $\varrho$ if $\chi$ is weakly unramified for $N$ and if the formula

$$ \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))^v) = \mathrm{Tr}(\mathrm{Fr}_{k_n}^v | \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, \mathbb{N}_\chi)) $$

holds for all integers $v \ge 1$, or equivalently if

$$ \det(1 - \varrho(\Theta_{M,k_n}(\chi))T) = \det(1 - T \mathrm{Fr}_{k_n} | \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, \mathbb{N}_\chi)). $$

The disjoint union over $n$ of the set of Frobenius-unramified characters is denoted $\mathcal{X}_F(\varrho)$.

**REMARK 3.37.** (1) The key point is that since $\varrho$ is a representation of $\mathbf{GL}_r$, we can consider the conjugacy class of $\varrho(\mathrm{Fr}_{M,k_n}(\chi))$ (in $\mathbf{GL}(V)$, where $\varrho$ is a representation on $V$); a priori, this is not meaningful for a representation of $\mathbf{G}_M^{\text{ari}}$, unless we know that elements of the conjugacy class of the Frobenius automorphism of $\mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, \mathbb{M}_\chi)$ are conjugate to some element of the arithmetic tannakian group, which is unique up to conjugacy in $\mathbf{G}_M^{\text{ari}}$.

(2) We will also sometimes write $\mathcal{X}_F(\varrho) = \mathcal{X}_F(N)$, although this set depends on $M$, since we view $N$ as an object of $\langle M \rangle^{\text{ari}}$. When confusion might arise, we may also write $\mathcal{X}_F(N)_M$.

Any unramified character for $M$ is Frobenius-unramified for all objects of $\langle M \rangle^{\text{ari}}$, by Lemma 3.35, (2). But in contrast to unramified characters, we can prove in all cases that the set of Frobenius-unramified characters is generic.

**PROPOSITION 3.38.** Let $M$ be an object of $\mathbf{Perv}(G)$ which is arithmetically semisimple and pure of weight zero and of tannakian dimension $r \ge 0$. For any representation $\varrho$ of $\mathbf{GL}_r$, the set $\mathcal{X}_F(\varrho)$ is generic.

**PROOF.** We first observe that it is straightforward that if two representations $\varrho_1$ and $\varrho_2$ of $\mathbf{GL}_r$ have the property that $\mathcal{X}_F(\varrho_1)$ and $\mathcal{X}_F(\varrho_2)$ are generic, then the sets $\mathcal{X}_F(\varrho_1 \oplus \varrho_2)$, $\mathcal{X}_F(\varrho_1 \otimes \varrho_2)$ and $\mathcal{X}_F(\varrho_1^\vee)$ are also generic. Indeed, consider the case of the tensor product, the others being similar (and in fact simpler). Let $N_i$ be the object corresponding to $\varrho_i$. For $\chi$ generic, we have

$$ \mathrm{H}_c^*(\mathbb{G}_{\bar{k}}, (N_1 *_{\text{int}} N_2)_\chi) = \mathrm{H}_c^*(\mathbb{G}_{\bar{k}}, (N_1 *_{!} N_2)_\chi) \simeq \mathrm{H}_c^*(\mathbb{G}_{\bar{k}}, N_{1,\chi}) \otimes \mathrm{H}_c^*(\mathbb{G}_{\bar{k}}, N_{2,\chi}) $$

as well as

$$ \mathrm{H}_c^*(\mathbb{G}_{\bar{k}}, N_{i,\chi}) = \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, N_{i,\chi}) $$

for $i=1$ and $i=2$, all these isomorphisms being compatible with Frobenius. Thus

$$ \det(1 - T \mathrm{Fr}_{k_n} | \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, (N_1 *_{\text{int}} N_2)_\chi)) = \det(1 - T \mathrm{Fr}_{k_n} | \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, N_{1,\chi})) \det(1 - T \mathrm{Fr}_{k_n} | \mathrm{H}_c^0(\mathbb{G}_{\bar{k}}, N_{2,\chi})) $$

for $\chi$ generic, which then establishes the claim concerning $\mathcal{X}_F(\varrho_1 \otimes \varrho_2)$ using the definition of $\mathcal{X}_F(\varrho_i)$ and the assumption that these are generic sets.

A first consequence of this observation is that we may assume that $\varrho$ is irreducible to prove the proposition. Recall next that every irreducible representation $\varrho$ of $\mathbf{GL}_r$ is isomorphic to one of the
form $\varrho = \varrho_0 \otimes \det(\cdot)^k$ for some representation $\varrho_0$ given by a Schur functor $S_\lambda$ and some integer $k \in \mathbb{Z}$ (see, e.g., [49, Prop. 15.47]). Since the determinant is itself a Schur functor, and $\det(\cdot)^{-k}$ is the contragredient of $\det(\cdot)^k$, the previous observation reduces the proof to the case where $\varrho = S_\lambda$ for some $\lambda$.

In this case, $\varrho$ is given by the image of an explicit projector (see, e.g., [49, § 6.1, th. 6.3]), and hence makes sense for any symmetric monoidal category where idempotents split. In particular, this applies to $D_c^b(G)$ with either of the two convolutions, since $D_c^b(G)$ is known to have this property (e.g., by combining the fact that $D_c^b(G)$ is equivalent to the bounded derived category of the category of perverse sheaves, by a theorem of Beilinson [7, Th. 1.3], and the fact that the bounded derived category of an abelian category is idempotent complete, by a result of Balmer and Schlichting [3, Cor. 2.10]). We will denote by $\varrho_!(M)$ (resp. $\varrho_*(M)$) the action of these functors on $M$ for the symmetric monoidal structures given by the convolution $(A, B) \mapsto A *_! B$ (resp. by $(A, B) \mapsto A *_* B$).

Since taking cohomology with compact support (resp. cohomology) is an additive monoidal functor for the convolution $A*_!B$ (resp. for $A*_*B$), by the Künneth formula, the explicit description of the idempotent defining $\varrho$ provides isomorphisms

$$ (3.16) \qquad H_c^*(G_{\bar{k}}, \varrho_!(M)_{\chi}) \simeq \varrho(H_c^*(G_{\bar{k}}, M_{\chi})) $$

$$ (3.17) \qquad H^*(G_{\bar{k}}, \varrho_*(M)_{\chi}) \simeq \varrho(H^*(G_{\bar{k}}, M_{\chi})) $$

for every $\chi$, which are also compatible with Frobenius, where the Schur functor acts on the right-hand sides in the category of bounded complexes of $\overline{\mathbb{Q}}_l$-vector spaces.

Let $N = \varrho(M)$ be the object of $\langle M \rangle^{\text{ari}}$ corresponding to $\varrho$. Since $\varrho$ is assumed to be a Schur functor, there exists an integer $l \ge 0$ and an embedding $N \to M_l = M^{*\text{int}}l$ (see, e.g., [49, § 6.1]). We obtain a commutative square

$$ \begin{tikzcd} \varrho_!(M) \arrow[r] & \varrho_*(M) \arrow[d] \\ M_{l, !} \arrow[r] & M_{l, *} \arrow[d] \end{tikzcd} $$

where $M_{l, !} = M^{*l}$ and $M_{l, *} = M^{*l}$. This implies, in particular, that the cone $C$ of the morphism

$$ \varrho_!(M) \to \varrho_*(M) $$

is negligible, since this is the case for the cone of the bottom morphism by Lemma 3.8. Applying Remark 3.13, there exists a generic set $\mathcal{X}$ of characters such that for $\chi \in \mathcal{X}$, we have isomorphisms

$$ (3.18) \qquad H_c^0(G_{\bar{k}}, N_{\chi}) \simeq H_c^*(G_{\bar{k}}, \varrho_!(M)_{\chi}) $$

which are compatible with Frobenius.

Let finally $\chi \in \mathcal{X}$ be a character which is weakly unramified for both $M$ and $N$. Then we have isomorphisms

$$ H_c^0(G_{\bar{k}}, N_{\chi}) \simeq H_c^*(G_{\bar{k}}, N_{\chi}) \simeq H_c^*(G_{\bar{k}}, \varrho_!(M)_{\chi}) \simeq \varrho(H_c^*(G_{\bar{k}}, M_{\chi})) \simeq \varrho(H_c^0(G_{\bar{k}}, M_{\chi})) $$

compatible with Frobenius (the first and fourth of these are given by the theorem, the second is (3.18) and the third is (3.16)), and hence

$$ \det(1 - T \mathrm{Fr}_{k_n} | H_c^0(G_{\bar{k}}, N_{\chi})) = \det(1 - \varrho(\Theta_{M, k_n}(\chi))T). $$

Since this holds for a generic set of characters (by Theorem 3.3), we obtain the desired result. $\square$
**COROLLARY 3.39.** Let M be an object of Perv(G) which is arithmetically semisimple and pure of weight zero and of tannakian dimension $r \ge 0$. If the group $G_M^{\text{ari}}$ is finite, then M is generically unramified.

**PROOF.** The fact that the tannakian group is finite implies that any object of $\langle M \rangle^{\text{ari}}$ is a subobject of a direct sum of copies of a single object $N = M^{*_{\text{int}}m} *_{\text{int}} (M^{\vee})^{*_{\text{int}}l}$ for some (fixed) integers $m$ and $l$ (see [32, Prop. 2.20 (a)]). Any Frobenius-unramified character for M is then an unramified character for M. $\square$

**3.11. Group-theoretic properties**

We continue with the notation of the previous sections.

The following basic proposition establishes the relation between the geometric and arithmetic tannakian groups.

**PROPOSITION 3.40.** Let M be a geometrically semisimple object of Perv(G). The geometric tannakian group $G_M^{\text{geo}}$ is a normal subgroup of the arithmetic tannakian group $G_M^{\text{ari}}$.

**PROOF.** The proof is identical with that of [75, Lemma 6.1]. $\square$

**PROPOSITION 3.41.** Let M be an arithmetically semisimple object of Perv(G). Assume that M is pure of weight zero.

(1) The quotient $G_M^{\text{ari}}/G_M^{\text{geo}}$ is of multiplicative type.

(2) Let V be a geometrically trivial object of $\langle M \rangle^{\text{ari}}$ which corresponds to a faithful representation of the group $G_M^{\text{ari}}/G_M^{\text{geo}}$. Any character $\chi \in \hat{G}$ is unramified for V, and the class $\xi$ of the Frobenius conjugacy class of any such character is independent of $\chi$ and generates a Zariski-dense subgroup of $G_M^{\text{ari}}/G_M^{\text{geo}}$.

(3) For any $n \ge 1$ and any character $\chi \in \hat{G}(k_n)$ unramified for M, the image in $G_M^{\text{ari}}/G_M^{\text{geo}}$ of the Frobenius conjugacy class $\text{Fr}_{M,k}(\chi)$ is equal to $\xi^n$.

**PROOF.** This follows by the same arguments as in [75, Lemma 7.1] (checking first that, using the structure of geometrically trivial objects as direct sums of $\alpha^{\text{deg}} \otimes \delta_1$ for suitable $\alpha$, it is indeed straightforward that all characters are unramified for such objects). $\square$

We will also use the following result in Chapter 9.

**PROPOSITION 3.42.** Let $G_1$ and $G_2$ be connected commutative algebraic groups over $k$ and let $p: G_1 \to G_2$ be a morphism of algebraic groups. Let M be a perverse sheaf on $G_1$ which is arithmetically semisimple and pure of weight zero.

Let $\chi_1 \in \hat{G}_1(k)$ be a character such that we have $\text{Rp}_!(M_{\chi_1}) = \text{Rp}_*(M_{\chi_1})$. Assume further that $N = \text{Rp}_!(M_{\chi_1})$ is perverse and arithmetically semisimple.

(1) The object N is pure of weight zero.

(2) Let $n \ge 1$ and let $\chi \in \mathcal{X}_w(N)(k_n)$ be a character such that $\chi_1 \cdot (\chi \circ p)$ is weakly unramified for M. Then the conjugacy classes $\Theta_{M,k_n}(\chi_1 \cdot (\chi \circ p))$ and $\Theta_{N,k_n}(\chi)$ satisfy

$$ \det(1 - \text{T}\Theta_{M,k_n}(\chi_1 \cdot (\chi \circ p))) = \det(1 - \text{T}\Theta_{N,k_n}(\chi)) \in \mathbf{C}[T] $$

and in particular

$$ \det(\Theta_{M,k_n}(\chi_1 \cdot (\chi \circ p))) = \det(\Theta_{N,k_n}(\chi)). $$
PROOF. It suffices to consider the case where $\chi \in \hat{G}(k)$. For any $n \ge 1$, the exponential sums

$$S_n = \sum_{x \in G_1(k_n)} t_M(x; k_n)(\chi_1 \cdot (\chi \circ p))(N_{k_n/k}(x))$$

$$S'_n = \sum_{y \in G_2(k_n)} t_N(y; k_n)\chi(N_{k_n/k}(y))$$

are equal by the trace formula. Hence, the corresponding L-functions

$$\exp\left(\sum_{n \ge 1} S_n \frac{T^n}{n}\right), \quad \exp\left(\sum_{n \ge 1} S'_n \frac{T^n}{n}\right)$$

are also equal. But these L-functions coincide with the (reversed) characteristic polynomials of the conjugacy classes $\Theta_{M,k}(\chi_1 \cdot (\chi \circ p))$ and $\Theta_{N,k}(\chi)$, by Lemma 3.35 (1), hence the result. $\square$

REMARK 3.43. If the morphism $p: G_1 \to G_2$ is affine, then the condition $Rp_!(M_{\chi_1}) = Rp_*(M_{\chi_1})$ implies that N is perverse.

We will give an application when the group $G_2$ is the multiplicative group. For this we need a lemma.

LEMMA 3.44. Let N be a simple perverse sheaf on $G_m$ over k which is an object of the category $\mathbf{P}^{\text{ari}}_{\text{int}}(G_m)$. Assume that N is pure of weight 0 and of tannakian dimension 1. Suppose that there exists an integer $d \ge 1$ and a finite set $\mathcal{Y} \subset \hat{G}_m$ such that for all $n \ge 1$ and for $\chi \in \hat{G}_m(k_n)-\mathcal{Y}(k_n)$, the determinant $\det(\Theta_{N,k_n})^d$ depends only on $n$. Then N is geometrically of finite order.

PROOF. If N is not geometrically of finite order, then the perverse sheaf N is a hypergeometric sheaf of generic rank at least 1 (see Section B.4 and Theorem B.4 for reminders of the definition of hypergeometric sheaves and for this result, due to Katz). But these hypergeometric sheaves do not have the indicated property, e.g. because the $\Theta_{N,k_n}(\chi)$ become equidistributed in $\mathbf{S}^1$ as $\chi$ varies among unramified characters in $\hat{G}_m(k_n)$ (see Theorem B.4, (3) and [75, Th. 7.2] or Theorem 4.11). $\square$

PROPOSITION 3.45. Let G be a connected commutative algebraic group over k and let p: G → $G_m$ be a non-trivial morphism of algebraic groups. Let M be a perverse sheaf on G which is arithmetically semisimple and pure of weight zero.

Let $\chi_1 \in \hat{G}(k)$ be a character such that the equality $Rp_!(M_{\chi_1}) = Rp_*(M_{\chi_1})$ holds. Assume further that the complex N = $Rp_!(M_{\chi_1})$ is a perverse sheaf on $G_m$ and is arithmetically semisimple. It is then pure of weight zero.

Suppose that the set of $\chi \in \hat{G}_m$ such that $\chi_1(\chi \circ p)$ is unramified for the object $\det(M)$ is generic, and that the tannakian determinant of N is arithmetically (resp. geometrically) of infinite order. Then the tannakian determinant of M is arithmetically (resp. geometrically) of infinite order.

PROOF. We begin by proving that the determinant is arithmetically of infinite order in both cases. Let $n \ge 1$ and let $\chi \in \hat{G}_m(k_n)$ be a character such that $\chi_1(\chi \circ p)$ is unramified for the object $\det(M)$. We then have

$$(3.19) \qquad \Theta_{\det(M),k_n}(\chi) = \det(\Theta_{M,k_n}(\chi_1(\chi \circ p))) = \det(\Theta_{N,k_n}(\chi))$$

by Proposition 3.42. By assumption this is valid for all but finitely many $\chi \in \hat{G}_m$, and moreover N has determinant which is arithmetically of infinite order, so that the arithmetic tannakian group of $\det(M)$ must be infinite.
It remains to deduce that the geometric tannakian determinant of M has infinite order if the same property holds for N. If not, then det(M)d would be geometrically trivial for some integer d ≥ 1. In this case, for any n ≥ 1 and any character χ ∈ Ḡ(k_n) which is Frobenius-unramified for det, the determinant det(Θ_{M,k_n}(χ))d only depends on n (see Proposition 3.41, (2)). By (3.19) and Lemma 3.44, the tannakian determinant of N (which is an object of tannakian dimension 1 on G_m) is geometrically of finite order, which contradicts the assumption. □

**REMARK 3.46.** If $G = T \times G_m$ for some torus $T$ and $p$ is the projection on $G_m$ then, according to Theorem 2.14 applied to $p$ and $M$, the assumption that $Rp_!M_{\chi_1} = Rp_*M_{\chi_1}$ and that this complex is a perverse sheaf is true for all $\chi_1$ outside of a finite union of tacs of $T$. Moreover, by varying $\chi_1$, we can always find such a character for which $\chi_1(\chi \circ p)$ is unramified for generic $\chi$, since $M$ is generically unramified by Theorem 3.27.

Using further work of Katz, we can give a sufficient criterion to apply this proposition.

**COROLLARY 3.47.** Let $G$ be a connected commutative algebraic group over $k$ and let $p: G \to G_m$ be a non-trivial morphism of algebraic groups. Let $M$ be a perverse sheaf on $G$ which is arithmetically semisimple and pure of weight zero.

Let $\chi_1 \in \hat{G}(k)$ be a character satisfying $Rp_!(M_{\chi_1}) = Rp_*(M_{\chi_1})$. Assume that $N = Rp_!(M_{\chi_1})$ is a perverse sheaf on $G_m$, which is arithmetically semisimple and of the form $\mathcal{F}[1]$ for some middle extension sheaf $\mathcal{F}$ (see Example A.12 for the definition of middle extension sheaves). Let

$$ (e_1, \dots, e_l), \quad (f_1, \dots, f_m) $$

be the sizes of the unipotent Jordan blocks in the tame monodromy representation of $\mathcal{F}$ at 0 and $\infty$ respectively.

Suppose that the set of $\chi \in \hat{G}_m$ such that $\chi_1(\chi \circ p)$ is unramified for the object $\det(M)$ is generic. If we have

$$ \sum_i e_i - \sum_j f_j \neq 0, $$

then the tannakian determinant of $M$ is geometrically of infinite order.

**PROOF.** According to the previous proposition, it suffices to show that the tannakian determinant of $N$ is geometrically of infinite order. By [75, Th. 16.1], the condition implies that the determinant of the Frobenius action on Deligne's fiber functor $\omega_{\mathrm{Del}}(N)$ is not unitary (see Section B.2 for the definition of this functor), and the result follows from Katz's classification of objects of tannakian dimension 1 on $G_m$ (Theorem B.4). □

## 3.12. External products

The following proposition concerns objects on a product $G = G_1 \times G_2$, and is useful for constructing various examples (see for instance Section 9.4).

**PROPOSITION 3.48.** Assume that $G = G_1 \times G_2$ for connected commutative algebraic groups $G_1$ and $G_2$. For any objects $M_i \in D_c^b(G_i)$, there exist natural isomorphisms

$$ M_1 \boxtimes M_2 \simeq (M_1 \boxtimes \mathbf{1}_{G_2}) *_! (\mathbf{1}_{G_1} \boxtimes M_2) \simeq (M_1 \boxtimes \mathbf{1}_{G_2}) *_* (\mathbf{1}_{G_1} \boxtimes M_2). $$

Moreover, if $M_1$ and $M_2$ are perverse sheaves on $G_1$ and $G_2$ with tannakian rank $r_1$ and $r_2$, respectively, then the object $M_1 \boxtimes M_2 \in D_c^b(G)$ is perverse and has tannakian rank $r_1 r_2$.
PROOF. We use coordinates $(x_1, x_2, y_1, y_2)$ on $G \times G$ with $x_i$ and $y_i$ coordinates on $G_i$. Let $m_{12}: G \times G \to G$ be the multiplication map for $G$, and $m_i: G_i \times G_i \to G_i$ those for $G_i$.

Let $N$ be the object on the right-hand side of the first isomorphism to be established. By definition, we have

$$N = (M_1 \boxtimes 1_{G_2}) *_! (1_{G_1} \boxtimes M_2) = m_{12, !}((M_1 \boxtimes 1_{G_2}) \boxtimes (1_{G_1} \boxtimes M_2)).$$

Let $s: G \to G$ be the involution given by $(x_1, x_2, y_1, y_2) \mapsto (x_1, y_2, y_1, x_2)$. We have $m_{12} = m_{12} \circ s$, and hence

$$N = m_{12, !} s!((M_1 \boxtimes 1_{G_2}) \boxtimes (1_{G_1} \boxtimes M_2)).$$

Since $s$ is an involution, we have $s! = s_* = s^*$, and therefore

$$s!((M_1 \boxtimes 1_{G_2}) \boxtimes (1_{G_1} \boxtimes M_2)) = p_1^*(M_1 \boxtimes M_2) \otimes p_2^*(1_G),$$

where $p_1, p_2: G \times G \to G$ are the two projections. Thus, using the definition again, we obtain an isomorphism

$$N \simeq m_{12!}(p_1^*(M_1 \boxtimes M_2) \otimes p_2^*(1_G)) = (M_1 \boxtimes M_2) *_! 1_G,$$

which is isomorphic to $M_1 \boxtimes M_2$ since $1_G$ is the unit for convolution.

We obtain similarly the second isomorphism

$$M_1 \boxtimes M_2 \simeq (M_1 \boxtimes 1_{G_2}) *_* (1_{G_1} \boxtimes M_2) \simeq (M_1 \boxtimes 1_{G_2}) *_* (1_{G_1} \boxtimes M_2).$$

It is classical that $M_1 \boxtimes M_2$ is perverse if $M_1$ and $M_2$ are, and the final assertion then results from the fact that

$$H_c^*(G_{\bar{k}}, M_1 \boxtimes M_2) \simeq H_c^*(G_{1, \bar{k}}, M_1) \otimes H_c^*(G_{2, \bar{k}}, M_2),$$

and the generic vanishing theorem.

<figure><img src="image_7.png" /></figure>

REMARK 3.49. Concretely, this proposition reflects the convolution formula

$$f_1(x_1)f_2(x_2) = \sum_{(y_1, y_2) \in G(k_n)} f_1(y_1)\delta_2(y_2)\delta_1(x_1y_1^{-1})f_2(x_2y_2^{-1})$$

for any functions $f_i: G_i(k_n) \to \mathbb{C}$, where the $\delta_i$ are Dirac masses at the unit element of $G_i$.

### 3.13. The rank 1 tannakian group

Given the group $G$ over $k$, we can form the subcategory $\overline{\mathbf{P}}^1(G)$ of $\overline{\mathbf{P}}(G)$ (resp. $\mathbf{P}_{\text{int}}^1(G)$ of $\mathbf{P}_{\text{int}}(G)$) additively generated by objects of tannakian rank 1. This is again a tannakian category, since the convolution (resp. dual) of objects of rank 1 is of rank 1.

