then the 2m-th moment satisfies the inequality

$$ (8.4) \qquad M_{2m}(\mathbf{G}, \mathbf{V}) \geq \sum_{i} n_{i}^{2}, $$

with equality if and only if the $V_i$ are pairwise non-isomorphic irreducible representations (see [72, 1.1.4]).

(5) If there exists a $\mathbf{G}$-invariant decomposition

$$ \mathrm{End}(\mathbf{V}) = \bigoplus_{i} m_{i} W_{i}, $$

then the fourth moment satisfies

$$ (8.5) \qquad M_4(\mathbf{G}, \mathbf{V}) \geq \sum_{i} m_i^2, $$

with equality if and only if the $W_i$ are pairwise non-isomorphic irreducible representations (see [72, 1.1.5]).

Since the tensor constructions involved in the definition of the moments are representations of the ambient group $\mathbf{GL}(\mathbf{V})$, Theorem 4.4 immediately yields a diophantine interpretation of the moments of the arithmetic tannakian group of a perverse sheaf.

**PROPOSITION 8.4.** Let $\mathbf{G}$ be a connected commutative algebraic group over $k$, and let $\mathbf{M}$ be an arithmetically semisimple $\ell$-adic perverse sheaf on $\mathbf{G}$ which is pure of weight zero. For each character $\chi \in \hat{\mathcal{G}}(k_n)$, consider the sum

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

Let $\mathcal{X} = \mathcal{X}_w(\mathbf{M})$ be the set of weakly unramified characters for $\mathbf{M}$ and let $m \geq 0$ be an integer.

The absolute moments of $\mathbf{M}$, viewed as a representation of the arithmetic tannakian group $\mathbf{G}_M^{\mathrm{ari}}$, satisfy the following:

$$ (8.6) \qquad M_{2m}(\mathbf{G}_M^{\mathrm{ari}}, \mathbf{M}) = \lim_{N \to +\infty} \frac{1}{N} \sum_{\substack{1 \le n \le N \\ \mathcal{X}(k_n) \ne \emptyset}} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} |S(\mathbf{M}, \chi)|^{2m}, $$

$$ (8.7) \qquad M_{2m}(\mathbf{G}_M^{\mathrm{ari}}, \mathbf{M}) \le \liminf_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|\mathbf{G}(k_n)|} \sum_{\chi \in \hat{\mathcal{G}}(k_n)} |S(\mathbf{M}, \chi)|^{2m}. $$

Moreover, if the limit

$$ (8.8) \qquad \lim_{n \to +\infty} \frac{1}{|\mathcal{X}(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} |S(\mathbf{M}, \chi)|^{2m} $$

exists, then it is equal to the 2m-th moment $M_{2m}(\mathbf{G}_M^{\mathrm{geo}}, \mathbf{M})$ of $\mathbf{M}$, viewed as a representation of the geometric tannakian group $\mathbf{G}_M^{\mathrm{geo}}$, and we have

$$ M_{2m}(\mathbf{G}_M^{\mathrm{geo}}, \mathbf{M}) = M_{2m}(\mathbf{G}_M^{\mathrm{ari}}, \mathbf{M}). $$

**PROOF.** We use the integral expression

$$ M_{2m}(\mathbf{G}_M^{\mathrm{ari}}, \mathbf{M}) = \int_K |\mathrm{Tr}(g)|^{2m} d\mu_K(g), $$

where $K \subset \mathbf{G}_M^{\mathrm{ari}}(\mathbf{C})$ is a maximal compact subgroup with Haar probability measure $\mu_K$. Recall that to each weakly unramified character $\chi \in \mathcal{X}(k_n)$ is associated the unitary conjugacy class $\Theta_{M,k_n}(\chi)$
such that the equality $S(M, \chi) = \mathrm{Tr}(\Theta_{M,k_n}(\chi))$ holds, and that these conjugacy classes become equidistributed on average as $n \to +\infty$ by Theorem 4.4. The first formula (8.6) follows from this result applied to the test function $g \mapsto |\mathrm{Tr}(g)|^{2m}$.