PROPOSITION 3.50. Let $\mathbf{L}(G)$ be the group of isomorphism classes of objects of $\mathbf{P}_{\text{int}}(G)$ of rank 1. The tannakian group of $\overline{\mathbf{P}}^1(G)$ and $\mathbf{P}_{\text{int}}^1(G)$ is the pro-algebraic subgroup of

$$\mathbf{GL}_1^{\mathbf{L}(G)}$$

defined by the equations

$$\prod_{1 \le i \le m} x_{L_i}^{n_i} = 1$$

for all integers $m \ge 0$, all families $(L_i)_{1 \le i \le m}$ of elements of $\mathbf{L}(G)$ and all families $(n_i)_{1 \le i \le m}$ of integers such that the object

$$L_1^{*\text{int}n_1} *_{\text{int}} \cdots *_{\text{int}} L_m^{*\text{int}n_m}$$

is isomorphic to $1_G$, or equivalently that the relation

$$L_1^{n_1} \cdots L_m^{n_m} = 1$$
holds in the group $\mathbf{L}(G)$.

PROOF. This amounts to proving that, for the object

$$M = L_1 \oplus \cdots \oplus L_m,$$

the tannakian group $\mathbf{G}$ is the subgroup of $\mathbf{GL}_1^m$ determined by the equations

$$(3.20) \qquad \prod_{1 \le i \le m} x_i^{n_i} = 1$$

for all $(n_1, \dots, n_m) \in \mathbb{Z}^m$ such that

$$(3.21) \qquad L_1^{n_1} \cdots L_m^{n_m} = 1.$$

Being a group of multiplicative type, $\mathbf{G}$ is characterized by its character group, and the character group of $\mathbf{G} \subset \mathbf{GL}_1^m$ is $\mathbb{Z}^m/\mathbb{H}$, where

$$H = \{ \chi : \mathbf{GL}_1^m \to \mathbf{G}_m \mid \mathbf{G} \subset \ker(\chi) \}.$$

A character $\chi$ of $\mathbf{GL}_1^m$ restricts to a character of $\mathbf{G}$, so we can form the object $\chi(M)$ in $\mathbf{P}_{\text{int}}^1(G)$. Since the image $\chi(\mathbf{G})$ of $\mathbf{G}$ by $\chi$ is the tannakian group of $\chi(M)$, we then have

$$\mathbf{G} \subset \ker(\chi) \text{ if and only if } \chi(M) \simeq \mathbf{1}_{\mathbf{G}}.$$

On the other hand, there is a natural isomorphism

$$\chi(M) \simeq L_1^{*\text{int}n_1} \cdots L_m^{*\text{int}n_m}$$

(indeed, this holds when $\chi$ is the character $(x_1, \dots, x_m) \mapsto x_i$, and then the general case follows by the compatibility with convolution and tensor product, which for characters is just the ordinary product), and therefore $H$ is the subgroup of $\mathbb{Z}^m$ formed by the tuples $(n_1, \dots, n_m)$ which satisfy (3.21). This means that $\mathbf{G}$ coincides with the subgroup of $\mathbf{GL}_1^m$ determined by the equations (3.20), and concludes the proof. $\square$

REMARK 3.51. This result is in fact valid, with the same proof, in any tannakian category. In particular, the analogue holds for the categories generated by objects of rank 1 in $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$. Of course, as usual, the arithmetic tannakian group may be bigger than the geometric one.

EXAMPLE 3.52. If $\mathbf{G}$ is a torus, then the group $\mathbf{L}(\mathbf{T})$ has been determined by Gabber and Loeser [50, Th 8.6.1], who denote it $\mathbf{H}_{\text{int}}(\mathbf{G})$.

Precisely, let $r \ge 0$ be such that $\mathbf{G}$ is isomorphic to $\mathbf{G}_m^r$. Let $\mathcal{S}$ be the set of one-dimensional subtori of $\mathbf{G}_{m, \bar{k}}^r$. For each torus $T \in \mathcal{S}$, denote by $i_T$ the closed immersion $T \to \mathbf{G}_{m, \bar{k}}^r$ and choose an isomorphism $\varphi_T: \mathbf{G}_{m, \bar{k}} \to \mathbf{T}$. Recall that $\Pi(\mathbf{G}_{m, \bar{k}}, \overline{\mathbf{Q}}_\ell)$ is the set of continuous tame characters of $\mathbf{G}_{m, \bar{k}}$ (see Section 1.10), and write the basis vectors of the free abelian group $\mathbf{Z}(\mathcal{S} \times \Pi(\mathbf{G}_{m, \bar{k}}, \overline{\mathbf{Q}}_\ell))$ as $(\mathbf{T}, \chi)$. Then Gabber and Loeser prove that there is an isomorphism

$$(\bar{k}^\times)^r \times \mathbf{Z}(\mathcal{S} \times \Pi(\mathbf{G}_{m, \bar{k}}, \overline{\mathbf{Q}}_\ell)) \to \mathbf{L}(\mathbf{G}_{m, \bar{k}}^r)$$

that maps $(\lambda, (\mathbf{T}, \chi))$ to the object

$$\delta_\lambda *_{\text{int}} \mathrm{R}(i_{\mathrm{T}} \circ \varphi_{\mathrm{T}})_*(j^* \mathcal{L}_\psi \otimes \mathcal{L}_\chi)[1],$$

where $j: \mathbf{G}_{m, \bar{k}} \to \mathbf{A}^1$ is the open immersion.
# Equidistribution theorems

## 4.1. Equidistribution on average

Along with the classical form of equidistribution that goes back in principle to Weyl and appears in Deligne's equidistribution theorem, we will apply a useful variant that allows us to avoid the assumption that the geometric and the arithmetic tannakian groups are equal, at the cost of getting slightly weaker statements.

**DEFINITION 4.1.** Let X be a locally compact topological space and let $\mu$ be a Borel probability measure on X. Let $(Y_n, \Theta_n)_{n \ge 1}$ be a sequence of pairs of finite sets $Y_n$ and maps $\Theta_n: Y_n \to X$.

(1) We say that $(Y_n, \Theta_n)$, or simply $(Y_n)$ when the maps $\Theta_n$ are clear from the context, becomes $\mu$-equidistributed on average as $n \to \infty$ if the sets $Y_n$ are non-empty for all large enough $n$ and if the sequence of probability measures

$$ \mu_N = \frac{1}{N'} \sum_{1 \le n \le N} \frac{1}{|Y_n|} \sum_{Y_n \ne \emptyset} \delta_{\Theta_n(y)}, \quad N' = |\{n \le N \mid Y_n \ne \emptyset\}|, $$

defined on X for large enough N, converges weakly to $\mu$ as N goes to infinity, i.e., for any bounded continuous function $f: X \to \mathbf{C}$, the following holds:

$$ (4.1) \qquad \lim_{N \to +\infty} \frac{1}{N'} \sum_{\substack{1 \le n \le N \\ Y_n \ne \emptyset}} \frac{1}{|Y_n|} \sum_{y \in Y_n} f(\Theta_n(y)) = \int_X f d\mu. $$

(2) The sequence $(Y_n, \Theta_n)$, or simply $(Y_n)$, becomes $\mu$-equidistributed as $n \to \infty$ if the sets $Y_n$ are non-empty for all large enough $n$ and if the sequence of probability measures

$$ \tilde{\mu}_n = \frac{1}{|Y_n|} \sum_{y \in Y_n} \delta_{\Theta_n(y)}, $$

defined on X for large enough $n$, converges weakly to $\mu$ as $n$ goes to infinity, i.e., for any bounded continuous function $f: X \to \mathbf{C}$, the following holds:

$$ (4.2) \qquad \lim_{n \to +\infty} \frac{1}{|Y_n|} \sum_{y \in Y_n} f(\Theta_n(y)) = \int_X f d\mu. $$

**REMARK 4.2.** (1) In practice, since $N' \sim N$ as $N \to +\infty$, we will sometimes not distinguish between N and N', and use the convention that those terms for which $Y_n$ is empty are omitted from the sum over $n$ when discussing equidistribution on average.

(2) Since convergence of a sequence $(x_n)$ of complex numbers implies that of its Cesàro means $(N^{-1} \sum_{1 \le n \le N} x_n)$, with the same limit, equidistribution implies equidistribution on average.
## 4.2. The basic estimate

We state here a preliminary estimate that will be the key analytic step in the proof of our equidistribution results, including Theorem 2 from the introduction.

We denote as usual by $k$ a finite field with algebraic closure $\bar{k}$, and by $k_n$ the extension of $k$ of degree $n$ in $\bar{k}$. We fix a prime $\ell$ distinct from the characteristic of $k$.

**PROPOSITION 4.3.** Let $G$ be a commutative connected algebraic group over $k$. Let $M$ be an $\ell$-adic perverse sheaf on $G$ that is arithmetically semisimple and pure of weight zero, and of tannakian dimension $r$. Let $N$ be an object of $\langle M \rangle^{\text{ari}}$.

For all $n \ge 1$ such that $\mathcal{X}_F(N)(k_n)$ is not empty, the following estimate holds:

$$ (4.3) \qquad \frac{1}{|\mathcal{X}_F(N)(k_n)|} \sum_{\chi \in \mathcal{X}_F(N)(k_n)} \mathrm{Tr}(\mathrm{Fr}_{k_n} | \mathcal{H}_c^0(G_{\bar{k}}, \mathcal{N}_\chi)) = t_N(e; k_n) + O(|k_n|^{-1/2}). $$

**PROOF.** We fix a quasi-projective embedding $u$ of $G$. Let $d$ denote the dimension of $G$, and put $\mathcal{X} = \mathcal{X}_F(N)$. For each non-zero integer $i$, consider the subset

$$ \mathcal{A}_i = \{ \chi \in \hat{G} \mid \mathcal{H}_c^i(G_{\bar{k}}, \mathcal{N}_\chi) \neq 0 \} $$

consisting of those characters $\chi$ such that $N_\chi$ has non-trivial cohomology with compact support in degree $i$. Then the left-hand side of (4.3) is equal to

$$ \begin{align} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\mathrm{Fr}_{k_n} | \mathcal{H}_c^0(G_{\bar{k}}, \mathcal{N}_\chi)) = & \nonumber \\ & \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \hat{G}(k_n)} \sum_{|i| \le d} (-1)^i \mathrm{Tr}(\mathrm{Fr}_{k_n} | \mathcal{H}_c^i(G_{\bar{k}}, \mathcal{N}_\chi)) \nonumber \\ & - \frac{1}{|\mathcal{X}(k_n)|} \sum_{0 < |i| \le d} (-1)^i \sum_{\chi \in \mathcal{A}_i(k_n)} \mathrm{Tr}(\mathrm{Fr}_{k_n} | \mathcal{H}_c^i(G_{\bar{k}}, \mathcal{N}_\chi)) \nonumber \\ & - \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \hat{G}(k_n) - \mathcal{X}(k_n)} \mathrm{Tr}(\mathrm{Fr}_{k_n} | \mathcal{H}_c^0(G_{\bar{k}}, \mathcal{N}_\chi)). \tag{4.4} \end{align} $$

By the Grothendieck–Lefschetz trace formula (see (A.5)), the equalities

$$ \sum_{|i| \le d} (-1)^i \mathrm{Tr}(\mathrm{Fr}_{k_n} | \mathcal{H}_c^i(G_{\bar{k}}, \mathcal{N}_\chi)) = \sum_{x \in G(k_n)} t_{N_\chi}(x; k_n) = \sum_{x \in G(k_n)} \chi(x) t_N(x; k_n) $$

hold for any character $\chi$. Combined with the orthogonality of characters of $G(k_n)$, this shows that the first summand in (4.4) is equal to

$$ \frac{|\hat{G}(k_n)|}{|\mathcal{X}(k_n)|} t_N(e; k_n) = t_N(e; k_n) + O(|k_n|^{-1}) $$

since the set $\mathcal{X}$ is generic, so that the estimate $\frac{|\hat{G}(k_n)|}{|\mathcal{X}(k_n)|} = 1 + O(|k_n|^{-1})$ holds.

We now turn to bounding the second and the third summands in the right-hand side of (4.4).

Since $M$ is pure of weight zero, the same holds for $N$ and $N_\chi$ by Theorem 3.33. It then follows from Deligne's Riemann Hypothesis (see Theorem A.19) that $\mathcal{H}_c^i(G_{\bar{k}}, \mathcal{N}_\chi)$ is mixed of weights $\le i$ for any $i$, in particular the eigenvalues of $\mathrm{Fr}_{k_n}$ acting on this space have modulus at most $|k_n|^{i/2}$. Moreover, using (1.1) and Theorem 1.5 (2), we get

$$ h_c^i(G_{\bar{k}}, \mathcal{N}_\chi) \le c_u(N_\chi) \ll c_u(N) c_u(\mathcal{L}_\chi) \ll c_u(N) $$
since the complexity $c_u(\mathcal{L}_\chi)$ is bounded independently of $\chi$ by Proposition 1.18. So the second term in (4.4) can be bounded by

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{0<i|\le d} \sum_{\chi \in \mathcal{A}_i(k_n)} h_c^i(G_{\bar{k}}, N_\chi) |k_n|^{i/2} \le \frac{1}{|\mathcal{X}(k_n)|} \sum_{0<i|\le d} \sum_{\chi \in \mathcal{A}_i(k_n)} |k_n|^{i/2}. $$

The Stratified Vanishing Theorem 2.3 applied to N gives the estimate

$$ (4.5) \qquad |\mathcal{A}_i(k_n)| \ll |k_n|^{d-|i|} $$

for $i$ such that $0 < |i| \le d$. We split the sum over $i$ into that over $1 \le i \le d$ and that over $-d \le i \le -1$, and obtain

$$ (4.6) \qquad \frac{1}{|\mathcal{X}(k_n)|} \sum_{0<i|\le d} \sum_{\chi \in \mathcal{A}_i(k_n)} |k_n|^{i/2} \ll \frac{1}{|\mathcal{X}(k_n)|} \sum_{1\le i\le d} |k_n|^{d-i/2} + \frac{1}{|\mathcal{X}(k_n)|} \sum_{-d\le i\le -1} |k_n|^{d+3i/2} \\ \ll \frac{|k_n|^{d-1/2}}{|\mathcal{X}(k_n)|}. $$

Thanks to the estimate $|\mathcal{X}(k_n)| = |k_n|^d + O(|k_n|^{d-1})$, the last term is $\ll |k_n|^{-1/2}$ and tends to 0 as $n \to +\infty$.

Finally, the third term in (4.4) satisfies

$$ (4.7) \qquad \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \hat{\mathcal{G}}(k_n) - \mathcal{X}(k_n)} \mathrm{Tr}(\mathrm{Fr}_{k_n} |H_c^0(G_{\bar{k}}, N_\chi)|) \ll \frac{|\hat{\mathcal{G}}(k_n) - \mathcal{X}(k_n)|}{|\mathcal{X}(k_n)|} \ll \frac{1}{|k_n|} $$

since $H_c^0(G_{\bar{k}}, N_\chi)$ is mixed of weights $\le 0$ and has dimension bounded for all $\chi$, and the set $\mathcal{X}$ is generic. This finishes the proof. $\square$

### 4.3. Equidistribution for characteristic polynomials

Let $k$ be a finite field, with an algebraic closure $\bar{k}$, and let $G$ be a connected commutative algebraic group over $k$. Let $\ell$ be a prime number distinct from the characteristic of $k$.

Our most general equidistribution result concerns the characteristic polynomials of the unitary Frobenius conjugacy classes for weakly unramified characters. Equivalently, this is about the conjugacy classes in the ambient unitary group.

**THEOREM 4.4.** Let $M$ be an $\ell$-adic perverse sheaf on $G$ that is arithmetically semisimple and pure of weight zero. Let $r \ge 0$ be the tannakian dimension of $M$. Let $K \subset \mathbf{U}_r(\mathbf{C})$ be a conjugate of a maximal compact subgroup of the arithmetic tannakian group $\mathbf{G}_M^{\mathrm{ari}}(\mathbf{C}) \subset \mathbf{GL}_r(\mathbf{C})$ of $M$, and denote by $\nu_{cp}$ the measure on the space $\mathbf{U}_r(\mathbf{C})^{\sharp}$ of conjugacy classes in the unitary group which is the direct image of the Haar probability measure $\mu$ on $K$ by the quotient map $K \to \mathbf{U}_r(\mathbf{C})^{\sharp}$. Then the families of unitary conjugacy classes $(\Theta_{M,k_n}(\chi))_{\chi \in \mathcal{X}_w(M)(k_n)}$ become $\nu_{cp}$-equidistributed on average in $\mathbf{U}_r(\mathbf{C})^{\sharp}$ as $n \to +\infty$.

**REMARK 4.5.** (1) To be precise, in terms of Definition 4.1, we consider the equidistribution on average of pairs $(\mathcal{X}_w(M)(k_n), \Theta_n)$ with $\Theta_n(\chi) = \Theta_{M,k_n}(\chi)$.

(2) The set $\mathbf{U}_r(\mathbf{C})^{\sharp}$ can be identified with the set of characteristic polynomials of unitary matrices of size $r$, or equivalently with the quotient topological space $(\mathbf{S}_1)^r/\mathcal{S}_r$ (by mapping a matrix to the set of eigenvalues, with multiplicity) so the statement means that the characteristic polynomials of the Frobenius automorphisms for weakly unramified characters tend to be distributed like the characteristic polynomials of random elements of $K$ (hence the notation $\nu_{cp}$).
PROOF. Let $\mathcal{X} = \mathcal{X}_w(M)$. It suffices to check the equality (4.1) for $f$ taken in a set of continuous functions on $\mathbf{U}_r(\mathbf{C})^\sharp$ that span a dense subset of the Banach space $\mathcal{C}(\mathbf{U}_r(\mathbf{C})^\sharp)$ of all continuous complex-valued functions on $\mathbf{U}_r(\mathbf{C})^\sharp$ (since probability measures on $\mathbf{U}_r(\mathbf{C})^\sharp$ are continuous functionals on $\mathcal{C}(\mathbf{U}_r(\mathbf{C})^\sharp)$ by the Riesz representation theorem). Thanks to the Peter-Weyl Theorem, it suffices to prove the equality

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = \int_K \mathrm{Tr}(\varrho(g)) d\mu(g) $$

for any irreducible unitary representation $\varrho$ of $\mathbf{U}_r(\mathbf{C})$. In fact, we will prove this for any unitary representation $\varrho$, not necessarily irreducible.

By the Peter-Weyl Theorem again, the right-hand side is the multiplicity of the trivial representation in the representation of $\mathbf{G}_M^{\mathrm{ari}}$ that corresponds to the restriction of $\varrho$ to $K$. We denote by $N = \varrho(M)$ the object of $\langle M \rangle^{\mathrm{ari}}$ that corresponds to this restriction of $\varrho$.

Let $\mathcal{X}_N = \mathcal{X}_F(N)_M$ be the set of Frobenius-unramified characters for $N$. We have

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in (\mathcal{X} - \mathcal{X}_N)(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = 0, $$

since $\mathcal{X}_N$ is generic (by Proposition 3.38) and the upper-bound

$$ |\mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi)))| \le \dim(\varrho) $$

holds for all $\chi \in \mathcal{X}(k_n)$.

By the definition of Frobenius-unramified characters, we have

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}_N(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}_N(k_n)} \mathrm{Tr}(\mathrm{Fr}_{k_n} | H_c^0(G_{\bar{k}}, N_{\chi})) $$

for $n \ge 1$. Since $\mathcal{X}$ and $\mathcal{X}_N$ are both generic, we have $\frac{|\mathcal{X}_N(k_n)|}{|\mathcal{X}(k_n)|} = 1 + O(\frac{1}{|k_n|})$. By Proposition 4.3, we deduce that

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}_N(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = t_N(e; k_n) + O(|k_n|^{-1/2}), $$

where $e$ is the identity of $G$.

We decompose the semisimple perverse sheaf $N$ as a direct sum

$$ N = \bigoplus_{r \ge 0} \bigoplus_{i \in I(r)} N_{r,i} $$

of pairwise non-isomorphic arithmetically simple perverse sheaves $N_{r,i}$ of support of dimension $r$. For $r \ge 1$, we get the pointwise bound

$$ t_{N_{r,i}}(e; k_n) \ll \frac{1}{\sqrt{|k_n|}}. $$

using Proposition 1.11.

The punctual objects $N_{0,i}$ are of the form $\alpha_i^{\deg} \otimes \delta_{x_i}$ for some unitary scalars $\alpha_i$ and some points $x_i$. If $x_i \ne e$, then

$$ t_{N_{0,i}}(e; k_n) = 0. $$
Thus, if we denote by $J \subset I(0)$ the subset where $x_i = e$ (which has cardinality equal to the multiplicity of the trivial representation in the restriction of $\varrho$ to $G_M^{\text{geo}}$), then the formula

$$ (4.8) \qquad \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}_N(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = \sum_{i \in J} \alpha_i^n + \mathrm{O}(|k_n|^{-1/2}) $$

holds. The subset $J^0 \subset J$ where $\alpha_i = 1$ has cardinality equal to the multiplicity of the trivial representation in the restriction of $\varrho$ to $G_M^{\text{ari}}$. Averaging over $n$ and using

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \alpha_i^n = 0 $$

for $i \in J - J^0$, we conclude that

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}_N(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = |J^0| + \mathrm{O}(|k_n|^{-1/2}), $$

which gives the desired result.

□

It is useful to state the following corollary of the proof, which is a diophantine version of Schur's Lemma in our context.

**COROLLARY 4.6 (Schur's Lemma).** Let $M$ and $N$ be geometrically simple $\ell$-adic perverse sheaves on $G$ which are pure of weight zero and are objects of $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$. Let $\mathcal{X}$ be the set of characters which are weakly unramified for $M \oplus N^{\vee}$. We have

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{n \le N} \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} S(M *_{\text{int}} N^{\vee}, \chi) = \begin{cases} 1 & \text{if } M \text{ is arithmetically isomorphic to } N, \\ 0 & \text{otherwise.} \end{cases} $$

**PROOF.** Proposition 4.3 applied to the perverse sheaf $M \oplus N^{\vee}$ and the object $Q = \text{Hom}(N, M)$ of the category $\langle M \oplus N^{\vee} \rangle^{\text{ari}}$ (the homomorphisms are in the category $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$) implies that

$$ \frac{1}{|\mathcal{X}_F(Q)(k_n)|} \sum_{\chi \in \mathcal{X}_F(Q)(k_n)} S(Q, \chi) = t_Q(e; k_n) + \mathrm{O}(|k_n|^{-1/2}) $$

for any $n \ge 1$, where

$$ S(Q, \chi) = \sum_{x \in G(k_n)} \chi(x) t_Q(x; k_n). $$

Since $\mathcal{X}_F(Q)$ is generic, and since there is a canonical isomorphism $Q \to M *_{\text{int}} N^{\vee}$, we deduce that

$$ \frac{1}{N} \sum_{n \le N} \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} S(M *_{\text{int}} N^{\vee}, \chi) = \frac{1}{N} \sum_{n \le N} t_Q(e; k_n) + \mathrm{O}(|k_n|^{-1/2}) $$

for all $N \ge 1$. Arguing as in the last part of the proof of Theorem 4.4, we see that the right-hand side converges to the multiplicity of the trivial representation in the representation corresponding to $Q$; by the classical form of Schur's Lemma, this is either 1 or 0, depending on whether $M$ is isomorphic to $N$ or not. □

**REMARK 4.7.** The proof of Theorem 4.4 allows us to see clearly what is involved in the use of the Cesàro mean in the average equidistribution.

First, we can see that it is necessary in general, unless $G_M^{\text{ari}} = G_M^{\text{geo}}$ (see Section 4.6 for statements under this assumption, in particular Proposition 4.18).
Second, we see that the use of the Cesàro average can be generalized to establish the convergence to the limit $\nu_{cp}$ of any sequence of average measures of the form

$$ \sum_{n \ge 1} \frac{\varphi_N(n)}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \delta_{\Theta_{M,k_n}(\chi)}, $$

where $\varphi_N(n)$ are non-negative coefficients that are bounded and satisfy the equality

$$ (4.9) \qquad \lim_{N \to +\infty} \sum_{n \ge 1} \varphi_N(n) \alpha^n = \begin{cases} 0 & \text{if } \alpha \ne 1, \\ 1 & \text{if } \alpha = 1 \end{cases} $$

for any complex number $\alpha$ of modulus 1. The Cesàro case corresponds to $\varphi_N(n) = 1/N$ for all $n \le N$ and $\varphi_N(n) = 0$ for $n \ge N$, but there are many other possibilities. (In classical terms, as expounded for instance by Hardy [57], these $\varphi_N$ define a “summation method”, and it is elementary that the requirements amounts essentially<fnref n="1" /> to asking that this summation method gives the “right” sum $1/(1-\alpha)$ to the geometric series for $|\alpha| = 1$ and $\alpha \ne 1$.)

It is also instructive to view the average probabilistically, interpreting $\varphi_N$ as the law of a random variable $X_N$ with values in positive integers. The condition above is the requirement that the equality

$$ \lim_{N \to +\infty} \mathbf{E}(e^{i\theta X_N}) = 0 $$

holds for all $\theta \in \mathbf{R}/2\pi\mathbf{Z} - \{0\}$.

Besides the Cesàro case, where $X_N$ is a random variable uniform on $\{1, \dots, N\}$, consider a Poisson distribution $X_N$ with parameter $\lambda_N > 0$, shifted to have support in the positive integers, i.e., let

$$ \mathbf{P}(X_N = n) = \varphi_N(n) = e^{-\lambda_N} \frac{\lambda_N^{n-1}}{(n-1)!} $$

for any positive integers $N$ and $n$. The condition above becomes the limit

$$ \mathbf{E}(e^{i\theta X_N}) = \exp(i\theta + \lambda_N(e^{i\theta} - 1)) \to 0 $$

as $N \to +\infty$ for $\theta \in \mathbf{R}/2\pi\mathbf{Z} - \{0\}$, which holds provided $\lambda_N \to +\infty$, since the modulus of the left-hand side is $\exp(\lambda_N(\cos(\theta) - 1))$.

Intuitively, this means that if we pick a positive integer $n$ according to a Poisson distribution with large parameter, then pick uniformly a random $\chi \in \mathcal{X}(k_n)$, then the Frobenius conjugacy class $\Theta_{M,k_n}(\chi)$ will be distributed like a random $\mathbf{U}_r(\mathbf{C})$-conjugacy class of an element of the maximal compact subgroup $K$. (A whimsical enough way to do this – according to the Rényi–Turan form of the Erdős–Kac Theorem, see e.g. [65, Prop. 4.14] – would be to pick a large integer $m \ge 1$ and to take $n$ to be the number of prime factors of $m$, which corresponds roughly to having $\lambda_N = \log \log N$.)

Note however that are also many cases where the condition (4.9) is not true. The most obvious is when $\varphi_N(N) = 1$ and $\varphi_N(n) = 0$ for $n \ne N$, corresponding to a limit without extra average at all. In addition, the condition implies that for any integers $q \ge 1$ and $a \in \mathbf{Z}$, we have

$$ \mathbf{P}(X_N \equiv a \pmod q) = \frac{1}{q} \sum_{b \pmod q} e^{-2i\pi ab/q} \mathbf{E}(e^{2i\pi b X_N / q}) \to \frac{1}{q}, $$

so there is a strong arithmetic restriction that $X_N \pmod q$ converge to the uniform probability measure modulo $q$ for all $q \ge 1$.

<footnote n="1">¹ Precisely, we need that the series $\sum a_n$ with $a_1 = \alpha$ and $a_n = \alpha^n - \alpha^{n-1}$ for $n \ge 2$ has “sum” $\alpha+(\alpha-1)/(1-\alpha) = 0$ for $|\alpha| = 1$ and $\alpha \ne 1$.</footnote>
Similar remarks apply in an obvious manner to our other equidistribution statements, e.g. to Theorem 2.

## 4.4. Equidistribution for arithmetic Fourier transforms

We now deduce from Theorem 4.4 the equidistribution of the exponential sums defined by

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

In fact, note that these sums make sense for all characters $\chi \in \hat{G}(k_n)$, and we can indeed prove equidistribution for all of them. This implies Theorem 2 from the introduction. As a final addition, we prove an equidistribution statement for the arithmetic Fourier transforms of all objects M of $D_c^b(G)$ which are mixed semiperverse sheaves of weights $\le 0$. This is of interest especially in more analytic applications, since the condition of being semiperverse and that of being mixed of weights $\le 0$ are much more flexible, and easier to check, than those of being perverse and pure.

**THEOREM 4.8.** Let $k$ be a finite field and let $G$ be a connected commutative algebraic group over $k$. Let $\ell$ be a prime number distinct from the characteristic of $k$.

Let $M$ be an object of $D_c^b(G)$. Assume that $M$ is semiperverse and mixed of weights $\le 0$. Let $N$ be the maximal perverse subsheaf of weight 0 of the arithmetic semisimplification of the perverse cohomology sheaf $\mathfrak{p}\mathcal{H}^0(M)$.

Let $r \ge 0$ be the tannakian dimension of $N$. Let $K \subset G_N^{\text{ari}}(C) \subset GL_r(C)$ be a maximal compact subgroup of the arithmetic tannakian group of $N$. Denote by $\mu$ the Haar probability measure on $K$ and by $\nu$ its image by the trace.

The families of exponential sums $S(M, \chi)$ for $\chi \in \hat{G}(k_n)$ become $\nu$-equidistributed on average as $n \to +\infty$.

**PROOF.** Up to conjugation, we may assume that $K \subset U_r(C)$.

We first assume that $M$ is perverse and pure of weight 0, so that the object $N$ coincides with $M$. We then observe that, by the generic vanishing theorem, it suffices to prove that the families of exponential sums associated to $\chi \in \mathcal{X}_w(M)$ become $\nu$-equidistributed on average, since for any bounded continuous function $f: C \to C$, we have

$$\left| \frac{1}{|\hat{G}(k_n)|} \sum_{\chi \in (\hat{G} - \mathcal{X}_w(M))(k_n)} f(\text{Tr}(\Theta_{M,k_n}(\chi))) \right| \le \|f\|_{\infty} \frac{|(\hat{G} - \mathcal{X}_w(M))(k_n)|}{|\hat{G}(k_n)|} \to 0$$

because $\mathcal{X}_w(M)$ is generic. But since $\text{Tr}_*(\nu_{cp}) = \nu$, this equidistribution follows from Theorem 4.4 by considering the composition $K \to U_r(C) \xrightarrow{\text{Tr}} C$.

We now consider the general case. We denote by $M_0$ the arithmetic semisimplification of the perverse sheaf $\mathfrak{p}\mathcal{H}^0(M)$, and by $N'$ the perverse sheaf such that $M_0 = N \oplus N'$, defined using the weight filtration on $M_0$; the perverse sheaf $N'$ is mixed of weights $\le -1$.

Since $M$ is semiperverse of weights $\le 0$, we have $\mathfrak{p}\mathcal{H}^i(M) = 0$ for $i \ge 1$, and $\mathfrak{p}\mathcal{H}^{-i}(M)$ is of weights $\le -i \le -1$ for all $i \ge 1$ (see [8, Th. 5.4.1]).

For any $n \ge 1$ and $\chi \in \hat{G}(k_n)$, we have the equality

$$(4.10) \qquad S(M, \chi) = S(N, \chi) + S(N', \chi) + \sum_{i \ge 1} (-1)^i S(\mathfrak{p}\mathcal{H}^{-i}(M), \chi)$$

by (A.4).
By generic vanishing and the trace formula (see Theorem 6.1 below, applied to $N'(-1/2)$ and $\mathfrak{p}\mathcal{H}^{-i}(M)(-1/2)$ for $i \ge 1$, which are mixed perverse sheaves of weights $\le 0$), there exists a generic subset $\mathcal{X} \subset \hat{\mathcal{G}}$ such that we have

$$ (4.11) \qquad S(N', \chi) + \sum_{i \ge 1} (-1)^i S(\mathfrak{p}\mathcal{H}^{-i}(M), \chi) \ll \frac{1}{|k_n|^{1/2}} $$

for all $n \ge 1$ and $\chi \in \mathcal{X}(k_n)$. This implies that the sequence $(\varpi_n)$ of probability measures defined as averages of delta masses at the points

$$ S(N', \chi) + \sum_{i \ge 1} (-1)^i S(\mathfrak{p}\mathcal{H}^{-i}(M), \chi) $$

for all $\chi \in \hat{\mathcal{G}}(k_n)$ converges to zero in probability, i.e., that for any fixed real number $\varepsilon > 0$, the limit

$$ \lim_{n \to +\infty} \varpi_n(\{|t| > \varepsilon\}) = 0 $$

holds.

By the first case applied to the perverse sheaf $N$, the sums $S(N, \chi)$ become $\nu$-equidistributed on average as $n \to +\infty$, and the formula (4.10) ensures then that the same holds for the $S(M, \chi)$ (see, e.g., [89, Cor. B.4.2] for the simple probabilistic argument that leads to this conclusion). $\square$

**REMARK 4.9.** (1) As we will see later, it is often of interest to attempt to apply equidistribution of exponential sums to the test function $z \mapsto z^m$ or $z \mapsto |z|^m$ for some integer $m \ge 1$. Such functions are continuous but not bounded on $C$, so that Theorem 4.8 does not apply, and Theorem 4.4 only gives the equidistribution for weakly unramified characters. In these attempts, the contribution of the other characters may therefore need to be handled separately (see for instance the proof of Theorem 9.11).

(2) See Chapter 7 for an application of this theorem to a question of independence of $\ell$ of tannakian groups.

(3) The measure $\nu$ is also the image by the trace of measure $\nu_{cp}$ on characteristic polynomials appearing in Theorem 4.4. It is often called the *Sato-Tate measure* associated to $M$.

**EXAMPLE 4.10.** Let $k = \mathbb{F}_p$, and let $\psi$ be the additive character on $k$ such that $\psi(x) = e(x/p)$ for $x \in k$. Let $X \subset G$ be a locally-closed subvariety of $G$ of dimension $d \ge 1$, and let $f: X \to \mathbb{A}^1$ be a non-zero function on $X$. Then there is a semiperverse sheaf $M$ on $G$, mixed of weights 0, such that the trace function of $M$ is given by the formula

$$ t_M(x; \mathbb{F}_{p^n}) = \begin{cases} (-1)^d p^{-nd/2} e(\mathrm{Tr}_{\mathbb{F}_{p^n}/\mathbb{F}_p}(f(x))/p) & \text{if } x \in X(\mathbb{F}_{p^n}) \\ 0 & \text{otherwise.} \end{cases} $$

for $n \ge 1$ and $x \in G(k_n)$, namely

$$ M = j_1 f^* \mathcal{L}_\psi[d](d/2), $$

where $j: X \to G$ is the natural immersion.

Hence Theorem 4.8 implies that the exponential sums

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

for $\chi \in \hat{\mathcal{G}}(\mathbb{F}_{p^n})$ always satisfy some equidistribution theorem on average.
A similar property holds if we fix a non-trivial multiplicative character $\eta$ of $\mathbf{F}_p^\times$ and an invertible function $g: X \to \mathbf{G}_m$, and consider the exponential sums

$$ \frac{1}{p^{nd/2}} \sum_{x \in \mathcal{X}(\mathbf{F}_{p^n}^\times)} \chi(x) \eta(g(x)) $$

(using the object $j_*g^*\mathcal{L}_\eta[d](d/2)$, which is also mixed and semiperverse of weights $\le 0$).

### 4.5. Equidistribution for conjugacy classes

We keep the notation of the previous sections. If the object $M$ that we consider is generically unramified, then we can prove equidistribution at the level of the Frobenius conjugacy classes in the maximal compact subgroup of the arithmetic tannakian group.

**THEOREM 4.11** (Equidistribution on average). Let $k$ be a finite field and let $G$ be a connected commutative algebraic group over $k$. Let $\ell$ be a prime number distinct from the characteristic of $k$.

Let $M$ be an $\ell$-adic perverse sheaf on $G$ that is arithmetically semisimple, pure of weight zero and generically unramified. Let $\mathcal{X} = \mathcal{X}(M)$ be the set of unramified characters for $M$. Let $K$ be a maximal compact subgroup of the arithmetic tannakian group $\mathbf{G}_M^{\text{ari}}(\mathbf{C})$ of $M$, and denote by $\mu^\sharp$ the direct image of the Haar probability measure $\mu$ on $K$ by the projection to the set $K^\sharp$ of conjugacy classes of $K$.

The families of unitary Frobenius conjugacy classes $(\Theta_{M,k_n}(\chi))_{\chi \in \mathcal{X}(k_n)}$ become $\mu$-equidistributed on average in $K^\sharp$ as $n \to +\infty$.

Precisely, we are considering here the equidistribution on average of the pairs $(\mathcal{X}(k_n), \Theta_n)$ where $\Theta_n(\chi) = \Theta_{M,k_n}(\chi)$.

**PROOF.** By Theorem 3.30 and the definition of generic sets, we know that $|\mathcal{X}(k_n)| \sim |G(k_n)|$ as $n \to +\infty$, and hence the sets of unramified conjugacy classes are non-empty for $n$ large enough.

By the Peter-Weyl theorem, any continuous central function $f: K \to \mathbf{C}$ is a uniform limit of linear combinations of characters of finite-dimensional unitary irreducible representations of $K$, and hence it suffices to prove the formula (4.1) when $f$ is such a character. For the trivial representation, both sides are equal to 1. If the representation is non-trivial, then the integral on the right-hand side vanishes, and we are reduced to showing that the limit on the left-hand side exists and is equal to 0. We thus consider a non-trivial irreducible representation $\varrho$ of $K$, which we identify with a non-trivial irreducible algebraic $\overline{\mathbb{Q}}_\ell$-representation of the arithmetic tannakian group $\mathbf{G}_M^{\text{ari}}$ by Weyl's unitarian trick (see, e.g., [69, 3.2] for this step); applying the next proposition then completes the proof. $\square$

**PROPOSITION 4.12.** With notation as in Theorem 4.11, let $\varrho$ be a non-trivial irreducible unitary representation of $K$, identified with a non-trivial irreducible representation of $\mathbf{G}_M^{\text{ari}}$.

$$ (4.12) \qquad \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) \ll \frac{1}{\sqrt{|k_n|}} $$

for all $n$ such that $\mathcal{X}(k_n)$ is not empty.

$$ (4.13) \qquad \lim_{N \to +\infty} \frac{1}{N} \sum_{\substack{1 \le n \le N \\ \mathcal{X}(k_n) \ne \emptyset}} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = 0. $$
PROOF. (1) We assume that the restriction of $\varrho$ to the geometric tannakian group is non-trivial.

Let $\varrho(M)$ denote the object of the tannakian category $\langle M \rangle^{\text{ari}}$ corresponding to the representation $\varrho$ of the arithmetic tannakian group $G_M^{\text{ari}}$; this is a simple perverse sheaf on $G$.

We have $\mathcal{X} \subset \mathcal{X}_F(\varrho(M))$. Applying Proposition 4.3 to the object $N = \varrho(M)$, we obtain

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = t_{\varrho(M)}(e; k_n) + O(|k_n|^{-1/2}) $$

since the conjugacy class $\Theta_{M,k_n}(\chi)$ coincides with $\Theta_{M,k_n}(\chi)$ when $\chi$ is unramified for $M$.

Since $\varrho(M)$ is a simple perverse sheaf on $G$, the classification of [8, Th. 4.3.1 (ii)] shows that there exist an irreducible closed subvariety $s: Y \to G$ of dimension $r$, an open dense smooth subvariety $j: U \to Y$, and an irreducible lisse $\overline{Q}_\ell$-sheaf $\mathcal{F}$ on $U$ such that $\varrho(M) = s_*j_*\mathcal{F}[r]$. Since the functors $s_*$ and $j_*$ are weight-preserving, the sheaf $\mathcal{F}$ is pure of weight $-r$.

If $r = 0$, then $Y$ consists of a closed point of $G$, which must be different from the neutral element $e$, since otherwise $\varrho(M)$ would be geometrically trivial, contrary to the assumption in (1). In that case, we have therefore $t_{\varrho(M)}(e; k_n) = 0$. On the other hand, if $r \ge 1$ we get

$$ (4.14) \qquad t_{\varrho(M)}(e; k_n) \ll \frac{1}{\sqrt{|k_n|}} $$

(by Proposition 1.11), which concludes the proof of (1).

(2) We assume that the restriction of the representation $\varrho$ to $G_M^{\text{geo}}$ is trivial. Then $\varrho$ has dimension 1 since the quotient $G_M^{\text{ari}}/G_M^{\text{geo}}$ is abelian (Proposition 3.41).

Let $Q$ be the set of integers $n \ge 1$ such that $\mathcal{X}(k_n)$ is not empty; it contains all sufficiently large integers. It follows from Proposition 3.41 that there exists an element $\xi$ of $G_M^{\text{ari}}/G_M^{\text{geo}}$, generating a Zariski-dense subgroup of this group, such that $\varrho(\Theta_{M,k_m}(\chi)) = \varrho(\xi)^n$ for any $n \ge 1$ and any $\chi \in \hat{G}(k_n)$ unramified for $M$. Moreover, we have $\varrho(\xi) \ne 1$, since otherwise the representation $\varrho$ would be trivial. We conclude that

$$ \frac{1}{N} \sum_{\substack{1 \le n \le N \\ \mathcal{X}(k_n) \ne \emptyset}} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\varrho(\Theta_{M,k_n}(\chi))) = \frac{1}{N} \sum_{\substack{1 \le n \le N \\ n \in Q}} \varrho(\xi)^n $$

converges to 0 as $N \to +\infty$ by summing a non-trivial geometric progression.

□

REMARK 4.13. For certain reductive groups $G \subset GL_r(C)$, a conjugacy class in a maximal compact subgroup $K$ of $G$ is determined by its characteristic polynomial (equivalently, the exterior powers of the standard representation generate the representation ring of $G$). If $G_M^{\text{ari}}(C)$ has this property, then Theorem 4.4 implies a version of Theorem 4.11, even if $M$ is not generically unramified.

If $G$ is semisimple, this property holds, for instance, for $SL_r(C) \subset GL_r(C)$, for $Sp_{2r}(C) \subset GL_{2r}(C)$, and for $G_2(C) \subset GL_7(C)$. Indeed, the first two cases are explained by Katz in [69, Lemma 13.1, Remark 13.2]; in the third case, we note that the second fundamental representation of $G_2(C)$ is virtually $\Lambda^2 \mathrm{Std} - \mathrm{Std}$ (see, e.g., [49, p. 353]) so that the exterior powers of the standard 7-dimensional representation generate the representation ring.

We deduce immediately from Theorem 4.11 a useful corollary, analogue to some classical consequences of the Chebotarev density theorem.
**COROLLARY 4.14.** Let $k$ be a finite field and let $G$ be a connected commutative algebraic group over $k$. Let $M$ be a perverse sheaf on $G$ which is arithmetically semisimple, pure of weight zero and generically unramified.

Let $S$ be any finite subset of $\hat{G}$. The union of the unitary Frobenius conjugacy classes of $M$ associated to unramified characters in $\hat{G} - S$ is dense in a maximal compact subgroup of $G_M^{\text{ari}}(C)$.

**4.6. Equidistribution without average**

We continue again with the previous notation. If we make the extra assumption that the geometric and the arithmetic tannakian groups coincide, then the equidistribution of Frobenius conjugacy classes holds without averaging over $n$. We summarize the variants of the previous theorems in this situation.

**THEOREM 4.15 (Equidistribution without average).** Let $M$ be an $l$-adic perverse sheaf on $G$ that is arithmetically semisimple, pure of weight zero. We assume that the inclusion $G_M^{\text{geo}} \subset G_M^{\text{ari}}$ is an equality.

Let $r \ge 0$ be the tannakian dimension of $M$. Let $K \subset U_r(C)$ be a conjugate of a maximal compact subgroup of the arithmetic tannakian group $G_M^{\text{ari}}(C) \subset GL_r(C)$ of $M$. Denote by $\mu$ the Haar probability measure on $K$, by $\nu_{cp}$ its direct image by the map $K \to U_r(C)^{\sharp}$, by $\nu$ its image by the trace, and by $\mu^{\sharp}$ its image by the map $K \to K^{\sharp}$.

(1) The families of unitary Frobenius conjugacy classes $(\Theta_{M,k_n}(\chi))_{\chi \in \mathcal{X}_w(M)(k_n)}$ become $\nu_{cp}$-equidistributed as $n \to +\infty$.

(2) The families of exponential sums $S(M, \chi)$ for $\chi \in \hat{G}(k_n)$ become $\nu$-equidistributed as $n \to +\infty$.

(3) If $M$ is generically unramified, then the family of conjugacy classes $(\Theta_{M,k_n}(\chi))_{\chi \in \mathcal{X}(M)(k_n)}$ become $\mu^{\sharp}$-equidistributed as $n$ goes to infinity.

**PROOF.** This follows from the Weyl Criterion as in the proof of Theorems 4.4, 4.8 and 4.11; in the case of the last statement, for instance, we use only the first part of Proposition 4.12 (as we may since a non-trivial irreducible representation of $G_M^{\text{ari}}$ is a non-trivial irreducible representation of $G_M^{\text{geo}}$ under the assumption). $\square$

**REMARK 4.16.** There is an obvious further variant of Theorems 4.15 and of the case of mixed semiperverse objects of weights $\le 0$ of 4.8: if $M$ is mixed semiperverse of weights $\le 0$, with $N$ as in Theorem 4.8 such that $G_N^{\text{ari}} = G_N^{\text{geo}}$, then the discrete Fourier transform becomes equidistributed towards the measure $\nu$ without average over $n$.

There is a converse to Theorem 4.15. In fact, there is a statement which is valid for an individual representation of the unitary group (this will be useful in Chapter 8).

**PROPOSITION 4.17.** Let $M$ be an $l$-adic perverse sheaf on $G$ that is arithmetically semisimple and pure of weight zero. Let $r$ be the tannakian dimension of $M$ and let $\mathcal{X} = \mathcal{X}_w(M)$ be the set of weakly unramified characters for $M$. Let $\rho$ be a finite-dimensional unitary representation of $U_r(C)$. Assume that the sequence

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \text{Tr}(\rho(\Theta_{M,k_n}(\chi))), $$

defined for all integers $n \ge 1$ such that $\mathcal{X}(k_n)$ is not empty, has a limit. Then this limit is equal to the multiplicity of the trivial representation in the restriction of $\rho$ to $G_M^{\text{geo}}$, and the latter equals the multiplicity of the trivial representation in $\rho$.
PROOF. We use the notation in the proof of Theorem 4.4. Taking the equality (4.8) into account, the assumption of the statement means that the limit

$$ \lim_{n \to +\infty} \sum_{i \in J} \alpha_i^n $$

exists, where the complex numbers $\alpha_i$ have modulus 1 and the set J has cardinality equal to the multiplicity of the trivial representation in the restriction of $\rho$ to $G_M^{\text{geo}}$. We claim that the existence of this limit implies the equality $\alpha_i = 1$ for all $i \in J$, so that the limit is equal to $|J|$, as desired.