Moreover, the inequality

$$ \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|G(k_n)|} \sum_{\chi \in \mathcal{X}(k_n)} |S(M, \chi)|^{2m} \le \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|G(k_n)|} \sum_{\chi \in \hat{\mathcal{G}}(k_n)} |S(M, \chi)|^{2m} $$

holds by positivity of $|S(M, \chi)|^{2m}$. Taking the equivalence $|G(k_n)| \sim |\mathcal{X}(k_n)|$ as $n \to +\infty$ from the generic vanishing theorem into account, we deduce the second formula (8.7).

Finally, the last assertion follows from Proposition 4.17, applied to the representation

$$ \varrho = \mathrm{Std}^{\otimes m} \otimes (\mathrm{Std}^{\vee})^{\otimes m}, $$

and from the fact that if the limit (8.8) exists, then its value is the same as the limit in (8.6). $\square$

We can combine this computation with Larsen's Alternative, a remarkable criterion that ensures that a reductive subgroup $G \subset GL_r$ is either finite or contains one of the standard classical groups, provided it has the correct fourth or eighth moment.

**THEOREM 8.5 (Larsen's Alternative).** Let $V$ be a vector space of dimension $r \ge 2$ over an algebraically closed field of characteristic zero and let $G \subset GL(V)$ be a reductive algebraic subgroup. Let $Z$ denote the center of $GL(V)$ and $G^\circ$ the connected component of the identity of $G$.

(1) The fourth moment satisfies $M_4(G, V) \ge 2$. Furthermore, if $V$ is self-dual and $r \ge 3$, then $M_4(G, V) \ge 3$.

(2) If $M_4(G, V) \le 5$, then the representation of $G$ on $V$ is irreducible.

(3) If $M_4(G, V) = 2$, then either $SL(V) \subset G$ or $G/(G \cap Z)$ is finite. If $G \cap Z$ is finite, for instance if $G$ is semisimple, then either $G^\circ = SL(V)$ or $G$ is finite.

(4) Assume $r \ge 5$. If $M_4(G, V) = 2$ and $M_8(G, V) = 24$, then $SL(V) \subset G$.

(5) Assume that there exists a non-degenerate symmetric bilinear form $B$ on $V$ such that $G$ lies in $O(B)$. If $M_4(G, V) = 3$, then either $G = SO(B)$, or $G = O(B)$, or $G$ is finite. If $r$ is 2 or 4, then $G$ is not contained in $SO(B)$.

(6) Assume that there exists a non-degenerate alternating bilinear form $B$ on $V$ such that $G$ lies in $Sp(B)$. If $r \ge 4$ and $M_4(G, V) = 3$, then either $G = Sp(B)$ or $G$ is finite.

**PROOF.** The first statement concerning the fourth moment is a straightforward consequence of the inequality (8.4). Indeed, since $V^{\otimes 2} \otimes (V^\vee)^{\otimes 2}$ always contains a trivial one-dimensional subrepresentation, the fourth moment can only be 1 for $V$ of dimension 1. Moreover, there is a $GL(V)$-invariant (and hence $G$-invariant) decomposition

$$ V^{\otimes 2} = \mathrm{Sym}^2 V \oplus \Lambda^2 V, $$

where the factors are distinct and non-trivial, and of dimension $\ge 2$ if $r \ge 3$. If $V$ is self-dual, one of the two summands contains a proper one-dimensional $G$-invariant subspace, so that the fourth moment is at least 3 using (8.4) again.

The other statements concerning the fourth moment are proved by Katz in [72, Th. 1.1.6]. The statement about the eighth moment was conjectured by Katz in [73, 2.3], and proved by Guralnick and Tiep in [55, Th. 1.4]. Indeed, according to loc. cit., a reductive subgroup $G$ of $GL(V)$ either satisfies $M_8(G) > M_8(GL(V))$ or contains the commutator subgroup $[GL(V), GL(V)] = SL(V)$,
and the eighth moment of GL(V) is equal to 24 for $r \ge 4$, for instance in view of the GL(V)-invariant decomposition

$$V^{\otimes 4} = \mathrm{Sym}^4 V \oplus \bigwedge^4 V \oplus 3S^{(3,1)}V \oplus 2S^{(2,2)}V \oplus 3S^{(2,1,1)}V$$

into pairwise non-isomorphic irreducible representations (see e.g. [49, Ex. 6.5]), where $S^\lambda$ denotes the Schur functor associated to a partition $\lambda$ of 4. □