Indeed, let $L \subset J$ be the set of $i$ where $\alpha_i \neq 1$. The sequence

$$ \sum_{i \in L} \alpha_i^n $$

converges as well, and its limit must be zero since it converges to 0 on average over $n \le N$. However, the lower bound

$$ \limsup_{n \to +\infty} \left| \sum_{i \in L} \alpha_i^n \right| \ge |L|^{1/2} $$

holds (see, e.g., [64, Lemma 11.41]), so we deduce that L is empty, which proves the claim. $\square$

A more global form of this converse, for generically unramified objects, is the following:

PROPOSITION 4.18. Let M be an $l$-adic perverse sheaf on $G$ that is arithmetically semisimple and pure of weight zero. Assume that M is generically unramified. Let $r$ be the tannakian dimension of M and let $\mathcal{X} = \mathcal{X}_w(M)$ be the set of unramified characters for M. If the sequence of probability measures

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \delta_{\Theta_{M,k_n}(\chi)}, $$

defined when $\mathcal{X}(k_n)$ is not empty, converges weakly to some probability measure, then we have the equality $G_M^{\text{ari}} = G_M^{\text{geo}}$.

PROOF. Suppose that $G_M^{\text{geo}} \neq G_M^{\text{ari}}$. By Proposition 3.41, there exists an element $\xi \neq 1$ of $G_M^{\text{ari}}/G_M^{\text{geo}}$ which generates a Zariski-dense subgroup of this group, which is abelian. Thus there exists an irreducible representation $\rho$ of the quotient $G_M^{\text{ari}}/G_M^{\text{geo}}$ such that $\rho(\xi) \neq 1$; for any $n \ge 1$ and any $\chi \in \hat{G}(k_n)$ unramified for M, the equality $\rho(\Theta_{M,k_n}(\chi)) = \rho(\xi)^n$ holds.

Let $\mathcal{X}$ be the set of characters unramified for M. Then

$$ \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\rho(\Theta_{M,k_n}(\chi))) = \rho(\xi)^n $$

for all $n \ge 1$ for which $\mathcal{X}(k_n)$ is not empty. Since $\rho(\xi) \neq 1$, this quantity does not converge as $n \to +\infty$, which implies the proposition by contraposition. $\square$

## 4.7. Horizontal equidistribution

The proof of Theorem 4.11 relies crucially on the estimates in the stratified vanishing theorem 2.3. We expect (see Remark 2.4) that the implied constants in these estimates depend only on the complexity of the perverse sheaf M (as is the case for unipotent groups).

Under the assumption that such a statement is valid, and in fact that this holds for the size of the set of unramified characters, one can obtain equidistribution statements for finite fields when their size tends to infinity (for instance, for $F_p$ as $p \to +\infty$; compare with [75, Ch. 28–29]).
We include a conditional statement of this type, anticipating some progress soon concerning the underlying uniformity question. We leave to the interested reader the task of formulating variants similar to Theorems 4.8 and 4.4.

**THEOREM 4.19 (Horizontal equidistribution).** Let $\ell$ be a prime number. Let $N \ge 1$ be an integer and let $(G, u)$ be a quasi-projective commutative group scheme over $\mathbb{Z}[1/\ell N]$ such that, for all primes $p \nmid \ell N$, the fiber $G_p$ of $G$ over $\mathbf{F}_p$ is a connected commutative algebraic group for which the estimate

$$|\hat{G}_p(\mathbf{F}_{p^n}) - \mathcal{X}(M)(\mathbf{F}_{p^n})| \ll c_u(M)p^{n(\dim(G_p)-1)}$$

holds for all primes $p$ and $n \ge 1$ and all arithmetically semisimple objects $M$ in $\mathrm{Perv}_{\mathrm{int}}(G_p)$ which are generically unramified.

Let $(M_p)_{p^\ell N\ell}$ be a sequence of arithmetically semisimple objects in $\mathrm{Perv}_{\mathrm{int}}(G_p)$ which are pure of weight zero. Suppose that the tannakian dimension $r$ of $M_p$ is independent of $p$, and that for all $p$, we have $G_{M_p}^{\mathrm{ari}} = G_{M_p}^{\mathrm{geo}}$, and that this common reductive group is conjugate to a fixed subgroup $G$ of $GL_r(\overline{\mathbb{Q}}_\ell)$.

Let $K$ be a maximal compact subgroup of $G(C)$ and let $\mu^\sharp$ be the direct image of the Haar probability measure on $K$ to $K^\sharp$.

Let $\mathcal{X}_p$ be the set of characters $\chi \in \hat{G}_p(\mathbf{F}_p)$ which are unramified for the object $M_p$.

If we have $c_u(M_p) \ll 1$ for all $p \nmid N\ell$, then the families of conjugacy classes $(\Theta_{M_p, \mathbf{F}_p}(\chi))_{\chi \in \mathcal{X}_p}$ become $\mu^\sharp$-equidistributed in $K^\sharp$ as $p \to +\infty$.

**PROOF.** The argument follows that of Theorem 4.11; it suffices to prove the estimate

$$\frac{1}{|\mathcal{X}_p|} \sum_{\chi \in \mathcal{X}_p} \mathrm{Tr} \left( \varrho(\Theta_{M_p, \mathbf{F}_p}(\chi)) \right) \ll \frac{1}{\sqrt{p}}$$

for all $p \nmid N\ell$. The proof of this is similar to the first part of Proposition 4.12, noting that, under our assumptions, the implied constants in the key bounds (4.14), (4.5), (4.6) and (4.7) are independent of $p$, since the complexity of $M_p$ is bounded independently of $p$, and hence also that of $\varrho(M_p)$ by [115, Prop. 6.33]. $\square$

**REMARK 4.20.** (1) For $G$ unipotent, results of this form are unconditional by Proposition 2.9 (the case of $G_a$ essentially goes back to Katz [69], whereas the case of an arbitrary power of $G_a$ follows from [115, Th. 7.22]). For $G = G_m$, a similar statement is proved by Katz in [75, Th. 28.1].

(2) The result is also unconditional in the case of abelian varieties (see Remark 2.4). We expect that a careful look at the proof of the generic vanishing theorem will also show that it is unconditional for $G_m \times G_a$. The case of tori of dimension $\ge 2$ is however not yet known.

## 4.8. Objects of rank 1

In this section, we apply the general equidistribution results to objects in the tannakian subcategory of $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G)$ additively generated by objects of tannakian rank 1. The corresponding arithmetic tannakian groups are computed (in principle) in Proposition 3.50.

**PROPOSITION 4.21.** Let $r \ge 1$ be an integer and let

$$M = L_1 \oplus \cdots \oplus L_r$$

where $L_i$ is an object of $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G)$ of tannakian rank 1. Let

$$H = \{(n_1, \dots, n_r) \in \mathbb{Z}^r \mid L_1^{*\mathrm{int} n_1} *_{\mathrm{int}} \cdots *_{\mathrm{int}} L_r^{*\mathrm{int} n_r} \simeq \mathbf{1}_G\},$$
and let

$$ K = \{(x_i) \in (\mathbf{S}^1)^r \mid \prod_{1 \le i \le r} x_i^{n_i} = 1 \text{ for all } (n_1, \dots, n_r) \in \mathbf{H}\}. $$

Then the unitary Frobenius conjugacy classes of M are equidistributed on average in K.

**PROOF.** This follows from Theorem 4.4, on noting that the arithmetic tannakian $\mathbf{G}_M^{\text{ari}}$ group of M is abelian (it can be viewed as a subgroup of the diagonal subgroup $\mathbf{GL}_1^r \subset \mathbf{GL}_r$); hence, the conjugacy classes of elements of $\mathbf{G}_M^{\text{ari}}$ are just singletons, and in particular are the same as the conjugacy classes in $\mathbf{GL}_r$. $\square$

As an application, we explain how to recover a theorem of Rojas-León [110, Th. 1], which concerns the equidistribution properties of Gauss sums.

**THEOREM 4.22 (Rojas-León).** Let $r \ge 1$ be an integer. Let $(\alpha_i)_{1 \le i \le r}$ be a family of non-constant morphisms $\alpha_i: \mathbf{G}_m \to \mathbf{G}_m^r$ defined over $k$. Let $t = (t_i) \in (k^\times)^r$ and let $(\eta_i)_{1 \le i \le r}$ be characters of $k^\times$.

The tuples

$$ \left( \frac{\chi(t)\tau(\psi, \eta_1 \cdot (\chi \circ \alpha_1))}{|k_n|^{1/2}}, \dots, \frac{\chi(t)\tau(\psi, \eta_r \cdot (\chi \circ \alpha_r))}{|k_n|^{1/2}} \right)_{\chi \in \widehat{\mathbf{G}}_m^r(k_n)} $$

of Gauss sums are equidistributed on average in $\mathbf{C}^r$ according to the probability Haar measure on a closed subgroup $K \subset (\mathbf{S}^1)^r \subset \mathbf{C}^r$.

Moreover, factor $\alpha_i = [x \mapsto x^{N_i}] \circ \beta_i$ for some closed immersion $\beta_i$ and some integer $N_i \ge 1$. If, for each $i$ with $1 \le i \le r$, the elements

$$ \sum_{\omega^{N_i} = \eta_i} \omega $$

of $\mathbf{Z}[\Pi(\mathbf{G}_{m,\bar{k}}, \overline{\mathbf{Q}}_\ell)]$ are linearly independent over $\mathbf{Z}$, then $K = (\mathbf{S}^1)^r$ and equidistribution holds without average.

**PROOF.** For simplicity of notation, we will assume that $t = 1$ and that each $N_i$ is coprime to $p$. Let $j: \mathbf{G}_m \to \mathbf{A}^1$ be the open immersion. We recall that $\chi \mapsto \tau(\psi, \eta_i \cdot (\chi \circ \alpha_i))$ is the discrete Mellin transform on $\mathbf{G}_m$ of the trace function of the perverse sheaf

$$ L_i = \alpha_{i*}(j^* \mathcal{L}_\psi *! \mathcal{L}_\chi[1]) (1/2) $$

on $\mathbf{G}_m^r$, which is pure of weight 0. This is an object of tannakian rank 1.

Thus the first statement is a direct application of Proposition 4.21, with $K$ a maximal compact subgroup for the arithmetic tannakian group of

$$ M = L_1 \oplus \dots \oplus L_r $$

(note that we consider the Gauss sums in $\mathbf{C}$ to avoid excluding those boundedly many $\chi$ where some $\eta_i \cdot (\chi \circ \alpha_i)$ is trivial, for which the modulus is $1/\sqrt{|k_n|}$ instead of 1; these do not affect the equidistribution property).

Using Proposition 4.21 again, for the second statement we need to prove that under the stated assumptions, there is no convolution relation between the objects $L_i$, or equivalently no relation between their classes in the group $L(\mathbf{G}_m^r)$; this implies that the arithmetic tannakian group of M is $\mathbf{G}_m^r$ (i.e., is as large as possible). In fact, we claim that the geometric tannakian group is already that large, which means that there are no geometric convolution relations between the objects $L_i$. This in particular also implies the equidistribution without average (see Theorem 4.15).
To prove the claim, we use the Gabber–Loeser isomorphism described in Example 3.52 to express the class of $L_i$ in $L(G_{m,\bar{k}}^r)$ as

$$ \lambda_i = (1, \sum_{\omega^{N_i} = \eta_i} (T_i, \omega)), $$

where $T_i$ is the image of $\beta_i$ and $(T_i, \omega)$ is one of the basis vectors in the free abelian group generated by pairs of a one-dimensional subtorus and a tame character (see loc. cit.). (This fact is a form of the Hasse–Davenport relation; see [70, Th. 8.9.1].)

By definition of a free abelian group, a non-trivial linear relation can only exist if, for some one-dimensional subtorus $T \subset G_m^r$, the elements $\lambda_i$ with $T_i = T$ are linearly dependent, and this in turn is equivalent with the elements

$$ \sum_{\omega^{N_i} = \eta_i} \omega $$

being linearly dependent, as claimed. $\square$
<figure><img src="image_8.png" /></figure>
Part 2

Applications
<figure><img src="image_9.png" /></figure>
Description of applications

The remainder of the book is devoted to applications of the theoretical results of the first part
of this book. We split these applications in further chapters as follows:

(1) We define in Chapter 5 the analogue of L-functions for the Fourier–Mellin transforms.
We establish with its help that the arithmetic tannakian group is infinite for many non-
punctual objects on abelian varieties.

(2) We present in Chapter 6 the concrete analytic translation of the stratified vanishing theo-
rem to stratification of estimates for exponential sums, in the spirit of Katz–Laumon [78]
and Fouvry–Katz [39]. We also present a statement of “generic Fourier invertibility”,
which shows that two semsimple perverse sheaves are isomorphic in the category $\overline{\mathbb{P}}^{\text{ari}}(\mathbb{G})$
if and only if the associated exponential sums coincide for a generic set of characters.

(3) In Chapter 7, we add a theoretical application of equidistribution in direction of independence of $\ell$ properties of the tannakian groups associated to a compatible system of $\ell$-adic complexes.

(4) In applications of equidistribution to concrete perverse sheaves, the main issue is to determine the tannakian group. The main tool that we will use for this purpose is Larsen's *Alternative*, and its link with equidistribution. We present this result (and a new variant for the exceptional group $E_6$) in Chapter 8.

(5) Then in the remaining chapters, we present examples of equidistribution for “concrete”
groups, namely:

- the product $G_m \times G_a$, which (apart from unipotent groups) is probably the simplest
group of dimension $\ge 2$ (Chapter 9); this corresponds to rather natural families of
exponential sums parameterized by both an additive character and a multiplicative
character.

- higher-dimensional tori, with applications to the study of the variance of arithmetic functions on $k[t]$ in arithmetic progressions modulo square-free polynomials (see Chapter 10).

- the jacobian of a curve (Chapter 11); the application we present is a generalization of an unpublished result of Katz (which answered a question of Tsimerman).

- in the same chapter, the intermediate jacobian of a smooth projective cubic hyper-
surface of dimension 3, which is an abelian variety of dimension 5 (see Chapter 11.2).
<figure><img src="image_10.png" /></figure>
# Über eine neue Art von L-Reihen

## 5.1. $\hat{L}$-functions

Let $k$ be a finite field, with algebraic closure $\bar{k}$ and intermediate extensions $k_n$. We fix as usual a prime $\ell$ different from the characteristic of $k$. Let $G$ be a connected commutative algebraic group over $k$, and let $d$ be its dimension. We denote by $e$ the neutral element of $G$.

By analogy with algebraic varieties over $k$, we can define “L-functions” for objects of $D_c^b(G)$, where suitable characters $\chi \in \hat{G}$ play the role of primes in an “Euler product”.

We denote by $\hat{G}^* \subset \hat{G}$ the set of characters such that $\chi \in \hat{G}^*(k_n)$ if and only if there is no $d \mid n$ with $d < n$ such that $\chi = \chi' \circ N_{k_n/k_d}$. We say that elements of $\hat{G}^*$ are *primitive*, and for $\chi \in \hat{G}^*(k_n)$, we put $\deg(\chi) = n$. We then denote by $[\hat{G}]$ the quotient set of $\hat{G}^*$ by the equivalence relation defined by $\chi_1 \sim \chi_2$ if and only if $\deg(\chi_1) = \deg(\chi_2)$ and

$$ \chi_2 = \chi_1 \circ \mathrm{Fr}_{k_{\deg(\chi_1)}}^j $$

for some integer $j \in \mathbb{Z}$. There are $\deg(\chi)$ primitive characters equivalent to a given $\chi \in \hat{G}^*$.

**DEFINITION 5.1 (L-function).** Let $M$ be an object of $D_c^b(G)$. The *Fourier-L-function*, or $\hat{L}$-function, of $M$ is the formal power series

$$ \hat{L}(M, T) = \prod_{\chi \in [\hat{G}]} \det(1 - T^{\deg(\chi)}) \mathrm{Fr}_{k_{\deg(\chi)}} \mid \mathrm{H}_c^*(\mathrm{G}_{\bar{k}}, \mathrm{M}_{\chi})^{-1} \in \overline{\mathcal{Q}}_\ell[[T]]. $$

This is similar to the definition

$$ \mathrm{L}(M, T) = \prod_{x \in [X]} \det(1 - T^{\mathrm{deg}(x)}) \mathrm{Fr}_{k_{\mathrm{deg}(x)}} \mid \mathrm{M}_x)^{-1} \in \overline{\mathcal{Q}}_\ell[[T]] $$

of the L-function of $M$ on an arbitrary algebraic variety $X$ over $k$, with primitive characters replacing the set $[X]$ of closed points of $X$.

Indeed, if $G$ is unipotent of dimension $d$, and $\mathrm{FT}(M)$ denotes the Fourier transform of $M$ on the (or “a”) Serre dual $G^\vee$ defined with respect to some additive character $\psi$, as in Section 2.2, then we obtain the identity

$$ \hat{L}(M, T) = \mathrm{L}(\mathrm{FT}(M), |k|^d T), $$

(e.g. by the formula (5.2) below, since the stalk of $\mathrm{FT}(M)$ at the origin is canonically isomorphic to $M$ by the proper base change theorem, and $|G(k_n)| = |k|^{nd}$ in this case).

In general, however, we obtain “new” L-functions. Their fundamental properties, including rationality, are given by the next proposition.

**PROPOSITION 5.2.** Let $M$ be an object of $D_c^b(G)$. We denote as usual

$$ S(M, \chi) = \sum_{x \in G(k_n)} \chi(x) t_M(x; k_n) $$
for $n \ge 1$ and $\chi \in \hat{G}(k_n)$.

(1) The $\hat{L}$-function satisfies

$$ (5.1) \qquad \hat{L}(M, T) = \exp\left(\sum_{n \ge 1} \left(\sum_{\chi \in \hat{G}(k_n)} S(M, \chi)\right) \frac{T^n}{n}\right) $$

$$ (5.2) \qquad = \exp\left(\sum_{n \ge 1} |G(k_n)| t_M(e; k_n) \frac{T^n}{n}\right). $$

(2) The $\hat{L}$-function is a rational function; if $M$ is a mixed complex, then the zeros and poles of $\hat{L}(M, T)$ are $|k|$-Weil numbers of some weights.

(3) For any $\chi \in \hat{G}(k)$, the equality $\hat{L}(M_\chi, T) = \hat{L}(M, T)$ holds.

PROOF. The proof of the formula (5.1), like in the classical case, is a simple consequence of the trace formula. Precisely, we apply the operator $f(T) \mapsto Td \log f(T)$ to both sides of this equality. On the left-hand side, after expressing the determinant as alternating product of the determinants on the various groups $H_c^i(G_{\bar{k}}, M_\chi)$, we obtain

$$ Td \log \hat{L}(M, T) = \sum_{\chi \in [\hat{G}]} \deg(\chi) \sum_{m \ge 1} T^{m \deg(\chi)} \mathrm{Tr}(\mathrm{Fr}_{k_{\deg(\chi)}}^m | H_c^*(G_{\bar{k}}, M_\chi)) = \sum_{n \ge 1} T^n \sum_{d|n} \sum_{\chi \in [\hat{G}](k_d)} d \mathrm{Tr}(\mathrm{Fr}_{k_d}^{n/d} | H_c^*(G_{\bar{k}}, M_\chi)). $$

On the right-hand side of (5.1), we obtain

$$ \sum_{n \ge 1} T^n \sum_{\chi \in \hat{G}(k_n)} S(M, \chi), $$

and hence the formula is equivalent with the fact that the identity

$$ (5.3) \qquad \sum_{d|n} \sum_{\chi \in [\hat{G}](k_d)} d \operatorname{Tr}(\operatorname{Fr}_{k_d}^{n/d} | H_c^*(G_{\bar{k}}, M_\chi)) = \sum_{\chi \in \hat{G}(k_n)} S(M, \chi) $$

holds for any integer $n \ge 1$.

Let $n \ge 1$. To establish (5.3) for $n$, we begin with the trace formula (A.5), which implies that

$$ S(M, \chi) = \mathrm{Tr}(\mathrm{Fr}_{k_n} | H_c^*(G_{\bar{k}}, M_\chi)), $$

for any $\chi \in \hat{G}(k_n)$.

There exists a unique divisor $d$ of $n$ and a character $\chi_0 \in \hat{G}^*(k_d)$ such that $\chi = \chi_0 \circ N_{k_n/k_d}$. The map sending $\chi$ to the equivalence class of $\chi_0$ in $[\hat{G}]$ has image the subset of classes $[\eta]$ of primitive characters $\eta$ with degree dividing $n$, and for any such class $[\eta]$, there are exactly $\deg([\eta])$ characters $\chi \in \hat{G}(k_n)$ mapping to $[\eta]$. Moreover, there are canonical isomorphisms

$$ H_c^*(G_{\bar{k}}, M_\chi) \simeq H_c^*(G_{\bar{k}}, M_\eta), $$

with the actions of $\mathrm{Fr}_{k_n}$ corresponding to that of $\mathrm{Fr}_{k_d}^{n/d}$, so that

$$ S(M, \chi) = \mathrm{Tr}(\mathrm{Fr}_{k_d}^{n/d} | H_c^*(G_{\bar{k}}, M_\eta)) $$

for all $\chi$ mapping to $[\eta]$. This implies the desired identity (5.3).
The second formula (5.2) for $\hat{L}(M, T)$ follows immediately from (5.1), since orthogonality of characters implies that the formula

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

holds for all $n \ge 1$.

Using next the trace formula and the Riemann Hypothesis to compute $|G(k_n)|$ as an alternating sum of $|k|$-Weil numbers, it follows that

$$ |G(k_n)|t_M(e; k_n) = \sum_{i \in I} \varepsilon_i \alpha_i^n $$

for some finite set $I$, some $\varepsilon_i \in \{-1, 1\}$, and some $|k|$-Weil numbers $\alpha_i$. The second assertion follows then from the usual power series expansion

$$ \exp\left(\sum_{n \ge 1} \frac{\alpha^n T^n}{n}\right) = \frac{1}{1 - \alpha T}. $$

The final assertion is clear either from the definition, or from the above, noting that $t_{M_\chi}(e; k_n) = t_M(e; k_n)$ for any $\chi \in \hat{G}(k)$ and $n \ge 1$. $\square$

REMARK 5.3. To illustrate the differences with L-functions, we note that if $G$ is not unipotent, then the $\hat{L}$-function is very rarely a polynomial or the inverse of a polynomial, and does not satisfy in general any functional equation of the form

$$ \hat{L}(M, T) = (\text{simple quantities}) \times \hat{L}(M^{\vee}, q^{\alpha} T^{-1}). $$

as is the case for the standard L-function of $M$ (this is related to the remark of Boyarchenko and Drinfeld [15, §1.6, Example 1.8]).

To give a concrete example, take $G = G_m$. In this case, we deduce from (5.2) the formula

$$ \hat{L}(M, T) = \exp\left(\sum_{n \ge 1} (|k|^n - 1) t_M(e; k_n) \frac{T^n}{n}\right) = \frac{L(M_e, |k|T)}{L(M_e, T)} $$

where $M_e$ is the stalk of $M$ at $e$ (where $L(M_e, T)$ is the L-function of the stalk of $M$ at $e$, viewed as a complex on $\{e\}$). If the L-function $L(M_e, T)$ is not constant, then there can never be cancellation in this quotient to obtain a polynomial or the inverse of a polynomial. If (say) we have

$$ L(M_e, T) = (1 - \alpha T)(1 - \alpha^{-1}T), $$

then

$$ \hat{L}(M, T) = \frac{(1 - |k|\alpha T)(1 - |k|\alpha^{-1}T)}{(1 - \alpha T)(1 - \alpha^{-1}T)}, $$

and this satisfies no simple functional relation.

We conclude with a result that will be useful in the next section when performing induction.

PROPOSITION 5.4. Let $G$ be a semiabelian variety over $k$. Let $S$ be a tac of $G_k$ defined by a morphism $\pi: G \to G'$ over $k$ and a character $\chi_0 \in \hat{G}(k)$, and let $[S]$ denote the classes in $[\hat{G}]$ of elements of $S$. Let $M$ be an object of $\mathrm{D}_c^b(G)$. We then have

$$ \prod_{\chi \in [S]} \det(1 - T^{\mathrm{deg}(\chi)} \mathrm{Fr}_{k_{\mathrm{deg}(\chi)}} | \mathrm{H}_c^*(\mathrm{G}_{\bar{k}}, \mathrm{M}_\chi))^{-1} = \hat{L}(\mathrm{R}\pi!\mathrm{M}_{\chi_0}, T). $$
PROOF. We have $\chi \in [S]$ if and only if $\chi = \chi_0 \cdot (\pi^*\eta)$ for some $\eta \in [\hat{G}']$, with $\deg(\chi) = \deg(\eta)$. By the projection formula, we have a canonical isomorphism

$$H_c^*(G_{\bar{k}}, M_{\chi}) = H_c^*(G_{\bar{k}}, M_{\chi_0} \otimes \pi^* \mathcal{L}_{\eta}) \simeq H_c^*(G'_{\bar{k}}, R_{\pi!}M_{\chi_0} \otimes \mathcal{L}_{\eta}),$$

from which the identity

$$\det(1 - T^{\deg(\chi)} \mathrm{Fr}_{k_{\deg(\chi)}} | H_c^*(G_{\bar{k}}, M_{\chi}))^{-1} = \det(1 - T^{\deg(\eta)} \mathrm{Fr}_{k_{\deg(\eta)}} | H_c^*(G'_{\bar{k}}, (R_{\pi!}M_{\chi_0})_{\eta}))^{-1}$$

follows for any $\chi \in [S]$.

□

## 5.2. Objects with finite arithmetic tannakian groups on abelian varieties

As a non-trivial application of $\hat{L}$-functions, we will show that they lead to a characterization of objects with finite arithmetic tannakian groups on abelian varieties. This is an analogue of a result of Katz (see [75, Th. 6.2], recalled in Theorem B.2, (1)) for $G_m$, where in fact the $\hat{L}$-function appears implicitly (more precisely, where the logarithmic derivative $Td \log \hat{L}(M, T)$ appears); similar results appear in a preprint of Weissauer [126].

More generally, inspired by the formulation used by Katz, we can prove a stronger statement.

**DEFINITION 5.5** (Quasi-unipotent object). Let $G$ be a connected commutative algebraic group over $k$. An object $M$ of $D_c^b(G)$ is said to be quasi-unipotent if it is generically unramified and if there exists an integer $m \ge 1$ such that for any unramified character $\chi \in \hat{G}$, the eigenvalues of Frobenius on $H^0(G_{\bar{k}}, M_{\chi})$ are roots of unity of order at most $m$.

**REMARK 5.6.** (1) Any perverse sheaf $M$ on $G$ with $G_M^{\mathrm{ari}}$ finite is quasi-unipotent. Indeed, first $M$ is generically unramified by Corollary 3.39. Let then $m$ be the size of $G_M^{\mathrm{ari}}$. For any unramified character $\chi \in \hat{G}$, the Frobenius action on $H^0(G_{\bar{k}}, M_{\chi})$ is “conjugate” to an element of $G_M^{\mathrm{ari}}$, so its eigenvalues are $m$-th roots of unity.

(2) If $M$ is a quasi-unipotent perverse sheaf on $G$, then it follows from the definition that any object of $\langle M \rangle$ is also quasi-unipotent.

(3) Let $M$ be a quasi-unipotent object of $D_c^b(G)$. Let $g_0 \in G(k)$. Then the translated object $M' = [g \mapsto gg_0]^*M$ is also quasi-unipotent. Indeed, since $M'$ is canonically isomorphic to the convolution $\delta_{g_0} *! M$, we obtain for any $\chi \in \hat{G}$ a canonical isomorphism

$$H_c^*(G_{\bar{k}}, M'_{\chi}) \simeq H_c^*(G_{\bar{k}}, (\delta_{g_0^{-1}})_{\chi}) \otimes H_c^*(G_{\bar{k}}, M_{\chi}).$$

Noting that $H_c^*(G_{\bar{k}}, (\delta_{g_0^{-1}})_{\chi}) = H_c^0(G_{\bar{k}}, (\delta_{g_0^{-1}})_{\chi})$, this shows already that $\chi$ is weakly-unramified for $M$ if and only if it is for $M'$.

If $\chi$ is weakly-unramified for $M'$, and belongs to $\hat{G}(k_n)$, then the Frobenius automorphism of $k_n$ acts on $H^0(G_{\bar{k}}, (\delta_{g_0^{-1}})_{\chi})$ by multiplication by $\chi(g_0^{-1})$, which is a root of unity of order bounded by the order of $g_0$ in $G(k)$. Since $M$ is quasi-unipotent, the eigenvalues of Frobenius on $H^0(G_{\bar{k}}, M'_{\chi})$ are roots of unity of order bounded independently of $\chi$.

**THEOREM 5.7.** Let $A$ be an abelian variety over $k$. Let $M$ be an arithmetically semisimple perverse sheaf of weight zero in $\mathbf{P}^{\mathrm{ari}}(A)$ which is non-zero. If $M$ is quasi-unipotent, for instance if the group $G_M^{\mathrm{ari}}$ is finite, then $M$ is punctual.

**REMARK 5.8.** As proved by Katz in the case of $G_m$, one may expect that the conclusion of the theorem extends to objects with finite geometric tannakian group (see [75, Th. 6.4] or Theorem B.2, (2)). We do not know how to prove this in general (Katz’s deduction of this fact from the analogue of Theorem 5.7 for $G_m$ uses the classification of objects of tannakian rank 1, for instance, which
we do not have in this setting). We will however prove a weaker statement in Section 8.7 which turns out to be sufficient for many applications, including those of Chapter 11.

Before giving the proof, we state two corollaries.

**COROLLARY 5.9.** Let $M$ be an arithmetically simple perverse sheaf of weight zero on an abelian variety $A$ over $k$ of dimension $g \ge 1$. Let $G$ be the neutral component of $G_M^{\text{ari}}$, and let $S$ be the support of $M$. The restriction of the standard representation of $G_M^{\text{ari}}$ to $G$ is irreducible unless there exists $x \neq e$ such that $M * \delta_x$ is isomorphic to $M$. In particular, this holds unless there exists $x \in A$ with $x \neq e$ such that $x + S = S$.

**PROOF.** Let $P$ be an object of $\langle M \rangle^{\text{ari}}$ which is a faithful representation of the finite component group $C = G_M^{\text{ari}}/G$. Its tannakian group is isomorphic to $C$, and hence the object $P$ is punctual by Theorem 5.7. The points appearing in the decomposition of $P$ generate a finite subgroup $B$ of $A(\bar{k})$, and each skyscraper sheaf for $x \in B$ corresponds to a character $\chi_x$ of $G_M^{\text{ari}}$ trivial on $G$.

By a simple application of Frobenius reciprocity, a representation $\varrho$ of $G_M^{\text{ari}}$ restricts to an irreducible representation of $G$ unless there exists $x \in C$ such that $x \neq e$ and $\varrho \otimes \chi_x$ is isomorphic to $\varrho$. In terms of perverse sheaves on $A$, this condition (for the standard representation) means that $M * \delta_x$ is isomorphic to $M$, which is the first assertion. Since it also implies that $S + x = S$, this concludes the proof. $\square$

**COROLLARY 5.10.** Let $A$ be an abelian variety over $k$. Let $M$ be a non-zero arithmetically semisimple perverse sheaf of weight zero in $P_{\text{int}}^{\text{ari}}(A)$. If $M$ is quasi-unipotent, for instance if the group $G_M^{\text{ari}}$ is finite, then $G_M^{\text{geo}}$ is a finite abelian group which is naturally isomorphic to the dual of the subgroup of $A(\bar{k})$ generated by the support of $M$.

**PROOF.** By Theorem 5.7, the object $M$ is punctual. If $F$ denotes its support, we have an isomorphism

$$M = \bigoplus_{x \in F} \alpha_x^{\text{deg}} \otimes \delta_x$$

for some unitary scalars $\alpha_x$, and therefore a geometric isomorphism of $M$ with the direct sum of the $\delta_x$ for $x \in F$. Let $\hat{H}$ be the subgroup of $A(\bar{k})$ generated by $F$, which is a finite abelian group and let $\hat{H} = \text{Hom}(H, \overline{Q}_\ell^\times)$ be the dual group of $H$. We obtain an additive functor from the finite-dimensional $\overline{Q}_\ell$-representations of $\hat{H}$ to $\langle M \rangle^{\text{geo}}$ by associating to the character "evaluation at $x$" of $\hat{H}$ the object $\delta_x$. Since $\delta_x * \delta_y \simeq \delta_{x+y}$, this is a tensor functor, and it gives an equivalence of categories. Hence $G_M^{\text{geo}}$ is isomorphic to $\hat{H}$. $\square$

We will prove Theorem 5.7 in the next two sections. In fact, since this case is somewhat easier, we will begin by assuming that the abelian variety $A$ is simple (which is in a reasonable sense the generic case) before handling the general situation. The reader may skip the first case to read directly the proof of the general result.

We first prove two lemmas that are used in both proofs.

**LEMMA 5.11.** Let $R$ be a commutative ring with unit and $\lambda$ a non-archimedean valuation on $R$. Assume that $R$ is complete with the topology given by $\lambda$.

Let $(\alpha_i)_{i \in I}$ be a family of elements of $R$ such that $|\alpha_i|_{\lambda} \le 1$ for all $i \in I$, and let $(d_i)_{i \in I}$ be a family of positive integers such that

$$\lim_{I} d_i = +\infty,$$

where the limit is along the filter of the complements of finite subsets of $I$.
The product

$$ \prod_{i \in I} (1 - \alpha_i T^{d_i}) $$

converges and is non-zero for T such that $|T|_{\lambda} < 1$.

PROOF. Let $J \subset K$ be finite subsets of $I$. Then for $|T|_{\lambda} \le 1$, we compute that

$$ \begin{aligned} \left| \prod_{i \in K} (1 - \alpha_i T^{d_i}) - \prod_{i \in J} (1 - \alpha_i T^{d_i}) \right|_{\lambda} &= \left| \prod_{i \in J} (1 - \alpha_i T^{d_i}) \left( \prod_{i \in K \setminus J} (1 - \alpha_i T^{d_i}) - 1 \right) \right|_{\lambda} \\ &\le \left| \prod_{i \in K \setminus J} (1 - \alpha_i T^{d_i}) - 1 \right|_{\lambda} = \left| \sum_{\emptyset \ne L \subset K \setminus J} (-1)^{|L|} \sigma_L T^{d_L} \right|_{\lambda} \end{aligned} $$

where

$$ \sigma_L = \prod_{i \in L} \alpha_i, \quad d_L = \sum_{i \in L} d_i. $$

We note that $|\sigma_L|_{\lambda} \le 1$ for all $L$. Moreover, since the lower-bound

$$ d_L \ge \min_{i \in I \setminus J} d_i $$

holds, the assumption that $d_i \to +\infty$ implies that for any integer $N \ge 1$, we can choose $J$ so that

$$ \left| \sum_{\emptyset \ne L \subset K \setminus J} (-1)^{|L|} \sigma_L T^{d_L} \right|_{\lambda} \le |T|_{\lambda}^N $$

for any finite set $K$ containing $J$. The absolute convergence of the product follows when $|T|_{\lambda} < 1$ using the Cauchy criterion. In particular, the product can only be zero if some term is zero, and this is not the case if $|T|_{\lambda} < 1$. $\square$

The next lemma gives basic structural information on zeros and poles of $\hat{L}(M, T)$, refining the last part of Proposition 5.2 in the case of abelian varieties.

DEFINITION 5.12. Let $f \in \overline{\mathbb{Q}}_{\ell}(X)$ be a non-zero rational function, $k$ a finite field and $r \in \mathbb{Z}$. We denote by $\mathrm{wt}_{k,r}(f)$ the rational function

$$ \prod_{\alpha \text{ of } k\text{-weight } -r} (1 - \alpha T)^{v_{\alpha}(f)} $$

where $\alpha$ runs over elements of $\overline{\mathbb{Q}}_{\ell}$ which are $k$-Weil numbers of weight $-r$, and $v_{\alpha}$ is the order of $f$ at $\alpha$.

In other words (note the minus sign), the rational function $\mathrm{wt}_{k,r}(f)$ is (up to leading terms) “the part of $f$ with zeros and poles of weight $r$”. Below, we will sometimes write $\mathrm{wt}_r$ when the finite field $k$ is clear from context.

The definition implies that the identity

$$ \mathrm{wt}_{k,r}(f_1 f_2) = \mathrm{wt}_{k,r}(f_1) \mathrm{wt}_{k,r}(f_2) $$

holds for any rational functions $f_1$ and $f_2$.

PROPOSITION 5.13. Let $M$ be a complex on an abelian variety $A$ over $k$ of dimension $g \ge 0$. Assume that $M$ is pure of weight zero and that $M_e$ has weights in $[a, b]$.
(1) The poles (resp. zeros) of $\hat{\mathcal{L}}(M, T)$ are $k$-Weil numbers. Their weights are of the form $-w - i$ for some even (resp. odd) integer $i$ with $0 \le i \le 2g$ and some integer $w$ with $a \le w \le b$.

If there exists such a zero or pole then there exists an eigenvalue of weight $w$ on $M_e$, and the formula

$$ \mathrm{wt}_{k,-w}(\hat{\mathcal{L}}(M, T)) = \mathrm{wt}_{k,-w}(\det(1 - T \mathrm{Fr}_k | M_e))^{-1} $$
holds.

(2) If $M$ is an arithmetically simple perverse sheaf, and if $e$ belongs to the open set of the support of $M$ where $M$ is lisse, then the poles (resp. zeros) of $\hat{\mathcal{L}}(M, T)$ have $k$-weights equal to $\dim \mathrm{Supp}(M) - i$ for some integers $i$ with $0 \le i \le 2g$ such that

$$ \dim \mathrm{Supp}(M) \equiv i \pmod{2}, $$

and there are poles and zeros of all these possible weights.

**PROOF.** (1) By Proposition 5.2, we have

$$ \hat{\mathcal{L}}(M, T) = \exp\left(\sum_{n \ge 1} |A(k_n)| t_M(e; k_n) \frac{T^n}{n}\right). $$

This expression, combined with the purity of $M$ and the structure of the cohomology of $A$, shows that $\hat{\mathcal{L}}(M, T)$ has:

(i) Poles of the form

$$ T = \frac{1}{\alpha\beta}, $$

where $\alpha$ is an eigenvalue of Frobenius on the stalk of $M$ at $e$, and $\beta$ is an eigenvalue of Frobenius on $H^i(A_{\bar{k}}, \bar{Q}_\ell)$ for some even integer $i$ with $0 \le i \le 2g$. Since $\alpha$ is pure of some weight $w$ where $a \le w \le b$, and $\beta$ is of weight $i$, such a pole is a $|k|$-Weil number of weight $-w - i$.

(ii) Zeros of the form

$$ T = \frac{1}{\alpha\beta}, $$

where $\alpha$ is an eigenvalue of Frobenius on the stalk of $M$ at $e$, and $\beta$ is an eigenvalue of Frobenius on $H^i(A_{\bar{k}}, \bar{Q}_\ell)$ for some odd integer $i$ with $1 \le i \le 2g - 1$. As above, such a zero is a $|k|$-Weil number of weight $-w - i$ where $a \le w \le b$.

The precise formula for the parts of weight $-w$ follows from the above since $\beta = 1$ is the unique eigenvalue of weight 0 on $H^*(A_{\bar{k}}, \bar{Q}_\ell)$.

(2) If $M$ is an arithmetically simple perverse sheaf and $e$ is a point where $M$ is lisse, then the eigenvalues $\alpha$ above have weight $w = -\dim(\mathrm{supp}(M))$, and there is at least one $\alpha$ since the stalk at $e$ is non-zero. Thus the poles and zeros above have weight $\dim(\mathrm{supp}(M)) - i$. $\square$

### 5.3. Perverse sheaves with finitely many ramified characters

In this section, we prove Theorem 5.7 in the case of an arithmetically semisimple perverse sheaf of weight zero in $\mathbf{P}^{\mathrm{ari}}(A)$ which has the property that the set of ramified characters for $M$ is finite. This applies in particular, for instance, if the abelian variety $A$ is simple, since the set of ramified characters is a finite union of tacs of $A$ (see Remark 3.29), and each tac is reduced to a single character if $A$ is simple.
Let M be an arithmetically semisimple perverse sheaf of weight zero such that the set $\mathcal{S}$ of ramified characters for M is finite. We will prove the following:

**PROPOSITION 5.14.** *Under the above assumptions, if M is quasi-unipotent and non-punctual, then it is negligible.*

In Theorem 5.7, we assume that M is quasi-unipotent and that $M_{\text{int}}$ is non-zero; comparing with the proposition, it implies that M must be punctual.

We now prove the proposition. After a finite extension of $k$, we may assume that each $\chi \in \mathcal{S}$ is in $\hat{A}(k)$.

One reduces using Lemma 1.28 to the case of M geometrically simple. We denote by S the support of M and by $r$ its dimension; we have $r \ge 1$ since M is not punctual. We denote by U a smooth open dense subset of S such that M is lisse on U.

Let $n \ge 1$ and let $a \in A(k_n)$. We denote $M^{(a)} = [x \mapsto x + a]^*M$, which is a simple perverse sheaf on $A_{k_n}$. The stalk of $M^{(a)}$ at $e$ is canonically isomorphic to the stalk $M_a$ of M at $a$. We note that the set of ramified characters for $M^{(a)}$ is also contained in $\mathcal{S}$, and that $M^{(a)}$ is quasi-unipotent (see Remark 5.6, (3)).

We then write

$$ \hat{\mathcal{L}}(M^{(a)}, T) = \hat{\mathcal{L}}_0(M^{(a)}, T) \prod_{\chi \in \mathcal{S}} \det(1 - T \mathrm{Fr}_k \mid \mathrm{H}^*(A_{\bar{k}}, M_{\chi}^{(a)}))^{-1} $$

where

$$ \hat{\mathcal{L}}_0(M^{(a)}, T) = \prod_{\chi \notin \mathcal{S}} \det(1 - T^{\mathrm{deg}(\chi)} \mathrm{Fr}_{k_{\mathrm{deg}(\chi)}} \mid \mathrm{H}^0(A_{\bar{k}}, M_{\chi}^{(a)}))^{-1}. $$

Note that $\hat{\mathcal{L}}_0(M^{(a)}, T)$ is a rational function since $\hat{\mathcal{L}}(M^{(a)}, T)$ is one (Proposition 5.2).

The quasi-unipotence property of $M^{(a)}$ shows that the infinite product $\hat{\mathcal{L}}_0(M^{(a)}, T)$ can be viewed as a formal power series in $\mathcal{O}[[T]]$ for some cyclotomic order $\mathcal{O}$. We can apply Lemma 5.11 to any non-archimedean place $\lambda$ of $\mathcal{O}$, since the eigenvalues of Frobenius on $\mathrm{H}^0(A_{\bar{k}}, M_{\chi}^{(a)})$ are roots of unity of bounded order for all unramified characters $\chi$. This implies that, for any non-archimedean place $\lambda$, the infinite product $\hat{\mathcal{L}}_0(M^{(a)}, T)$ converges in the disc defined by $|T|_{\lambda} < 1$. Taking $\lambda$ to correspond to places above the characteristic of $k$, this implies that the rational function $\hat{\mathcal{L}}_0(M^{(a)}, T)$ cannot have a zero or pole which is a $|k|$-Weil number of positive weight.

Suppose that $a \in (A - S)(\bar{k})$. Then $M_e^{(a)} = 0$. Hence we deduce that

$$ (5.4) \qquad 1 = \prod_{\chi \in \mathcal{S}} \mathrm{wt}_r(\det(1 - T \mathrm{Fr}_k \mid \mathrm{H}^*(A_{\bar{k}}, M_{\chi}^{(a)}))). $$

Thus, the Frobenius automorphism has no eigenvalue of weight $-r$ acting on any of the cohomology spaces $\mathrm{H}^i(A_{\bar{k}}, M_{\chi}^{(a)})$. By purity, this translates to the condition

$$ \mathrm{H}^{-r}(A_{\bar{k}}, M_{\chi}^{(a)}) = 0 $$

for all $\chi \in \mathcal{S}$.

On the other hand, suppose that $a \in U(\bar{k})$. Then $M_e^{(a)} = M_a$ is pure of weight $-r$. From the above and Proposition 5.13, (2), we deduce that

$$ (5.5) \qquad \hat{\mathcal{L}}(M^{(a)}, T) = \mathrm{wt}_r(\hat{\mathcal{L}}(M^{(a)}, T)) = \prod_{\chi \in \mathcal{S}} \mathrm{wt}_r(\det(1 - T \mathrm{Fr}_k \mid \mathrm{H}^*(A_{\bar{k}}, M_{\chi}^{(a)}))), $$
and since the left-hand side is not 1, there exists (by purity again) at least one $\chi \in \mathcal{S}$ such that

$$H^{-r}(A_{\bar{k}}, M_{\chi}^{(a)}) \neq 0.$$

If we combine these two statements, we conclude that $S = A$. Indeed, the spaces $H^{-r}(A_{\bar{k}}, M_{\chi}^{(a)})$ are independent of $a \in A(\bar{k})$ up to isomorphism. Hence, since there exists some $a_0 \in U(\bar{k})$, if one of these spaces is non-zero, then no $a \in A(\bar{k})$ can satisfy the condition required to have $a \notin S(\bar{k})$.

Fixing again $a_0 \in U(\bar{k})$, let $\chi \in \mathcal{S}$ be such that

$$H^{-r}(A_{\bar{k}}, M_{\chi}^{(a_0)})$$

is non-zero. Since $M^{(a_0)}$ is a simple perverse sheaf supported on $S = A$, and $r = \dim(S) = \dim(A)$, it follows from Lemma A.15 that $M_{\chi}^{(a_0)}$ is geometrically trivial. This implies that $M$ is negligible.

## 5.4. The general case

In this section, we prove Theorem 5.7 in the general case. Thus let $M$ be an arithmetically semisimple perverse sheaf of weight zero in $\mathbf{P}_{\text{int}}^{\text{ari}}(A)$, which we assume is quasi-unipotent and not punctual. We will show that $M$ is negligible.

It suffices to treat the case of a simple perverse sheaf $M$ (Lemma 1.28).

We denote by $S$ the support of $M$ and by $r$ its dimension; we have $r \ge 1$ by our assumption that $M$ is not punctual. Let $U$ be an open dense subset of $S$ contained in the smooth locus of $S$ such that $M$ is lisse on $U$.