In practice, computing a given moment of the arithmetic tannakian group $G_M^{\mathrm{ari}}$ by means of the limit (8.6) is feasible if there are sufficiently many independent variables of summation, corresponding to the characters of G, in comparison with the number of variables involved in the object M, that is, the dimension of its support. It is then possible, at least in some cases, to detect a diagonal behavior that can lead to the asymptotic formula for the moment. This limitation explains why it is difficult to apply Larsen’s alternative when G is one-dimensional, but starting from two-dimensional groups it can be sometimes implemented for objects supported on curves.

**REMARK 8.6.** (1) Using typical terminology from geometric group theory, it is convenient to summarize the third part of Theorem 8.5 by saying that if $G \subset GL(V)$ has fourth moment equal to 2, then either $G \supset SL(V)$ or $G$ is virtually central in $GL(V)$.

(2) The book [73] of Katz develops applications of Larsen’s alternative which involve sums of the type

$$S(f) = \sum_{x \in X(k)} t_1(x) t_2(f(x)),$$

for suitable trace functions $t_1$ and $t_2$ (on X and some affine space $A^r$, respectively), parameterized by elements $f: X \to A^r$ of a “function space” $\mathcal{F}$. One of the conditions that are shown by Katz to ensure that the 2m-th moment can be computed is that the evaluation maps

$$f \mapsto (f(x_1), \dots, f(x_{2m}))$$

be surjective for distinct $x_i$ in $X(k)$ (see [73, §1.15, Th. 1.20.2] for a precise and more general statement).

## 8.4. Sidon morphisms

**DEFINITION 8.7 (Sidon sets and Sidon morphisms).** Let A be an abelian group. A subset $S \subset A$ is called a Sidon set if all solutions $x_1, x_2, x_3, x_4$ in S of the equation $x_1x_2 = x_3x_4$ satisfy $x_1 \in \{x_3, x_4\}$.

More generally, let $r \ge 2$ be an integer. We say that S is an r-Sidon set if all tuples $(x_i)_{1 \le i \le r}$ and $(y_i)_{1 \le i \le r}$ in $S^r$ such that the equality

$$x_1 \cdots x_r = y_1 \cdots y_r$$

holds satisfy $\{x_1, \dots, x_r\} = \{y_1, \dots, y_r\}$. A Sidon set is thus the same as a 2-Sidon set.

Let $\alpha \in A$. A subset $S \subset A$ is called an $\alpha$-symmetric Sidon set if $S = \alpha S^{-1}$ and all solutions $x_1, x_2, x_3, x_4 \in S$ of the equation $x_1x_2 = x_3x_4$ satisfy $x_1 \in \{x_3, x_4\}$ or $x_2 = \alpha x_1^{-1}$.

Let G be a connected commutative algebraic group over a field k, and let s: X → G be a locally-closed immersion of k-schemes. We say that s is a Sidon morphism, or that s(X) is a Sidon subvariety of G if, for any extension k' of k, the subset s(X)(k') ⊂ G(k') is a Sidon set. We define similarly r-Sidon morphisms for any r ≥ 2.

Let i be an involution on X and a ∈ G. We say that s is an i-symmetric Sidon morphism if the product morphism $(s \circ i) \cdot s: X \to G$ is a constant morphism, say equal to $\alpha \in G(k)$, and if, for any extension k' of k, the set s(X(k')) is an $\alpha$-symmetric Sidon set in G(k').
The interest of a Sidon morphism $X \to G$ is that it leads to computations of the fourth moment for objects M on G that are pushed from X. We have two versions, depending on whether we have a Sidon morphism or a symmetric Sidon morphism.

**PROPOSITION 8.8.** Let G be a connected commutative algebraic group over a finite field $k$ and let $s: X \to G$ be a closed immersion of $k$-schemes. Let $N$ be a geometrically simple $\ell$-adic perverse sheaf on $X$ which is pure of weight 0, so that the object $M = s_*N = s!N$ on $G$ is a geometrically simple perverse sheaf on $G$ and is pure of weight 0.

(1) If $s$ is a Sidon morphism, then the equality

$$M_4(\mathbf{G}_M^{\text{ari}}, M) = 2$$

holds unless M has tannakian dimension $\le 1$.