Let $(\mathcal{S})_{i \in I}$ be a finite family of tacs such that the set of ramified characters is contained in the union $\mathcal{S}$ of the $\mathcal{S}_i$. After a finite extension of $k$, we may assume that each $\mathcal{S}_i$ is defined by a quotient morphism $\pi_i: A \to A_i$ defined over $k$ and a character $\chi_i \in \hat{A}(k)$.

For any subset $J$ of $I$, we denote by $\mathcal{S}_J$ the intersection of $\mathcal{S}_i$ for $i \in J$; this is either empty or a tac of $A$, also defined over $k$ (Lemma 1.25), in which case we denote by $\pi_J: A \to A_J$ and $\chi_J$ the corresponding quotient morphism and character; these are all defined over $k$. From Lemma 1.25, it follows also that $\ker(\pi_J)$ is the algebraic subgroup of $A$ generated by the family of subgroups $(\ker(\pi_i))_{i \in I}$. We write $[\mathcal{S}_J]$ for the set of classes in $[\hat{A}]$ of characters in $\mathcal{S}_J$.

Let $a \in A(k)$. We denote $M^{(a)} = [x \mapsto x+a]^*M$, so that the stalk $M_e^{(a)}$ is canonically isomorphic to the stalk $M_a$ of $M$ at $a$. The ramified characters for $M^{(a)}$ are also contained in the tac $\mathcal{S}$, and the perverse sheaf $M^{(a)}$ is quasi-unipotent (see Remark 5.6, (3)).

We define

$$\hat{L}_0(M^{(a)}, T) = \prod_{\chi \in [\hat{A}] - \mathcal{S}} \det(1 - T^{\deg(\chi)} \mathrm{Fr}_{k_{\deg(\chi)}} \mid \mathrm{H}^0(A_{\bar{k}}, M_{\chi}^{(a)}))^{-1}.$$

By an application of inclusion-exclusion, we have

$$\hat{L}(M^{(a)}, T) = \hat{L}_0(M^{(a)}, T) \prod_{\emptyset \neq J \subseteq I} \prod_{\mathcal{S}_J \neq \emptyset} \det(1 - T^{\deg(\chi)} \mathrm{Fr}_{k_{\deg(\chi)}} \mid \mathrm{H}^*(A_{\bar{k}}, M_{\chi}^{(a)}))^{(-1)^{|J|}}.$$

For any $J \subset I$ such that $\mathcal{S}_J$ is not empty, we denote

$$Q_J^{(a)} = R\pi_{J*}M_{\chi J}^{(a)}.$$
Proposition 5.4 implies the formula

$$
(5.6) \qquad \prod_{\chi \in [\mathcal{S}_J]} \det(1 - T^{\deg(\chi)} \mathrm{Fr}_{k_{\deg(\chi)}} \mid \mathrm{H}^*(A_{\bar{k}}, \mathrm{M}^{(a)}))^{-1} = \hat{\mathrm{L}}(\mathrm{Q}_J^{(a)}, \mathrm{T}),
$$

so that we can rewrite the above expression as

$$
(5.7) \qquad \hat{\mathrm{L}}(\mathrm{M}^{(a)}, \mathrm{T}) = \hat{\mathrm{L}}_0(\mathrm{M}^{(a)}, \mathrm{T}) \prod_{\substack{\emptyset \neq J \subset I \\ \mathcal{S}_J \neq \emptyset}} \hat{\mathrm{L}}(\mathrm{Q}_J^a, \mathrm{T})^{(-1)^{|J|+1}}.
$$

By Proposition 5.2 (2), this shows in particular that $\hat{\mathrm{L}}_0(\mathrm{M}^{(a)}, \mathrm{T})$ is a rational function. The quasi-unipotence property of $\mathrm{M}^{(a)}$ shows that the infinite product $\hat{\mathrm{L}}_0(\mathrm{M}^{(a)}, \mathrm{T})$ can be viewed as a formal power series in $\mathcal{O}[[\mathrm{T}]]$ for some cyclotomic order $\mathcal{O}$. We can apply Lemma 5.11 to any non-archimedean place $\lambda$ of $\mathcal{O}$, since the eigenvalues of Frobenius on $\mathrm{H}^0(A_{\bar{k}}, \mathrm{M}^{(a)}_\chi)$ are roots of unity of bounded order for all unramified characters $\chi$. This implies that, for any non-archimedean place $\lambda$, the infinite product $\hat{\mathrm{L}}_0(\mathrm{M}^{(a)}, \mathrm{T})$ converges in the disc defined by $|\mathrm{T}|_\lambda < 1$. Taking $\lambda$ to correspond to places above the characteristic of $k$, this implies that the rational function $\hat{\mathrm{L}}_0(\mathrm{M}^{(a)}, \mathrm{T})$ cannot have a zero or pole which is a $|k|$-Weil number of positive weight.

Since $r \ge 1$, the formula (5.7) therefore implies the formula

$$
(5.8) \qquad \mathrm{wt}_r(\hat{\mathrm{L}}(\mathrm{M}^{(a)}, \mathrm{T})) = \prod_{\substack{\emptyset \neq J \subset I \\ \mathcal{S}_J \neq \emptyset}} \mathrm{wt}_r \left( \hat{\mathrm{L}}(\mathrm{Q}_J^a, \mathrm{T})^{(-1)^{|J|+1}} \right).
$$

Let $J \subset I$. By proper base change, we have a canonical isomorphism

$$
\mathcal{Q}_{J,e}^{(a)} = (\mathrm{R}\pi_{J*}\mathcal{M}_{\chi_J}^{(a)})_e \simeq \mathrm{H}^*(\ker(\pi_J)_{\bar{k}}, \mathcal{M}_{\chi_J}^{(a)}).
$$

Since $M^{(a)}$, hence also $M_{\chi_J}^{(a)}$, is a perverse sheaf, the complex $M_{\chi_J}^{(a)}$ is concentrated in degrees between $-r$ and $r$. Its support is $S-a$, and consequently, the cohomology group

$$
\mathrm{H}^i(\ker(\pi_J)_{\bar{k}}, \mathrm{M}_{\chi_J}^{(a)}) = \mathrm{H}^i((\ker(\pi_J) \cap (S-a))_{\bar{k}}, \mathrm{M}_{\chi_J}^{(a)})
$$

vanishes unless $0 \le i + r \le 2 \dim(\ker(\pi_J) \cap (S-a))$. Since $M^{(a)}$ has weight 0, this space has weight $i$ when it is non-zero. Using the formula

$$
\hat{\mathrm{L}}(\mathrm{Q}_J^{(a)}, \mathrm{T}) = \exp\left(\sum_{n \ge 1} |\mathrm{A}_J(k_n)| t_{\mathrm{Q}_J^{(a)}}(e; k_n) \frac{\mathrm{T}^n}{n}\right)
$$

of Proposition 5.2, this means that

$$
\begin{align*}
\mathrm{wt}_r(\hat{\mathrm{L}}(\mathrm{Q}_J^{(a)}, \mathrm{T})) &= \det(1 - \mathrm{T} \mathrm{Fr}_k \mid \mathrm{H}^{-r}((\ker(\pi_J) \cap (S-a))_{\bar{k}}, \mathrm{M}_{\chi_J}^{(a)}))^{-1} \\
&= \det(1 - \mathrm{T} \mathrm{Fr}_k \mid \mathrm{H}^{-r}(((a + \ker(\pi_J)) \cap \mathrm{S})_{\bar{k}}, \mathrm{M}_{\chi_J}^{(a)}))^{-1} \\
&= \det(1 - \mathrm{T} \mathrm{Fr}_k \mid (\mathrm{R}^{-r} \pi_{J*} \mathrm{M}_{\chi_J})_{\pi_J(a)})^{-1} = \det(1 - \mathrm{T} \mathrm{Fr}_k \mid (\pi_J^* \mathrm{R}^{-r} \pi_{J*} \mathrm{M}_{\chi_J})_a)^{-1}.
\end{align*}
$$

Let $X = S - U$, so that $A - X = U \cup (A - S)$. If $a \in (A - X)(k)$, the left-hand side of (5.8) is the part of weight $-r$ of

$$
\hat{\mathrm{L}}(\mathrm{M}^{(a)}, \mathrm{T}) = \exp\left(\sum_{n \ge 1} |\mathrm{A}(k_n)| t_{\mathrm{M}}(a; k_n) \frac{\mathrm{T}^n}{n}\right).
$$
Since $M_a$ is $|k|$-pure of weight $-r$ (either because $a \in U(k)$, so that $M$ is lisse and of weight $-r$ at $a$, or because $a \in (A-S)(k)$, so that $M_a$ is zero, hence pure of any weight), we deduce that the equality

$$ (5.9) \qquad \det(1 - T \mathrm{Fr}_k | M_a) = \prod_{\substack{\emptyset \neq J \subset I \\ \mathcal{S}_J \neq \emptyset}} \det(1 - T \mathrm{Fr}_k | (\pi_J^* \mathrm{R}^{-r} \pi_{J_*} M_{\chi_J})_a)^{(-1)^{|J|+1}} $$

holds. In particular, this gives the equality

$$ t_M(a; k) = \sum_{\substack{\emptyset \neq J \subset I \\ \mathcal{S}_J \neq \emptyset}} (-1)^{|J|+1} t_{\pi_J^* \mathrm{R}^{-r} \pi_{J_*} M_{\chi_J}}(a; k) $$

of values of trace functions for $a \in (A - X)(k)$.

Let $n \ge 1$. Applying this argument to the base change of $M$ to $k_n$, we see that the formula

$$ t_M(a; k_n) = \sum_{\substack{\emptyset \neq J \subset I \\ \mathcal{S}_J \neq \emptyset}} (-1)^{|J|+1} t_{\pi_J^* \mathrm{R}^{-r} \pi_{J_*} M_{\chi_J}}(a; k_n) $$

holds for $a \in (A - X)(k_n)$. By the injectivity of trace functions (see [98, Th. 1.1.2]), this means that we have an equality

$$ (5.10) \qquad M = \sum_{\substack{\emptyset \neq J \subset I \\ \mathcal{S}_J \neq \emptyset}} (-1)^{|J|+1} \pi_J^* \mathrm{R}^{-r} \pi_{J_*} M_{\chi_J} $$

in the Grothendieck group $K(A - X)$.

If $U = S$ (e.g. if $M$ is the extension by zero of a lisse sheaf of weight 0 placed in degree $-r$ on a smooth closed subvariety $S$, which will be the case in the applications of Theorem 5.7 in Chapter 11), then $X$ is empty, so this equality holds in $K(A)$. The right-hand side is a linear combination of negligible objects (see Example 3.5) so we deduce that $M$ is negligible by taking the Euler-Poincaré characteristic (see Corollary 3.23).

We now consider the general case. Let $j$ be the open immersion of $A - X$ in $A$. Recall that the classes of simple perverse sheaves form a basis of the $\mathbb{Z}$-module $K(A - X)$ (see Proposition A.22). Thus, the equality (5.10) implies that there exists some $J$ such that the simple perverse sheaf $j^*M$ appears in the decomposition in simple perverse sheaves of the class of $j^*N$ in $K(A - X)$, where

$$ N = \pi_J^* \mathrm{R}^{-r} \pi_{J_*} M_{\chi_J}. $$

Furthermore, this means that there exists $i \in \mathbb{Z}$ such that $j^*M$ occurs in the decomposition of the perverse sheaf $\mathfrak{p}\mathcal{H}^i(j^*N)$, since

$$ j^*N = \sum_{i \in \mathbb{Z}} (-1)^i \mathfrak{p}\mathcal{H}^i(j^*N) $$

in $K(A - X)$.

The functor $j^*$ is $t$-exact (since $j$ is smooth of relative dimension 0) so that there exists a canonical isomorphism $\mathfrak{p}\mathcal{H}^i(j^*N) \to j^*\mathfrak{p}\mathcal{H}^i(N)$. Since $j^*M$ and $j^*\mathfrak{p}\mathcal{H}^i(N)$ are pure, hence geometrically semisimple, this implies the existence of an injective morphism

$$ f: j^*M \to j^*\mathfrak{p}\mathcal{H}^i(N) $$

of perverse sheaves. Applying the functor $j_{!*}$, which preserves injectivity (e.g., by [71, §.2.17]) and satisfies $j_{!*} \circ j^* = \mathrm{Id}$ on perverse sheaves, we deduce that there exists an injective morphism $j_{!*}f: M \to \mathfrak{p}\mathcal{H}^i(N)$. Since $N$ is negligible, so is $\mathfrak{p}\mathcal{H}^i(N)$, and hence also $M$.
REMARK 5.15. A similar argument leads to a proof of the following fact: if M is a negligible arithmetically simple perverse sheaf of weight zero in $\mathbf{P}_{\text{int}}^{\text{ari}}(A)$, and if $\mathcal{N}(M)$ is contained in a finite union of tacs of $A$, then there exists a morphism $\pi: A \to B$ of abelian varieties with $\dim(\ker(\pi)) \ge 1$, a character $\chi \in \hat{A}$ and an object $N$ of $\mathcal{D}_c^b(B)$ such that $M$ is geometrically isomorphic to $(\pi^*N)_\chi$.

This fact is equivalent (for pure perverse sheaves of weight 0) to the characterization of negligible objects by Weissauer [127, Th. 3], since it is known that the assumption on $\mathcal{N}(M)$ is always true (Corollary 3.24). However, the proof that this is so relies on the generic vanishing theorem (Theorem 2.19), which appeals to this result of Weissauer, so this remark does not provide a different proof of this characterization.

We sketch the argument nevertheless for the sake of illustration. It is relatively elementary that it suffices to prove that the isomorphism class of M is invariant under translation by a non-trivial abelian subvariety (this is [127, Lemma 6]), and we will establish this fact.

To simplify matters, we assume that $S = U$ in the notation of the previous proof. Since M is negligible, it is quasi-unipotent; arguing as in the previous proof, we obtain a finite decomposition

$$M = \sum_{i \in I} n_i \pi_i^* M_i$$

in $K(A)$ for some morphisms $\pi_i: A \to A_i$ with $\dim(\ker(\pi_i)) \ge 1$, some objects $M_i \in \mathcal{D}_c^b(A_i)$ and some non-zero $n_i \in \mathbf{Z}$.

Since the classes of simple perverse sheaves form a basis of the $\mathbf{Z}$-module $K(A)$, there exists some $i \in I$ such that

$$\pi_i^* M_i = mM + \sum_{j \in J} m_j M_{i,j}$$

in $K(A)$ for some non-zero integers $m$ and $m_j$ and some simple perverse sheaves $M_{i,j}$ not isomorphic to $M$.

The isomorphism class of the complex $\pi_i^* M_i$ is invariant under translation by elements of $\ker(\pi_i)$, and a fortiori by the abelian subvariety $A' = \ker(\pi_i)^\circ$. We claim that this implies that the same property holds for $M$ and the other constituents $M_{i,j}$. Indeed, the $\bar{k}$-valued points of $A'$ act on the finite set of isomorphism classes of the simple perverse sheaves $(M, M_{i,j})$, and thus the stabilizer of any of them is a finite index subgroup. Since it is also an algebraic subgroup, it is equal to $A'(\bar{k})$, and the assertion follows. Thus the isomorphism class of $M$ is invariant under translation by the non-trivial abelian variety $\ker(\pi_i)^\circ$, as desired.
# Stratification and generic Fourier invertibility

As usual, $k$ is a finite field, with an algebraic closure $\bar{k}$ and finite extensions $k_n$ of $k$ in $\bar{k}$ of degree $n$. We fix a prime $\ell$ distinct from the characteristic of $k$.

Let $G$ be a connected commutative algebraic group over $k$, with dimension $d$. Given an object $\mathrm{M}$ of $\mathrm{D}_c^b(G)$, an integer $n \ge 1$ and a character $\chi \in \hat{G}(k_n)$, we set

$$S(\mathrm{M}, \chi) = \sum_{x \in \hat{G}(k_n)} \chi(x) t_{\mathrm{M}}(x; k_n).$$

## 6.1. Stratification for exponential sums

The results of this section are straightforward consequences of Theorem 2.3 and Deligne’s Riemann Hypothesis. We spell them out since some of them are likely to be useful for applications to analytic number theory.

**THEOREM 6.1.** Let $\mathrm{M}$ be an object of $\mathrm{D}_c^b(G)$. Assume that $\mathrm{M}$ is semiperverse and mixed of weights $\le 0$. There exist subsets $\mathcal{S}_d \subset \dots \subset \mathcal{S}_0 = \hat{G}$ such that

(1) For $0 \le i \le d$, the estimate

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

holds for $n \ge 1$.

(2) The set $\mathcal{S}_d$ is empty if $\mathrm{M}$ belongs to the category $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G)$.

(3) For any $n \ge 1$, any integer $i$ with $1 \le i \le d$ and any $\chi \in \hat{G}(k_n) - \mathcal{S}_i(k_n)$, the estimate

$$\sum_{x \in \hat{G}(k_n)} \chi(x) t_{\mathrm{M}}(x; k_n) \ll c_u(\mathrm{M}) |k|^{n(i-1)/2},$$

holds, where the implied constant is independent of $\mathrm{M}$.

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

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

**PROOF.** For $1 \le i \le d$, let $\mathcal{S}_i$ be the set of characters such that there exists some $l \ge i$ with $H_c^l(\mathrm{G}_{\bar{k}}, \mathrm{M}_\chi) \ne 0$.

If $\chi \in \hat{G} - \mathcal{S}_i$, then we deduce from the trace formula and the Riemann Hypothesis of Deligne, combined with Lemma 1.3, that the estimate

$$\sum_{x \in \hat{G}(k_n)} \chi(x) t_{\mathrm{M}}(x; k_n) \ll c_u(\mathrm{M}) |k|^{n(i-1)/2}$$

holds for $n \ge 1$, which is (3). We will check that these sets also satisfy conditions (1) and (2).
Fix an integer $i$ with $1 \le i \le d$. For any $l$ and $j$, we have the perverse spectral sequence

$$H_c^j(G_{\bar{k}}, \mathfrak{p}\mathcal{H}^{l-j}(M_{\chi})) \implies H_c^j(G_{\bar{k}}, M_{\chi})$$

(see (A.3)) so that the condition $\chi \in \mathcal{S}_i$ implies that

$$H_c^j(G_{\bar{k}}, \mathfrak{p}\mathcal{H}^{l-j}(M_{\chi})) \neq 0$$

for some $l \ge i$. Since $M$ is semiperverse, so is $M_{\chi}$, which means that this condition implies $j \ge l \ge i$.

Thus, if we denote by $(\mathcal{S}_{j,i})_{0 \le i \le d}$ the sets provided by the Stratified Vanishing Theorem 2.3 applied to $\mathfrak{p}\mathcal{H}^j(M)$, we have shown that

$$\mathcal{S}_i \subset \bigcup_{i \le j \le d} \bigcup_{i \le l \le j} \mathcal{S}_{l-j,j}.$$

The set $\mathcal{S}_{l-j,j}$ has character codimension at least $j$, so that $\mathcal{S}_i$ has the same property, establishing (1). Point (2) follows from the fact that

$$H_c^d(G_{\bar{k}}, N_{\chi}) = 0$$

for a geometrically simple perverse sheaf $N$ which is not geometrically isomorphic to $\mathcal{L}_{\chi^{-1}}$.

Points (4) and (5) follow from the strengthened versions of the Stratified Vanishing Theorem for tori, abelian varieties and unipotent groups, which are stated in Theorem 2.3, (4) or Proposition 2.9. $\square$

REMARK 6.2. The following elementary estimate can also sometimes be useful. Fix a locally-closed immersion $u: G \to \mathbf{P}^m$ for some integer $m \ge 1$. Let $M$ be an $\ell$-adic perverse sheaf on $G$ that is pure of weight zero. Then by orthogonality of characters, we derive that the formula

$$\frac{1}{|G(k_n)|} \left| \sum_{\chi \in \hat{G}(k_n)} t_M(x; k_n) \chi(x) \right|^2 = \sum_{x \in G(k_n)} |t_M(x; k_n)|^2$$

holds for $n \ge 1$. By the Riemann Hypothesis, it follows that the estimate

$$\frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} \left| \sum_{\chi \in \hat{G}(k_n)} t_M(x; k_n) \chi(x) \right|^2 \ll c_u(M)$$

holds for $n \ge 1$ (see Theorem 1.10). Fix then a sequence $T = (T_n)$ of positive real numbers, and let $\mathcal{X}_T \subset \hat{G}$ be the set such that $\chi \in \mathcal{X}_T(k_n)$ if and only if

$$\left| \sum_{x \in G(k_n)} t_M(x; k_n) \chi(x) \right| > T_n.$$

Then we find by positivity that

$$|\mathcal{X}_T(k_n)| \ll c_u(M) |G(k_n)| T_n^{-2}.$$

COROLLARY 6.3. Let $k$ be a finite field, and let $G$ be a connected commutative algebraic group of dimension $d$ over $k$. Let $\ell$ be a prime distinct from the characteristic of $k$ and let $M$ be an $\ell$-adic perverse sheaf on $G$ which is pure of weight zero.

For any generic subsets $\mathcal{X}$ and $\mathcal{Y}$ of $\hat{G}$, the estimate

$$\frac{1}{|G(k_n)|} \sum_{\chi \in (\mathcal{X} - \mathcal{Y})(k_n)} \left| \sum_{x \in k_n} \chi(x) t_M(x; k_n) \right| \ll \frac{1}{|k|^{n/2}}$$

holds for all $n \ge 1$.
PROOF. We may assume that $\mathcal{X} = \hat{\mathbb{G}}$. Let $(\mathcal{S}_i)_{0\le i\le d}$ be sets of characters as in Theorem 6.1. We have

$$ \frac{1}{|\mathbb{G}(k_n)|} \sum_{\chi \in (\hat{\mathbb{G}} - \mathcal{Y})(k_n)} \left| \sum_{x \in k_n} \chi(x) t_{\mathbb{M}}(x; k_n) \right| = S_0 + \dots + S_{d-1}, $$

where for each integer $i$ with $0 \le i < d$, we put

$$ S_i = \frac{1}{|\mathbb{G}(k_n)|} \sum_{\chi \in (\mathcal{S}_i - (\mathcal{S}_{i+1} \cup \mathcal{Y}))(k_n)} \left| \sum_{x \in k_n} \chi(x) t_{\mathbb{M}}(x; k_n) \right|. $$

For $i=0$, the exponential sums in the inner sum are $\ll 1$, and since $\mathcal{Y}$ is generic, the set $\mathcal{S}_0 - (\mathcal{S}_1 \cup \mathcal{Y})$ has character codimension at least 1, so that

$$ S_0 \ll |k|^{n(-d+(d-1))} = |k|^{-n}. $$

For $i \le i < d$, we have $\mathcal{S}_i - (\mathcal{S}_{i+1} \cup \mathcal{Y}) \subset \mathcal{S}_i - \mathcal{S}_{i+1}$, so that by Theorem 6.1, (1) (for the size of $\mathcal{S}_i$) and (3) (estimating the exponential sums for $\chi \notin \mathcal{S}_{i+1}$), we obtain

$$ S_i \ll |k|^{-nd+n((d-i)+i/2)} = |k|^{-ni/2}. $$

□

The next corollary states, intuitively, that for the purpose of computing the arithmetic Fourier transform of a semiperverse complex (mixed of weights $\le 0$), the contribution of any closed (suitably “transverse”) subvariety is negligible.