(2) If $X$ is a curve and $s: X \to G$ is a 4-Sidon morphism, then the equality

$$M_8(\mathbf{G}_M^{\text{geo}}, M) = M_8(\mathbf{G}_M^{\text{ari}}, M) = 24$$

holds unless N is geometrically isomorphic to $s^*\mathcal{L}_\chi[1]$ for some character $\chi \in \hat{G}$.

**PROOF.** Let $n \ge 1$ be an integer. The formula

$$\frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} |S(M, \chi)|^4 = \sum_{\substack{y_1, \dots, y_4 \in X(k_n) \\ s(y_1)s(y_2) = s(y_3)s(y_4)}} t_N(y_1, k_n) t_N(y_2; k_n) \overline{t_N(y_3; k_n) t_N(y_4; k_n)}$$

holds by orthogonality of characters. If $s$ is a Sidon morphism, then we obtain by definition

$$\frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} |S(M, \chi)|^4 = 2 \left( \sum_{y \in X(k_n)} |t_N(y, k_n)|^2 \right)^2 - \sum_{y \in X(k_n)} |t_N(y, k_n)|^4,$$

where the second term accounts for the double-counting of the solutions of the equation $s(y_1)s(y_2) = s(y_3)s(y_4)$ where $y_1 = y_2 = y_3 = y_4$. In particular, we deduce that the inequality

$$\frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} |S(M, \chi)|^4 \le 2 \left( \sum_{y \in X(k_n)} |t_N(y, k_n)|^2 \right)^2$$

holds for all $n \ge 1$. Since N is geometrically simple, the right-hand side of this expression converges to 2 as $n \to +\infty$ by Proposition 8.1. Using the inequality (8.7) from Proposition 8.4, we deduce that

$$M_4(\mathbf{G}_M^{\text{ari}}, M) \le 2$$

in the setting of (1). Hence, the fourth moment is either $\le 1$ or equal to 2. By Theorem 8.5 (1), the former is only possible if $M_{\text{int}}$ is of tannakian dimension $\le 1$.

Now we assume that $s$ is a 4-Sidon morphism. We obtain similarly

$$\frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} |S(M, \chi)|^8 \le 24 \left( \sum_{y \in X(k_n)} |t_N(y, k_n)|^2 \right)^4$$

where the right-hand side converges to 24 for the same reason as before.

Assume now that $X$ is a curve and $s$ is a 4-Sidon morphism. We apply the Riemann Hypothesis (Theorem 1.10) to the simple perverse sheaves $s^*\mathcal{L}_{\chi^{-1}[1]}(1/2)$ (of weight 0) and to N. By assumption, these are not geometrically isomorphic, and therefore the estimate

$$S(M, \chi) = \sum_{y \in X(k_n)} \chi(s(y)) t_N(y; k_n) \ll 1$$
holds for all characters $\chi$. We deduce then that the formula

$$ \lim_{n \to +\infty} \frac{1}{|G(k_n)|} \sum_{\chi \notin \mathcal{X}_w^c(M)(k_n)} |S(M, \chi)|^{2m} = 0 $$

holds for any integer $m \ge 1$; we finally conclude from the previous computations and the last assertion of Proposition 8.4 that $M_8(\mathbf{G}_M^{\text{geo}}, M) = M_8(\mathbf{G}_M^{\text{ari}}, M) = 24$. $\square$

We now state the version involving symmetric Sidon morphisms.

**PROPOSITION 8.9.** Let $G$ be a connected commutative algebraic group over a finite field $k$. Let $X$ be a smooth irreducible algebraic variety over $k$ and $i$ an involution on $X$. Let $s: X \to G$ be an $i$-symmetric Sidon morphism which is a closed immersion. Let $\alpha$ be the constant value of the morphism $(s \circ i) \cdot s$.

Let $N$ be a geometrically simple $\ell$-adic perverse sheaf on $X$ which is pure of weight 0, so that the object $s_*N = s_1N$ on $G$ is a geometrically simple perverse sheaf on $G$ and is pure of weight 0.

(1) If $i^*N$ is isomorphic to $D(N)$, then we have $(s_*N)^\vee = [\times \alpha^{-1}]^*(s_*N)$, and

$$ M_4(\mathbf{G}_{s_*N}^{\text{ari}}, s_*N) = 3, $$

unless $(s_*N)_{\text{int}}$ has tannakian dimension $\le 2$.

(2) If $i^*N$ is not isomorphic to $D(N)$, then

$$ M_4(\mathbf{G}_{s_*N}^{\text{ari}}, s_*N) = 2, $$

unless $(s_*N)_{\text{int}}$ has tannakian dimension $\le 2$.

**PROOF.** Let $M = s_*N$. In the situation of (1), the definition of $\alpha$ means that there is an equality $s \circ i = [\times \alpha] \circ (\text{inv} \circ s)$. Therefore, we obtain canonical isomorphisms

$$ \begin{aligned} (8.9) \quad M^{\vee} &= \text{inv}^*(D(s_*N)) = \text{inv}^*(s_*(D(N))) = \text{inv}^*((s \circ i)_*N) \\ &= \text{inv}^*([\times \alpha]_* (\text{inv} \circ s)_* N) = (\text{inv} \circ [\times \alpha] \circ \text{inv})_* (s_*N) = [\times \alpha^{-1}]^*M. \end{aligned} $$

We go back to the general case. Arguing as in the proof of the previous proposition, we obtain the inequality

$$ \frac{1}{|G(k_n)|} \sum_{\chi \in \hat{G}(k_n)} |S(M, \chi)|^4 \le 2 \left( \sum_{y \in X(k_n)} |t_N(y, k_n)|^2 \right)^2 + \sum_{(y,z) \in X(k_n)^2} t_N(y, k_n) t_N(i(y); k_n) \overline{t_N(z; k_n) t_N(i(z); k_n)} $$

for all $n \ge 1$, by the definition of symmetric Sidon sets. The second sum is equal to the quantity

$$ \left| \sum_{y \in X(k_n)} t_N(y, k_n) t_N(i(y); k_n) \right|^2, $$

which converges to 1 under the assumption (1) (using (A.6)), by Proposition 8.1, and to 0 under the assumption (2), by the Riemann Hypothesis. Thus we deduce from Proposition 8.4 that

$$ M_4(\mathbf{G}_M^{\text{ari}}, M) \le 3, \quad \text{resp.} \quad M_4(\mathbf{G}_M^{\text{ari}}, M) \le 2, $$

in case (1) (resp. (2)), and we conclude as before from Theorem 8.5, (1).

$\square$
REMARK 8.10. The caveats concerning the tannakian dimension of $s_*N$ in these statements are necessary. We will indeed see concrete examples (see Example 9.12 (1) and Remark 11.2 (1)) where the fourth moment does not coincide with the limit

$$ \lim_{n \to +\infty} \frac{1}{|\mathcal{G}(k_n)|} \sum_{\chi \in \hat{\mathcal{G}}(k_n)} |\mathcal{S}(M, \chi)|^4 $$

(although the latter exists) because of the contribution of some special ramified characters.

The result of Propositions 8.8 and 8.9 will be the basis of applications in Chapters 9, 10 and 11. Here are the relevant cases of Sidon morphisms, together with some further elementary examples.

PROPOSITION 8.11. Let $k$ be a field, not necessarily finite.

(1) For any $\alpha \in k^\times$, the embedding $x \mapsto (x, \alpha x)$ of $G_m$ in $G_m \times G_a$ is a Sidon morphism.

(2) Let $C$ be a smooth projective connected algebraic curve of genus $g \ge 2$ over $k$. Let $D$ be a divisor of degree 1 on $C$, and let $A = \mathrm{Jac}(C)$ be the jacobian of $C$. The closed immersion $s: x \mapsto x - D$ of $C$ in $A$ is a Sidon morphism unless $C$ is hyperelliptic, in which case it is an $i$-symmetric Sidon morphism, where $i$ is the hyperelliptic involution.

(3) With notation as in the previous item, if the gonality of $C$ is at least 5, then $s$ is a 4-Sidon morphism.