COROLLARY 6.4. Let $k$ be a finite field, and let $G$ be a connected commutative algebraic group of dimension $d$ over $k$. Let $\ell$ be a prime distinct from the characteristic of $k$ and let $M$ be an object of $\mathrm{D}^b_c(G)$. Assume that $M$ is semiperverse and mixed of weights $\le 0$.

Let $X \subset G$ be a closed subvariety of $G$ and let $i: X \to G$ be the corresponding closed immersion.

Let $m \ge 0$ be an integer such that for each $j \in \mathbb{Z}$, the estimate

$$ \dim(X \cap \mathrm{Supp}(\mathcal{H}^j(M))) \le \dim \mathrm{Supp}(\mathcal{H}^j(M)) - m $$

holds.

There exists a generic subset $\mathcal{X}$ of $\hat{\mathbb{G}}$ such that the estimate

$$ \sum_{x \in X(k_n)} \chi(x) t_{\mathbb{M}}(x; k_n) \ll \frac{c_u(M)}{|k_n|^{m/2}} $$

holds for all $n \ge 1$ and all $\chi \in \mathcal{X}(k_n)$.

Alternatively, we have

$$ S(M, \chi) = \sum_{x \in (G-X)(k_n)} \chi(x) t_{\mathbb{M}}(x; k_n) + O(|k_n|^{-m/2}), $$

which explains the interpretation that $X$ does not contribute “systematically” to the arithmetic Fourier transform.

PROOF. The assumption implies that the complex $N = i_i i^* M[-m](-m/2)$ is semiperverse on $G$, since $M$ is semiperverse and, for any $j \in \mathbb{Z}$, the support of $\mathcal{H}^j(N)$ is $X \cap \mathrm{Supp}(\mathcal{H}^{j-m}(M))$ so that

$$ \dim(\mathrm{Supp}(\mathcal{H}^j(N))) = \dim(X \cap \mathrm{Supp}(\mathcal{H}^{j-m}(M))) \le \dim(\mathrm{Supp}(\mathcal{H}^{j-m}(M))) - m \\ \le -(j-m) - m = -j. $$
Moreover, the complex N has weights $\le 0$. Thus we may apply Theorem 6.1 to N. Let $\mathcal{S}_0, \dots, \mathcal{S}_d$ be the corresponding sets of characters, and let $\mathcal{X} = \hat{\mathcal{G}} - \mathcal{S}_1$. This is a generic subset of $\hat{\mathcal{G}}$, and for $n \ge 1$ and $\chi \in \mathcal{X}(k_n)$, we have

$$(-1)^m |k_n|^{m/2} \sum_{x \in \mathcal{X}(k_n)} \chi(x) t_{\mathcal{M}}(x; k_n) = \sum_{x \in \mathcal{G}(k_n)} \chi(x) t_{\mathcal{N}}(x; k_n) \ll c_u(\mathcal{M}),$$

hence the result.

<figure><img src="image_11.png" /></figure>

**EXAMPLE 6.5.** Let $\mathcal{F}$ be a non-zero lisse sheaf on $\mathcal{G}$, pure of weight 0, and let $M = \mathcal{F}[d](d/2)$. We then have $\mathrm{Supp}(\mathcal{H}^j(\mathcal{M})) = \emptyset$ except when $j = -d$, in which case the support of $\mathcal{H}^{-d}(\mathcal{M})$ is $\mathcal{G}$. We can therefore apply the corollary to any closed subvariety $X$ of $\mathcal{G}$ of codimension at least $m$. In particular, for any closed subvariety $X \ne \mathcal{G}$, hence of codimension at least 1, there exists a generic set of characters $\mathcal{X}$ for which the estimate

$$\sum_{x \in \mathcal{X}(k_n)} \chi(x) t_{\mathcal{M}}(x; k_n) \ll \frac{c_u(\mathcal{M})}{|k_n|^{1/2}}$$

holds for $n \ge 1$ and $\chi \in \mathcal{X}(k_n)$.

A uniform version of the stratified vanishing theorem, as in Remark 2.4, would be especially welcome for stratification estimates, as it would lead to strong potential applications in analytic number theory (compare with the results of Fouvry and Katz [39] based on stratification for the additive Fourier transform). We state a conditional result of this kind for emphasis.

**THEOREM 6.6.** Let $\ell$ be a prime number. Let $N \ge 1$ be an integer and let $(G, u)$ be a quasi-projective commutative group scheme over $\mathbb{Z}[1/\ell N]$.

Assume that, for all primes $p \nmid \ell N$, the fiber $G_p$ of $G$ over $\mathbf{F}_p$ is a connected commutative algebraic group such that Theorem 2.3 holds uniformly with respect to the complexity $c_{u_p}(\mathcal{M})$ where $u_p$ is the locally-closed immersion of $G_p$ deduced from $u$, i.e., such that for a perverse sheaf $M$ on $G_p$, the sets $\mathcal{S}_i$ in loc. cit. satisfy

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

where the implied constant depends only on $c_{u_p}(\mathcal{M})$.

Let $(M_p)_{p \nmid N\ell}$ be a sequence of arithmetically semisimple sheaves on $G_p$, pure of weight zero, such that $c_u(M_p) \ll 1$ for all $p$.

For each prime $p$, there exist subsets $\mathcal{S}_d(\mathbf{F}_p) \subset \dots \subset \mathcal{S}_0(\mathbf{F}_p) = \hat{\mathcal{G}}_p(\mathbf{F}_p)$ such that

(1) For $0 \le i \le d$ and $p$ prime, we have

$$|\mathcal{S}_i(\mathbf{F}_p)| \ll p^{d-i}.$$

(2) The set $\mathcal{S}_d(\mathbf{F}_p)$ is empty if $M_p$ belongs to the category $\mathbf{P}_{\mathrm{int}}^{\mathrm{ari}}(G_p)$.

(3) For any prime $p$, any integer $i$ with $0 \le i \le d$ and any $\chi \in \hat{\mathcal{G}}_p(\mathbf{F}_p) - \mathcal{S}_i(\mathbf{F}_p)$, we have

$$\sum_{x \in \mathcal{G}(\mathbf{F}_p)} \chi(x) t_p(x) \ll p^{(i-1)/2},$$

where $t_p$ is the trace function of $M_p$ over $\mathbf{F}_p$.

(4) If $G$ is a torus or an abelian variety, then the sets $\mathcal{S}_i(\mathbf{F}_p)$ are contained in the union of a bounded number of tacs of $G_{\mathbf{F}_p}$ of dimension $\le d-i$.

**REMARK 6.7.** For $G = \mathbf{G}_a^d$, results of this kind are unconditional; see for instance [39, Th. 1.1, Th. 3.1] (note that there the sets $\mathcal{S}_i$ are points of subschemes defined over $\mathbb{Z}$, which we cannot hope in the general situation where $M_p$ is allowed to vary with $p$).
In the case of $G_m^d$ (which is currently conditional), this would give (for instance) stratification and generic square-root cancellation for sums of the type

$$ \sum_{x_1, \dots, x_d \in \mathbf{F}_p^\times} \chi_1(x_1) \cdots \chi_d(x_d) e\left(\frac{f(x_1, \dots, x_d)}{p}\right), $$

where $f \in \mathbb{Z}[X_1, \dots, X_d]$ is a polynomial and $\chi_1, \dots, \chi_d$ are Dirichlet characters modulo $p$, together with an a priori algebraic description of the sets of characters where the sum has size $\asymp p^{i/2}$.

Over finite fields, we can still derive some applications, such as the following proposition, similar to [39, Cor. 1.4] (although the vertical direction means that equidistribution is only in the finite set $(1/p\mathbb{Z})/\mathbb{Z} \subset \mathbf{R}/\mathbb{Z}$, or equivalently modulo $p$, as we phrase it.)

**PROPOSITION 6.8.** Let $p$ be the characteristic of $k$. Let $d \ge 1$ and $r \le d$ be integers and let $f = (f_i): G_m^d \to \mathbf{A}^r$ be a morphism whose image is not contained in an affine hyperplane. For a sequence $(w_n)$ such that $0 \le w_n < |k|^n - 1$ and $w_n/(|k_n|^{1/2} \log |k_n|) \to +\infty$, and for an arbitrary generator $y_n$ of $k_n^\times$, the family of residue classes

$$ \mathrm{Tr}_{k_n/\mathbf{F}_p}(f(y_n^{v_1}, \dots, y_n^{v_d})) (\mathrm{mod} \, p), \quad 0 \le v_i \le w_n \text{ for all } i $$

become uniformly distributed in $(\mathbb{Z}/p\mathbb{Z})^r$.

**PROOF.** Let $G = G_m^d$ and $q = |k|$; for $n \ge 1$, denote by $\psi_n$ the additive character $x \mapsto e(\mathrm{Tr}_{k_n/\mathbf{F}_p}(x)/p)$ of $k_n$. Using the generator $y_n$, we can identify the group $G(k_n) = (k_n^\times)^d$ with the group $(\mathbb{Z}/(q^n-1)\mathbb{Z})^d$ and we also identify $\hat{G}(k_n)$ with $(\mathbb{Z}/(q^n-1)\mathbb{Z})^d$, the element $\beta \in (\mathbb{Z}/(q^n-1)\mathbb{Z})^d$ corresponding to the character $\chi$ such that

$$ \chi(y_n^{v_1}, \dots, y_n^{v_d}) = e\left(\frac{1}{q^n-1}(\beta_1 v_1 + \dots + \beta_d v_d)\right). $$

By the Weyl Criterion, we need to prove that

$$ \lim_{n \to +\infty} \frac{1}{w_n^d} \sum_{0 \le v_i \le w_n} \psi_n \left( \sum_{i=1}^d h_i f_i(y_n^{v_1}, \dots, y_n^{v_d}) \right) = 0 $$

for any $h \in (\mathbb{Z}/p\mathbb{Z})^r - \{0\}$. Detecting the interval $0 \le v \le w_n$ by Fourier expansion, we have to study the limit of

$$ \frac{1}{w_n^d} \sum_{\chi \in \hat{G}(k_n)} \hat{\alpha}_n(\chi) \sum_{x \in G(k_n)} \chi(x) \psi_n \left( \sum_{i=1}^d h_i f_i(x) \right) $$

where

$$ \hat{\alpha}_n(\chi) = \frac{1}{(q^n - 1)^d} \sum_{0 \le v_i \le w_n} \overline{\chi(y_n^{v_1}, \dots, y_n^{v_d})}. $$

Define $g_h: G \to \mathbf{A}^1$ by

$$ g_h(x) = \sum_{i=1}^{d} h_i f_i(x). $$

We can write

$$ \frac{1}{w_n^d} \sum_{\chi \in \hat{G}(k_n)} \hat{\alpha}_n(\chi) \sum_{x \in G(k_n)} \chi(x) \psi_n \left( \sum_{i=1}^r h_i f_i(x) \right) = \frac{1}{w_n^d} \sum_{\chi \in \hat{G}(k_n)} \hat{\alpha}_n(\chi) q^{nd/2} S(M, \chi) $$

for the complex $M = g_h^* \mathcal{L}_{\psi_1}[d](d/2)$ on $G$, which is a simple perverse sheaf, pure of weight 0, on $G$.
We apply Theorem 6.1 to M. Let ($\mathcal{S}_i$) be the subsets described there. We have $\mathcal{S}_d = \emptyset$ because the image of $f$ is not contained in an affine hyperplane, which implies that $g_h$ is non-constant, and hence M is non-trivial, from which the fact that it does not coincide with $\mathcal{L}_\chi$ for any character $\chi \in \hat{\mathbb{G}}$ follows. Moreover, we also know that each $\mathcal{S}_i$ is a finite union of tacs of G of dimension $\le d - i$.

The contribution of all $\chi \in (\mathcal{S}_0 - \mathcal{S}_1)(k_n)$ to the previous sum satisfies the bound

$$ \frac{1}{w_n^d} \sum_{\chi \in (\hat{\mathbb{G}} - \mathcal{S}_1)(k_n)} \hat{\alpha}_n(\chi) q^{nd/2} S(M, \chi) \ll \frac{q^{nd/2}}{w_n^d} \sum_{\chi \in \hat{\mathbb{G}}(k_n)} |\hat{\alpha}_n(\chi)|. $$

It is well-known that the bound

$$ (6.1) \qquad \sum_{\chi \in \hat{\mathbb{G}}(k_n)} |\hat{\alpha}_n(\chi)| \ll (\log q)^d $$

holds for all $n \ge 1$ (see Remark 6.10 below), where the implied constant depends on $d$, so that

$$ \frac{1}{w_n^d} \sum_{\chi \in (\hat{\mathbb{G}} - \mathcal{S}_1)(k_n)} \hat{\alpha}_n(\chi) q^{nd/2} S(M, \chi) \ll \left( \frac{q^{n/2} \log(q)}{w_n} \right)^d, $$

which converges to 0 as $n \to +\infty$ by assumption.

We now handle the remaining terms. Let $1 \le j \le d - 2$. By Theorem 6.1, the estimate

$$ \frac{1}{w_n^d} \sum_{\chi \in (\mathcal{S}_j - \mathcal{S}_{j+1})(k_n)} \hat{\alpha}_n(\chi) q^{nd/2} S(M, \chi) \ll \frac{q^{n(d+j)/2}}{w_n^d} \sum_{\chi \in \mathcal{S}_j(k_n)} |\hat{\alpha}_n(\chi)| $$

holds for all $n \ge 1$. From Lemma 6.9 below and the fact that $\mathcal{S}_j$ is a finite union of tacs of codimension at least $j$, we deduce that the estimate

$$ \sum_{\chi \in \mathcal{S}_j(k_n)} |\hat{\alpha}_n(\chi)| \ll \left(\frac{w_n}{q^n}\right)^j (\log q)^d $$

holds for $n \ge 1$. It follows that

$$ \frac{1}{w_n^d} \sum_{\chi \in (\mathcal{S}_j - \mathcal{S}_{j+1})(k_n)} \hat{\alpha}_n(\chi) q^{nd/2} S(M, \chi) \ll \frac{q^{n(d+j)/2}}{w_n^d} \left(\frac{w_n}{q^n}\right)^j (\log q)^d = \left(\frac{q^{n/2}}{w_n}\right)^{d-j} (\log q)^d. $$

The conclusion follows.

<figure><img src="image_12.png" /></figure>

LEMMA 6.9. With notation as above, for any tac $\mathcal{S}$ of $\mathbf{G}_m^d$ of dimension $d - j < d$, we have

$$ \sum_{\chi \in \mathcal{S}(k_n)} |\hat{\alpha}_n(\chi)| \ll \left(\frac{w_n}{q^n}\right)^j (\log q)^d. $$

PROOF. Mutatis mutandis, this is very close to [39, Lemma 9.5], in the (simpler) case where the variety $\mathcal{V}$ of loc. cit. is an affine hyperplane (but with the primes $p$ replaced by the sequence $q^n - 1$). Indeed, let $f: \mathbf{G}_m^d \to \mathbf{G}_m^{d-j}$ and $\chi_0 = (\chi_{0,1}, \dots, \chi_{0,d})$ be the morphism of tori and the character $\chi_0$ defining $\mathcal{S}$. There exists a matrix $m = (m_{k,l})$ of size $d \times (d - j)$ with integral coefficients of rank $j$ such that a character $\eta = (\eta_1, \dots, \eta_d)$ of $\mathbf{G}_m^d$ belongs to $\mathcal{S}(k_n)$ if and only if

$$ \prod_{l=1}^{d} \eta_l^{m_{k,l}} = \chi_{0,k}^{-1} $$
for $1 \le k \le d-j$. When we identify $\hat{G}_m^d(k_n)$ with $(\mathbb{Z}/(q^n-1)\mathbb{Z})^d$, this means that $\mathcal{S}(k_n)$ is identified with the set of solutions $(\xi_1, \dots, \xi_d)$ in $(\mathbb{Z}/(q^n-1))^d$ of the linear equation

$$ \sum_{l=1}^{d} m_{k,l} \xi_l = y_k $$

for some $y_k \in (\mathbb{Z}/(q^n-1)\mathbb{Z})$. $\square$

REMARK 6.10. We recall the proof of (6.1). We denote $N = q^n - 1$ so that $\tilde{G}(k_n)$ is isomorphic to $(\mathbb{Z}/N\mathbb{Z})^d$. The sum to estimate is

$$ \begin{aligned} \frac{1}{N^d} \sum_{\xi \in (\mathbb{Z}/N\mathbb{Z})^d} \left| \sum_{0 \le v_i \le w_n} e\left(\frac{\xi_1 v_1 + \dots + \xi_d v_d}{N}\right) \right| &= \frac{1}{N^d} \sum_{\xi \in (\mathbb{Z}/N\mathbb{Z})^d} \prod_{i=1}^d \left| \sum_{0 \le v \le w_n} e\left(\frac{\xi_i v}{N}\right) \right| \\ &= \frac{1}{N^d} \prod_{i=1}^d \left| \sum_{\xi \in (\mathbb{Z}/N\mathbb{Z})} \sum_{0 \le v \le w_n} e\left(\frac{\xi v}{N}\right) \right|, \end{aligned} $$

which shows that it is enough to handle the case $d=1$. In this case, one uses the bound

$$ \left| \sum_{0 \le v \le w_n} e\left(\frac{\xi v}{N}\right) \right| \le \min\left(w_n + 1, \frac{e((w_n + 1)\xi/N) - 1}{e(\xi/N) - 1}\right) \le \min\left(w_n + 1, \frac{1}{2\|\xi/N\|}\right) $$

where $\|\xi/N\|$ is the distance to the nearest integer of $\xi/N$ (the sum is a finite geometric sum, and in the last step, we used the lower-bound $|\sin(x)| \ge 2\|x\|$, valid for $x \in \mathbb{R}$). We then sum over the range $0 \le \xi \le N-1$; for $\xi = 0$, the bound is $\le (w_n+1) \le N$, and for $1 \le \xi \le N-1$, we have

$$ \|\xi/N\| \ge \min\left(\frac{\xi}{N}, \frac{N-\xi}{N}\right), $$

hence

$$ \frac{1}{N} \sum_{\xi \in (\mathbb{Z}/N\mathbb{Z})} \left| \sum_{0 \le v \le w_n} e\left(\frac{\xi v}{N}\right) \right| \le 1 + \sum_{\xi=1}^{(N-1)/2} \frac{1}{\xi} \ll \log N, $$

as claimed.

## 6.2. Generic Fourier invertibility

For two semisimple perverse sheaves M and N, Proposition 1.20 implies that if the arithmetic Fourier transforms of M and N coincide, in the sense that $S(M, \chi) = S(N, \chi)$ for any $\chi \in \hat{G}$, then the trace functions of M and N coincide over $k_n$ for all $n \ge 1$, which implies that M and N are isomorphic (Proposition A.22; see also [103, Prop. 4.2.3] for tori).

The stratified vanishing theorem allows us to prove a statement of “generic Fourier invertibility” for pure perverse sheaves, which relaxes the condition of equality of all sums $S(M, \chi)$ and $S(N, \chi)$ to a condition for a generic set of characters.

**THEOREM 6.11 (Generic Fourier invertibility).** Let G be a connected commutative algebraic group over k. Let M and N be arithmetically semisimple ℓ-adic perverse sheaves on G which are pure of weight zero.

The perverse sheaves $M_{\text{int}}$ and $N_{\text{int}}$ are arithmetically isomorphic if and only if there exists a generic set $\mathcal{X} \subset \hat{G}$ such that $S(M, \chi) = S(N, \chi)$ for all $\chi \in \mathcal{X}$.
PROOF. If $M_{\text{int}}$ is isomorphic to $N_{\text{int}}$, then the sums $S(M, \chi)$ and $S(N, \chi)$ coincide for a generic set of characters because $S(P, \chi)$ vanishes generically for a negligible object $P$.

To prove the converse, we may assume that $M = M_{\text{int}}$ and $N = N_{\text{int}}$, i.e., that $M$ and $N$ are objects of $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$. We then argue by induction on the sum $m$ of the lengths of $M$ and $N$.

If $m = 0$, then the perverse sheaves $M$ and $N$ are both zero.

Suppose now that $m \ge 1$ and that the statement holds for all pairs $(M_1, N_1)$ of perverse sheaves in $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$ such that the sum of the lengths of $M_1$ and $N_1$ is $\le m - 1$. One at least of the perverse sheaves $M$ and $N$ is non-zero, and (up to exchanging $M$ and $N$) we may assume that $M$ is non-zero. Let $(Q_i)_{i \in I}$ be the simple components (without multiplicity) of the perverse sheaf $M \oplus N$, and for $i \in I$, let $\mu_M(i)$ (resp. $\mu_N(i)$) be the multiplicity of $Q_i$ in $M_{\text{int}}$ (resp. $N_{\text{int}}$).

Let $\mathcal{Y}$ be the set of Frobenius-unramified characters for the perverse sheaf

$$(M *_{\text{int}} N^{\vee}) \oplus (M *_{\text{int}} M^{\vee}),$$

viewed as an object of $\langle M \oplus N \rangle^{\text{ari}}$.

For any integer $n \ge 1$, we consider the sum

$$T_n = \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{Y}(k_n)} S(M *_{\text{int}} N^{\vee}, \chi).$$

Applying Corollary 4.6 after decomposing $M$ and $N$ in terms of the simple perverse sheaves $Q_i$, we obtain the formula

$$\lim_{N \to +\infty} \frac{1}{N} \sum_{n \le N} T_n = \sum_{i \in I} \mu_M(i) \mu_N(i).$$

On the other hand, we can write

$$T_n = \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{Y}(k_n)} S(M *_{\text{int}} M^{\vee}, \chi) + \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{Y}(k_n)} \left( S(M *_{\text{int}} N^{\vee}, \chi) - S(M *_{\text{int}} M^{\vee}, \chi) \right)$$

for any $n \ge 1$. For $\chi$ in the generic set $\mathcal{X} \cap \mathcal{Y}$, the assumption implies that

$$S(M *_{\text{int}} N^{\vee}, \chi) = S(M *_{\text{int}} M^{\vee}, \chi).$$

Thus, using Corollary 6.3, the assumption implies that the bound

$$\frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{Y}(k_n)} \left( S(M *_{\text{int}} N^{\vee}, \chi) - S(M *_{\text{int}} M^{\vee}, \chi) \right) \ll |k_n|^{-1/2}$$

holds for $n \ge 1$. Applying Corollary 4.6 once more and comparing with the previous computation, we deduce that

$$\sum_{i \in I} \mu_M(i) \mu_N(i) = \sum_{i \in I} \mu_M(i)^2.$$

The right-hand side is $\ge 1$ since $M$ is non-zero. Hence, there exists $i$ such that $\mu_M(i)\mu_N(i) \ge 1$, which means that $Q_i$ appears with positive multiplicity in both $M$ and $N$. Removing one occurrence of $Q_i$ from $M$ and $N$, we obtain perverse sheaves $M_1$ and $N_1$ in $\mathbf{P}_{\text{int}}^{\text{ari}}(G)$ for which we can apply the induction hypothesis, so that $M_1$ is isomorphic to $N_1$, and adding the simple perverse sheaf $Q_i$ to both sides, we deduce that $M$ is isomorphic to $N$. $\square$

REMARK 6.12. In the case of tori, this theorem can be compared with a conditional result of Loeser [103, Prop. 4.2.5].
CHAPTER 7

Independence of $\ell$

We consider in this section a connected commutative algebraic group $G$ over a finite field $k$. Let $p$ be the characteristic of $k$. Since we will vary the prime $\ell \neq p$, we will indicate it in the notation. For an object $M$ of $\mathrm{D}_c^b(G, \overline{\mathbb{Q}}_\ell)$, we will now denote by $t_M(x; k_n)$ the $\overline{\mathbb{Q}}_\ell$-valued trace function of $M$, and we will also specify explicitly the isomorphisms $\iota$ used to define their complex-valued analogues. In particular, we write $\hat{G}^{(\ell)}$ for the set of $\ell$-adic characters.