(4) Let $d \ge 1$ be an integer and let $f$ be a separable polynomial of degree $d$ over $k$. Let $Z$ be the set of zeros of $f$. The closed immersion $x \mapsto (z-x)_{z \in Z}$ of $\mathbf{A}^1[1/f]$ in $\mathbf{G}_m^Z$ is a Sidon morphism if $d \ge 2$. It is a 4-Sidon morphism if $d \ge 4$.

(5) Suppose that the characteristic of $k$ is not 3. The graph $s: x \mapsto (x, x^3)$ from $G_a$ to $G_a^2$ is an $i$-symmetric Sidon morphism, where $i$ is the involution $x \mapsto -x$.

(6) The morphism $x \mapsto (x, 1-x)$ from $G_m - \{1\}$ to $G_m \times G_m$ is a Sidon morphism.

PROOF. (1) For $x_1, \dots, x_4$ in $G_m$, the equation

$$ (x_1, \alpha x_1) \cdot (x_2, \alpha x_2) = (x_3, \alpha x_3) \cdot (x_4, \alpha x_4) $$

in $G_a \times G_m$ means that $x_1 + x_2 = x_3 + x_4$ and $x_1 x_2 = x_3 x_4$, which implies that $\{x_1, x_2\} = \{x_3, x_4\}$, both sets being the solutions of the same quadratic equation.

(2) Let $x_1, \dots, x_4$ in $C$ be solutions of

$$ s(x_1) + s(x_2) = s(x_3) + s(x_4). $$

Assume $x_1 \notin \{x_3, x_4\}$. Then the equation implies the existence of a rational function on $C$ with zeros $\{x_1, x_2\}$ and poles $\{x_3, x_4\}$, which corresponds to a morphism $f: C \to \mathbf{P}^1$ of degree at most 2. This is not possible unless $C$ is hyperelliptic.

With the same notation, if $C$ is hyperelliptic with hyperelliptic involution $i$, then the uniqueness of the morphism $f: C \to \mathbf{P}^1$ of degree 2 up to automorphisms (see, e.g., [100, Rem. 4.30]) shows that $i$ exchanges the points of the fibers of $f$, or in other words, that the equalities $x_2 = i(x_1)$ and $x_4 = i(x_3)$ hold.

(3) The argument is similar: the equation

$$ s(x_1) + s(x_2) + s(x_3) + s(x_4) = s(x_5) + s(x_6) + s(x_7) + s(x_8) $$

where $\{x_i\} \ne \{y_i\}$ implies the existence of a non-constant morphism $f: C \to \mathbf{P}^1$ of degree at most 4, and hence implies that $C$ has gonality at most 4.

(4) Suppose that $x_1, \dots, x_4$ satisfy $s(x_1)s(x_2) = s(x_3)s(x_4)$. Then we get

$$ (x_1 - z)(x_2 - z) = (x_3 - z)(x_4 - z) $$
for all $z \in \mathbb{Z}$, i.e., the monic polynomials $(x_1 - X)(x_2 - X)$ and $(x_3 - X)(x_4 - X)$ take the same values at the points of $\mathbb{Z}$. By interpolation, they are equal if $|\mathbb{Z}| = d \ge 2$. The case of the 4-Sidon property is analogous with polynomials of degree 4.

(5) Suppose that $x_1, \dots, x_4 \in \mathbb{G}_a^4$ satisfy

$$ \begin{cases} x_1 + x_2 = x_3 + x_4 \\ x_1^3 + x_2^3 = x_3^3 + x_4^3. \end{cases} $$

If $x_2 \neq -x_1$, then these imply that

$$ (x_1 + x_2)^2 - 3x_1x_2 = (x_3 + x_4)^2 - 3x_3x_4, $$

and therefore $x_1x_2 = x_3x_4$ when the characteristic is not 3. Now we conclude as in (1).

(6) This is again about quadratic equations: let $x_1, \dots, x_4$ in $\mathbb{G}_m - \{1\}$ be such that

$$ \begin{cases} x_1x_2 = x_3 \\ (1-x_1)(1-x_2) = (1-x_3)(1-x_4). \end{cases} $$

Then we get further $x_1 + x_2 = x_3 + x_4$, and conclude as before. $\square$

**REMARK 8.12.** (1) Example (1) is classical: it is often attributed to Ruzsa [111], but it was pointed out by Eberhard and Manners [34] that it occurs previously in a paper of Ganley [52, p. 323], where it is attributed to Spence.

Example (5) was also indicated to us by Eberhard and Manners.

(2) There is much work in combinatorics in trying to find the largest possible Sidon sets in a finite abelian group A (for instance, see the classification in [34] of known examples of size $\sim |A|^{1/2}$, which they show are all related to finite projective planes). A natural analogue geometric question is to classify the Sidon morphisms $s: X \to G$ such that $\dim(X)$ is maximal. The best possible value for a given group G is $\dim(X) = \lfloor \frac{\dim(G)}{2} \rfloor$. When can this be achieved?

Note that a subset S of an abelian group A is a Sidon set if and only if the induced map $S^2/\mathfrak{S}_2 \to A$ defined by $(x, y) \mapsto x+y$ is injective (where $\mathfrak{S}_2$ acts by permuting the two coordinates). Consider the variant definition of a Sidon morphism $s: X \to G$ where we ask that $s$ be a morphism such that the induced map $s^{(2)}: X^{(2)} \to G$ from the symmetric square of X to G is a closed immersion. Again we have $2\dim(X) \le \dim(G)$, but we can see in this case that if G is an abelian variety, then equality is not possible. Indeed, this would imply that $s^{(2)}$ is an isomorphism, which is impossible (if $\dim(X) \ge 2$, because $X^{(2)}$ is then singular, and if X is a curve, because it would have to be smooth of genus 2, so G is an abelian surface, but for instance the second cohomology groups do not have the same dimension).

The result concerning jacobians of smooth projective curves can be generalized by considering either Rosenlicht's generalized jacobians (which appear in geometric class field theory, see the book of Serre [118]), or the Picard group of certain singular curves. The case of generalized jacobians is analyzed in complete generality in our paper [37]. We state the result here (see [37, Th. 1]).

**PROPOSITION 8.13.** Let $k$ be a (not necessarily finite) field and let $C$ be a smooth projective geometrically connected curve of genus $g$ over $k$. Let $\mathfrak{m}$ be an effective divisor on $C$ and $J_{\mathfrak{m}}$ the associated generalized jacobian, which is a commutative algebraic group of dimension $g + \max(\deg(\mathfrak{m}) - 1, 0)$. Let $\delta$ be a divisor of degree 1 on $C$ whose support does not intersect that of $\mathfrak{m}$. Let $s: C - \mathfrak{m} \to J_{\mathfrak{m}}$ be the morphism induced by the map $x \mapsto (x) - \delta$ on divisors.

If $\dim(J_{\mathfrak{m}}) \ge 2$, then $s$ is either a Sidon morphism or a symmetric Sidon morphism.
If, moreover, $(C - \mathfrak{m})(k)$ is non-empty, then it is a symmetric Sidon set if and only if one of the following conditions hold:

(1) $g = 1$ and $\deg(\mathfrak{m}) = 2$; in this case, writing $\mathfrak{m} = (p) + (q)$ (where $p$ and $q$ are not necessarily $k$-points of $C$, but the divisor $\mathfrak{m}$ is assumed to be defined over $k$), the value $\alpha$ of $s(x) + s(p + q - x)$ for $x \in (C - \mathfrak{m})(k)$ is independent of $x$ and $s((C - \mathfrak{m})(k))$ is an $\alpha$-symmetric Sidon set.

(2) $g \ge 2$, the curve $C$ is hyperelliptic, and either $\deg(\mathfrak{m}) \le 1$ or $\mathfrak{m} = (p) + (i(p))$ for some $p \in C$, where $i$ is the hyperelliptic involution on $C$. In both of these cases, the value $\alpha$ of $s(x) + s(i(x))$ for $x \in (C - \mathfrak{m})(k)$ is independent of $x$ and $s((C - \mathfrak{m})(k))$ is an $\alpha$-symmetric Sidon set.

REMARK 8.14. (1) Both the generalized jacobians $J_S$ and the Picard group scheme of an irreducible curve are connected commutative algebraic groups over $k$ which may involve all types of groups (unipotent groups, abelian varieties and tori).