We recall (see, e.g, [48, Def. 1.2] with $E = C$) that if $A$ is a set of pairs $(\ell, \iota)$ consisting of a prime number $\ell$ different from the characteristic of $k$ and an isomorphism $\iota: \overline{\mathbb{Q}}_\ell \to C$, a family $(M_\alpha)_{\alpha \in A}$ of objects $M_\alpha$ of $\mathrm{Perv}(G, \overline{\mathbb{Q}}_\ell)$ is said to be a compatible system if for any $n \ge 1$ and $x \in G(k_n)$, the complex numbers $\iota(t_{M_\alpha}(x; k_n))$ are independent of $\alpha = (\ell, \iota) \in A$. This is equivalent to asking that the eigenvalues of Frobenius for the stalk of $M_\alpha$ at $x$ are independent of $\alpha$.

The question we wish to address is the following:

**QUESTION.** Suppose that we have a compatible system $(M_\alpha)_{\alpha \in A}$; to what extent are the arithmetic and geometric tannakian groups of $M_\alpha$ independent of $\alpha$?

We note that the analogue question for the monodromy groups of a compatible system of lisse sheaves on an algebraic variety $X$ over $k$ (especially a curve) has been considered in depth by, among others, Serre [120, p. 1–21], Larsen–Pink [97] and Chin [20]. Using Deligne’s Fourier transform, this gives corresponding answers to our question in the case of the group $G_a$. We note also that the deepest results (such as that of Chin) depend on the global Langlands correspondance over function fields.

In this section, we take a first step in addressing the question. We will only compare two objects, so for the remainder of this section, we let $(\ell_1, \iota_1)$ and $(\ell_2, \iota_2)$ be pairs of prime numbers and isomorphisms $\iota_j: \overline{\mathbb{Q}}_{\ell_j} \to C$. For $j = 1, 2$, we fix an $\ell_j$-adic arithmetically semisimple perverse sheaf $M_j$ on $G$ which is pure of $\iota_j$-weight zero. We assume that $M_1$ and $M_2$ are compatible, that is we assume that the system with $A = \{(\ell_1, \iota_1), (\ell_2, \iota_2)\}$ is compatible.

**LEMMA 7.1.** *The following properties hold:*

(1) For any $n \ge 1$, the map

$$ \eta: \hat{G}^{(\ell_1)}(k_n) \to \hat{G}^{(\ell_2)}(k_n) \\ \eta \mapsto \iota_2^{-1} \circ \iota_1 \circ \chi $$

is a bijection such that $\iota_1(S(M_1, \chi)) = \iota_2(S(M_2, \eta(\chi)))$ holds for all $\chi$.

(2) For any $\chi \in \hat{G}^{(\ell_1)}$, the objects $(M_1)_\chi$ and $(M_2)_{\eta(\chi)}$ are compatible.

(3) The set of weakly unramified characters $\chi \in \hat{G}^{(\ell_1)}$ for $M_1$ such that $\eta(\chi)$ is weakly unramified for $M_2$ is generic.
PROOF. This boils down to the computation

$$
\begin{align*}
\iota_1(t_{(M_1)\chi}(x; k_n)) &= \iota_1(t_{M_1}(x; k_n)\chi(N_{k_n/k}(x))) \\
&= \iota_1(t_{M_1}(x; k_n))\iota_1(\chi(N_{k_n/k}(x))) \\
&= \iota_2(t_{M_2}(x; k_n))\iota_2(\eta(\chi)(N_{k_n/k}(x))) = \iota_2(t_{(M_2)_{\eta(\chi)}}(x; k_n))
\end{align*}
$$

for any $n \ge 1$ and $x \in G(k_n)$, which follows from the definitions, and the fact that $\mathcal{X}_w(M_1)$ and $\eta^{-1}(\mathcal{X}_w(M_2))$ are both generic, and hence so is their intersection in $\hat{G}^{(\ell_1)}$. $\square$

REMARK 7.2. We will say that two sets $\mathcal{A}_j \subset \hat{G}^{(\ell_j)}$, defined for $j = 1$ and $j = 2$, are compatible if the bijection $\eta$ induces bijections $\mathcal{A}_1(k_n) \to \mathcal{A}_2(k_n)$ for all $n \ge 1$.

LEMMA 7.3. The tannakian dimensions of M₁ and M₂ coincide.

PROOF. By Proposition 3.17 and the generic vanishing theorem, the tannakian dimension of $M_j$ is equal to the Euler-Poincaré characteristic of $(M_j)_\chi$ for $\chi$ in a generic set $\mathcal{X}_j \subset \hat{G}^{(\ell_j)}$. By Lemma 7.1, we can find $\chi \in \mathcal{X}_1$ such that $\eta(\chi) \in \mathcal{X}_2$. The result then follows from the fact that, since $(M_1)_\chi$ and $(M_2)_{\eta(\chi)}$ are compatible, they have the same Euler-Poincaré characteristic (see, e.g., [115, Lemma 6.38]). $\square$

We denote from now on by $r$ the common tannakian dimension of $M_1$ and $M_2$. We further denote by $K_j$ a maximal compact subgroup of $\iota_j(\mathbf{G}_{M_j}^{\text{ar})}(\mathbf{C}))$, and by $\mu_j$ the probability Haar measure on $K_j$. We define the measures $\nu_{cp,j}$ on $\mathbf{U}_r(\mathbf{C})^\sharp$ and the measures $\nu_j$ on $\mathbf{C}$ as in Theorems 4.4 and 4.8, respectively (the latter is the Sato-Tate measure of $M_j$).

LEMMA 7.4. With notation and assumptions as above, the objects $\varrho(M_1)$ and $\varrho(M_2)$ are compatible for any representation $\varrho$ of $\mathbf{GL}_r$.

PROOF. This is clear from the definition since the character of $\varrho$ is a symmetric polynomial of the eigenvalues of the matrix argument in $\mathbf{GL}_r$. $\square$

The basic information we have is the following consequence of equidistribution.

PROPOSITION 7.5. With notation and assumptions as above, we have $\nu_{cp,1} = \nu_{cp,2}$ and $\nu_1 = \nu_2$.

PROOF. It suffices to prove the equality $\nu_{cp,1} = \nu_{cp,2}$, and this is essentially because the measures $\nu_{cp,j}$ are both determined by equidistribution of “the same data”.

To be precise, we first note that by the Peter-Weyl Theorem, it is enough to prove that

$$
\int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,1}(g) = \int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,2}(g)
$$

holds for all finite-dimensional representations $\varrho$ of $\mathbf{U}_r(\mathbf{C})$. By Theorem 4.4, applied to the bounded test function $f = \mathrm{Tr}(\varrho)$, the equality

$$
\int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,j}(g) = \lim_{N \to +\infty} \frac{1}{N} \sum_{n \le N} \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{X}_j(k_n)} \mathrm{Tr}(\varrho(\Theta_{M_j, k_n}(\chi)))
$$

holds, where $\mathcal{X}_j \subset \hat{G}^{(\ell_j)}$ is the set of weakly unramified characters for $M_j$.

By Lemma 7.4, the objects $\varrho(M_1)$ and $\varrho(M_2)$ are compatible; hence, by Lemma 7.1 applied to these two perverse sheaves, we have

$$
(7.1) \qquad \mathrm{Tr}(\varrho(\Theta_{M_1, k_n}(\chi))) = \mathrm{Tr}(\varrho(\Theta_{M_2, k_n}(\eta(\chi))))
$$
if $\chi \in \mathcal{X}_1(k_n)$ is such that $\eta(\chi) \in \mathcal{X}_2(k_n)$. Therefore, the difference

$$ \int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,1}(g) - \int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,2}(g) $$

is equal to

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{n \le N} \frac{1}{|\mathcal{G}(k_n)|} \left( \sum_{\chi \in \mathcal{Y}_1(k_n)} \mathrm{Tr}(\varrho(\Theta_{M_1, k_n}(\chi))) - \sum_{\chi \in \mathcal{Y}_2(k_n)} \mathrm{Tr}(\varrho(\Theta_{M_2, k_n}(\chi))) \right), $$

where $\mathcal{Y}_1$ (resp. $\mathcal{Y}_2$) is the set of $\chi \in \mathcal{X}_1$ satisfying $\eta(\chi) \notin \mathcal{X}_2$ (resp. the set of $\chi \in \mathcal{X}_2$ satisfying $\eta^{-1}(\chi) \notin \mathcal{X}_1$).

Both of the sets $\mathcal{Y}_1$ and $\mathcal{Y}_2$ have positive character codimension, and hence we deduce

$$ \int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,1}(g) - \int_{\mathbf{U}_r(\mathbf{C})^{\sharp}} \mathrm{Tr}(\varrho(g)) d\nu_{cp,2}(g) = 0, $$

which implies the theorem. $\square$

<figure><img src="image_13.png" /></figure>

The equality of the characteristic polynomial measure or of the Sato-Tate measures of objects in a compatible system can provide a considerable amount of information. In ideal cases, this equality may be enough to imply that $G_{M_1}^{\mathrm{ari}}$ and $G_{M_2}^{\mathrm{ari}}$ are isomorphic. This does happen, but it is far from being always the case.

**EXAMPLE 7.6.** Let $H$ be a finite group and $H \subset \mathbf{U}_{|H|}(\mathbf{C})$ be its regular representation. Then the Sato-Tate measure is

$$ \left(1 - \frac{1}{|H|}\right) \delta_0 + \frac{1}{|H|} \delta_{|H|}, $$

where $\delta_z$ denotes a Dirac mass at a point $z \in \mathbf{C}$. Thus the Sato-Tate measure only determines the order of $H$ in that case.

For characteristic polynomials, Sutherland [123, Remark 1.5] gives examples of non-isomorphic transitive finite permutation groups with the same distributions of characteristic polynomials. We refer the reader to Sutherland's survey [123] for more examples and discussion of Sato-Tate measures in a more traditional context.

**COROLLARY 7.7.** We continue with the notation and assumptions above.

(1) The reductive ranks of the reductive groups $G_{M_1}^{\mathrm{ari}}$ and $G_{M_2}^{\mathrm{ari}}$ are the same.

(2) The group $G_{M_1}^{\mathrm{ari}}$ is finite if and only if the group $G_{M_2}^{\mathrm{ari}}$ is finite, and in this case, both groups have the same order.

**PROOF.** (1) The reductive rank of $G_{M_j}^{\mathrm{ari}}$ is the dimension of the space of characteristic polynomials of $G_{M_j}^{\mathrm{ari}}$ (see Serre [120, p. 17] for this fact), and hence it is equal to the dimension of the support of the measure $\nu_{cp,j}$ on $\mathbf{U}_r(\mathbf{C})^{\sharp}$. The result therefore follows from the proposition.

(2) This holds because the group $G_{M_j}^{\mathrm{ari}}$ is finite if and only if the measure $\nu_j$ is a finite sum of Dirac masses (for the “if” direction, one can use the same result of Serre as in (1)), and if that is true, then the size of $G_{M_j}^{\mathrm{ari}}$ is equal to the inverse of $\nu_j(\{r\})$.

(3) This follows from another result of Serre [120, p. 19], according to which a connected compact subgroup of $\mathbf{U}_r(\mathbf{C})$ cannot induce the same measure on characteristic polynomials as a non-connected subgroup. $\square$

<figure><img src="image_14.png" /></figure>
### REMARK 7.8.
In the next chapter, we will also see results which imply that $M_1$ has tannakian group containing $SL_r$ if and only $M_2$ has the same property, and some related statements, following from Larsen's Alternative (see Theorem 8.5 or Proposition 8.17 below).

Using another result of Serre, we can prove that connectedness of the arithmetic tannakian group is also independent of $\ell$.

### PROPOSITION 7.9.
We continue with the notation and assumptions above. The group $G_{M_1}^{\text{ari}}$ is connected if and only if the group $G_{M_2}^{\text{ari}}$ is connected.

This follows directly by combining Proposition 7.5 and the following lemma, which is the measure-theoretic version of Serre's "zero-one law" of [120, p. 18, Théorème].

### LEMMA 7.10.
Let $K$ be a compact subgroup of $U_r(C)$. Let $\nu$ denote the measure on $U_r(C)^{\#}$ image of the probability Haar measure of $K$ by the natural map $K \to U_r(C)^{\#}$. Then $K$ is connected if and only if, for all functions $f \in Z[a_1, \dots, a_r]$, the measure $\nu(\{g \in K | f(g) = 0\})$ is equal to 0 or 1, where $f(g)$ is computed with $a_i$ replaced by the coefficients of the characteristic polynomial of $g$.

### PROOF.
(1) If $K$ is connected, then either the set $\{g \in K | f(g) = 0\}$ is all of $K$, or it has codimension $\ge 1$, and has measure zero.

(2) If $K$ is not connected, and $g \in K$ is an element which is not in the neutral component, then Serre [120, p. 17, Lemme 1] proves that there exists $f \in Z[a_1, \dots, a_r]$ such that $f$ vanishes on the connected component of $g$, but is non-zero at the identity. Then the set

$$ \{g \in K \mid f(g) = 0\} $$

contains some connected component of $K$, say $p$ of them, but not all of them. Its measure is then $p/|\pi^0(K)|$, which is neither 0 nor 1. $\square$

This last proposition suggests that the groups of connected components of compatible objects should be isomorphic, as Serre proved is the case for classical monodromy groups (see [120, p. 15]). We hope to come back to this problem soon.

Finally, under rather strong "connectedness" assumptions, we can get a definitive answer by exploiting deep results of Larsen and Pink [96].

### PROPOSITION 7.11.
We continue with the notation and assumptions above. Denote by $G_j$ the connected derived subgroup of $G_{M_j}^{\text{ari}}$. Assume that for all representations $\varrho: GL_r \to GL(V)$, the multiplicity of the trivial representation in the restrictions of $\varrho$ to $G_{M_j}^{\text{ari}}$ and to $G_j$ are equal.

Then the complex semisimple Lie groups $\iota_1(G_1)$ and $\iota_2(G_2)$ are isomorphic.

Moreover, if $M_1$ or $M_2$ is arithmetically simple, then the groups $\iota_1(G_1)$ and $\iota_2(G_2)$ are conjugate in $GL_r(C)$.

### PROOF.
Let $\varrho: GL_r \to GL(V)$ be any finite-dimensional representation of $GL_r$ and let $\mu_j(\varrho)$ be the multiplicity of the trivial representation in the restriction of $\varrho$ to $G_{M_j}^{\text{ari}}$. By Proposition 7.5, the following equality holds:

$$ \mu_1(\varrho) = \int_{U_r(C)^{\#}} \mathrm{Tr}(\varrho) d\nu_{cp,1} = \int_{U_r(C)^{\#}} \mathrm{Tr}(\varrho) d\nu_{cp,2} = \mu_2(\varrho). $$

By our assumption, the multiplicity $\mu_j(\varrho)$ is also the multiplicity $\nu_j(\varrho)$ of the trivial representation in the restriction of $\varrho$ to $G_j$, and thus we have $\nu_1(\varrho) = \nu_2(\varrho)$.
Since this equality holds for all representations $\rho$, and $G_1$ and $G_2$ are connected semisimple algebraic groups, a theorem of Larsen and Pink [96, Th. 1] implies that $\iota_1(G_1)$ and $\iota_2(G_2)$ are isomorphic.

Assume now that $M_1$ is arithmetically simple. Denoting by $\text{Ad}$ the adjoint representation of $GL_r$, this is equivalent to $\mu_1(\text{Ad}) = 1$ by Schur's Lemma, and hence we also have $\nu_1(\text{Ad}) = 1$ and $\nu_2(\text{Ad}) = 1$ by the previous results. The result then follows from another theorem of Larsen and Pink [96, Th. 2]. □

REMARK 7.12. (1) Proposition 7.11 applies for instance if one knows that, for $j = 1$ and $j = 2$, the groups $G_{M_j}^{\text{ari}}$ are connected and semisimple. However, it does not apply in a situation where, say $G_{M_1}^{\text{ari}} = SL_r$ and $G_{M_2}^{\text{ari}} = GL_r$, since the determinant is an example of a representation where the multiplicities for $G_{M_2}^{\text{ari}}$ and for its derived connected subgroup are not the same. (We thank one referee for pointing out this issue in our previous version.)

(2) Larsen and Pink [96, Th. 3 and § 3] give examples showing that, in general, the assumption that $M_1$ (or $M_2$) is simple cannot be omitted in the second part of the proposition.
<figure><img src="image_15.png" /></figure>
Diophantine group theory

In order to determine the tannakian (or monodromy) group associated to a perverse sheaf, Katz has developed essentially two different sets of methods. The first one (see, e.g., [69, 70]) relies on local monodromy information, and applies mostly to the additive group, although there is also a weaker analogue for the multiplicative group (see [75, Ch. 16] and Corollary 3.47). However, we are not currently aware of any similar tools for other groups. The second method, expounded in [73], is much more global, and exploits the diophantine potential of the equidistribution of exponential sums to reveal properties of the underlying group. It turns out that this global method adapts very well to the tannakian framework, and this will be our fundamental tool.

We denote as usual by $k$ a finite field, with an algebraic closure $\bar{k}$, and finite extensions $k_n$ of degree $n$ in $\bar{k}$. We fix a prime $\ell$ different from the characteristic of $k$.

**8.1. The diophantine irreducibility criterion**

We first state Katz's criterion for a perverse sheaf to be geometrically simple in terms of its trace functions.

**PROPOSITION 8.1.** Let $X$ be a quasi-projective algebraic variety over $k$, and $M$ an $\ell$-adic perverse sheaf on $X$ which is pure of weight zero. Then the equality

$$ (8.1) \qquad \lim_{n \to +\infty} \sum_{x \in X(k_n)} |t_M(x; k_n)|^2 = 1 $$

holds if and only if $M$ is geometrically simple.

See [73, Th. 1.7.2 (3)] for the proof.

**REMARK 8.2.** This can be seen as a version of Schur's Lemma (compare with Corollary 4.6): intuitively, by equidistribution, the limit in the proposition should converge to the multiplicity of the trivial representation in the representation End(Std), where Std is the standard representation of the (usual) geometric monodromy group of the lisse sheaf on an open dense subset of the support of $M$ that is associated to $M$. The classical version of Schur's Lemma states that this multiplicity is equal to 1 if and only if the standard representation is irreducible.

**8.2. The Frobenius–Schur indicator**

Recall that if $G$ is an arbitrary group and $\rho: G \to GL_r(C)$ is a finite-dimensional representation, one says that $\rho$ is of *orthogonal type* (resp. of *symplectic type*) if there exists a $G$-invariant non-degenerate symmetric (resp. alternating) bilinear form on $C^r$. Suppose that $\rho$ is irreducible. The Frobenius–Schur indicator $FS(\rho)$ is defined to be 1 if $\rho$ is of orthogonal type, $-1$ if $\rho$ is of symplectic type, and 0 otherwise. If $G = K$ is a compact group, with probability Haar measure $\mu_K$, and if $\rho$ is irreducible and continuous, then one has an integral formula

$$ FS(\rho) = \int_K \mathrm{Tr}(\rho(g^2)) d\mu_K(g) $$
(see, e.g., [87, Th. 6.2.3]).

As in previous works of Katz (see, e.g., [75, Th. 9.1] or [73, Th. 1.9.6]), there is a diophantine interpretation of the Frobenius–Schur indicator.

**PROPOSITION 8.3.** Let $\mathcal{G}$ be a connected commutative algebraic group over $k$, and let $M$ be an arithmetically irreducible $\ell$-adic perverse sheaf on $\mathcal{G}$ which is pure of weight zero. Let $\mathcal{X} = \mathcal{X}_w(M)$ be the set of weakly unramified characters for $M$.

The Frobenius–Schur indicator of $M$, viewed as a representation of the arithmetic tannakian group $\mathcal{G}_M^{\text{ari}}$, is given by the formula

$$ \mathrm{FS}(M) = \lim_{N \to +\infty} \frac{1}{N} \sum_{\substack{1 \le n \le N \\ \mathcal{X}(k_n) \ne \emptyset}} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} \mathrm{Tr}(\Theta_{M, k_n}(\chi)^2). $$

The proof is straightforward using the integral formula above and the equidistribution theorem (Theorem 4.4).

### 8.3. Larsen's Alternative

In this section, $r \ge 1$ is an integer and $\mathcal{G}$ is a reductive algebraic subgroup of $\mathcal{GL}_r$ over an algebraically closed field of characteristic zero (recall that reductive groups are not required to be connected). For each integer $m \ge 1$, the absolute $2m$-th moment of an algebraic representation $V$ of $\mathcal{G}$ is defined as

$$ M_{2m}(\mathcal{G}, V) = \dim(V^{\otimes m} \otimes (V^{\vee})^{\otimes m})^{\mathcal{G}}. $$

When $V$ is the “standard” $r$-dimensional representation given by the inclusion $\mathcal{G} \subset \mathcal{GL}_r$ (also denoted by Std), we will simply write $M_{2m}(\mathcal{G})$.

If the base field is $\mathbf{C}$, so that $\mathcal{G}$ is a reductive subgroup of $\mathcal{GL}_r(\mathbf{C})$, the moments can be written as integrals over a maximal compact subgroup $K$ of $\mathcal{G}$ with Haar probability measure $\mu_K$. Namely, for all $m \ge 1$, they are given by the integral expression

$$ (8.2) \qquad M_{2m}(\mathcal{G}) = \int_K |\mathrm{Tr}(g)|^{2m} d\mu_K(g). $$

We first note some elementary properties of the moments.

(1) Given a surjective homomorphism $f: \mathcal{H} \to \mathcal{G}$ and a representation $\varrho: \mathcal{G} \to \mathcal{GL}(V)$, the equality

$$ M_{2m}(\mathcal{H}, \varrho \circ f) = M_{2m}(\mathcal{G}, \varrho) $$

holds for all $m \ge 1$ (since $(\varrho^{\otimes m} \otimes (\varrho^{\vee})^{\otimes m})^{\mathcal{G}} = ((\varrho \circ f)^{\otimes m} \otimes (\varrho \circ f)^{\vee})^{\otimes m})^{\mathbf{H}}$ by definition).

(2) For groups $\mathcal{G}_1$ and $\mathcal{G}_2$ with representations $V_1$ and $V_2$, the equality

$$ (8.3) \qquad M_{2m}(\mathcal{G}_1 \times \mathcal{G}_2, V_1 \boxtimes V_2) = M_{2m}(\mathcal{G}_1, V_1)M_{2m}(\mathcal{G}_2, V_2) $$

holds for all $m \ge 1$ (this might be easiest to see using the integral expression (8.2)).

(3) If $\mathcal{G} \subset \mathcal{GL}(V)$, and $Z$ is a subgroup of scalar matrices in $\mathcal{GL}(V)$, then the equality

$$ M_{2m}(\mathcal{G}, V) = M_{2m}(Z\mathcal{G}, V) $$

holds (because $Z$ acts trivially on the whole space $V^{\otimes m} \otimes (V^{\vee})^{\otimes m}$).

(4) If there exists a $\mathcal{G}$-invariant decomposition

$$ V^{\otimes m} = \bigoplus_i n_i V_i, $$