More precisely, the following results hold:

(1) Let $C$ be a smooth projective curve of genus $g \ge 0$ over $k$ and $\mathfrak{m}$ an effective divisor on $C$. Write $\mathfrak{m}$ in the form

$$ \mathfrak{m} = \sum_{x \in \text{Supp}(\mathfrak{m})} n_x(x) $$

with $n_x \ge 1$. The generalized jacobian $J_{\mathfrak{m}}$ is an extension

$$ 0 \to L_{\mathfrak{m}} \to J_{\mathfrak{m}} \to \text{Jac}(C) \to 0 $$

of the (usual) jacobian of $C$, with kernel $L_{\mathfrak{m}} = R_{\mathfrak{m}}/G_{\mathfrak{m}}$, where $R_{\mathfrak{m}}$ is isomorphic to a product

$$ R_{\mathfrak{m}} = \prod_{x \in \text{Supp}(\mathfrak{m})} (G_m \times V_x) $$

with $V_x$ unipotent of dimension $n_x - 1$, and with $G_m$ embedded diagonally in $R_{\mathfrak{m}}$ (see, e.g., [118, p. 2 and V.13, V.14]).

In particular, assuming that $g \ge 1$, the group $J_{\mathfrak{m}}$ has non-trivial abelian, toric and unipotent parts as soon as the support of $\mathfrak{m}$ contains two distinct points, one of which at least has coefficient $\ge 2$.

(2) Let $C$ be an irreducible projective curve over an algebraically closed field. Let $\tilde{C} \to C$ be the normalization of $C$, and for $x \in C(k)$, define $m_x$ to be the cardinality of the fiber of $\tilde{C} \to C$ over $x$. Then $\text{Pic}^0(C)$ has dimension $\dim H^1(C, \mathcal{O}_C)$, and it is an extension

$$ 0 \to K_C \to \text{Pic}^0(C) \to \text{Jac}(\tilde{C}) \to 0 $$

of the jacobian of the normalization $\tilde{C}$, with kernel $K_C$ which is an extension of a torus of dimension

$$ \sum_{x \in C - U} (m_x - 1) $$

by a unipotent group, of dimension therefore equal to

$$ \dim H^1(C, \mathcal{O}_C) - g(\tilde{C}) - \sum_{x \in C - U} (m_x - 1) $$

(see, e.g., [100, Def. 5.13, Th. 7.5.19, Lemma 5.18]).
Note furthermore that these two classes of algebraic groups are closely related (e.g., any generalized jacobian $J_m$ is the Picard group of some singular curve).

(2) All the examples of Sidon morphisms in Proposition 8.11 can be interpreted in terms of generalized jacobians. For instance, consider the curve $C = \mathbf{P}^1$ over $k$, and the effective divisor $S = (0) + 2(\infty)$, so that $U = \mathbf{P}^1 - \{0, \infty\} = G_m$. According to the above, the generalized jacobian $J_S$ is isomorphic to $G = (G_m \times (G_m \times G_a))/G_m^\Delta$, where the subgroup $G_m^\Delta$ is embedded diagonally by $x \mapsto (x, (x, 0))$. An isomorphism $\varphi: J_S \to G$ is given as follows: given a divisor E of degree 0 on $\mathbf{P}^1$, represent it as the divisor of a rational function $g: \mathbf{P}^1 \to \mathbf{P}^1$, and let

$$ \varphi(E) = (g(0), (g(\infty), \frac{g'}{g}(\infty))) $$

(this can be checked from the description in [118, p. 2 and V.13, V.14]). The morphism $G \to G_m \times G_a$ given by $(x, (y, a)) \mapsto (xy^{-1}, a)$ is an isomorphism, and using it to identify $G$ with $G_m \times G_a$, the formula above becomes $\varphi(E) = (\frac{g(1)}{g(\infty)}, \frac{g'}{g}(\infty))$.

Consider then the morphism $U = G_m \to J_S$ defined using the divisor $D = (1)$. Then the morphism $s_D: x \mapsto (x) - (1)$ is given by $s_D(x) = (x, 1 - x)$ (take $g(t) = (t - x)/(t - 1)$ to compute $\varphi((x) - (1))$). This is a Sidon morphism, the argument for this being identical with that of Proposition 8.11 (1).

We refer again to [37, § 2] for more discussion, in particular in comparison with the paper of Eberhard and Manners.

## 8.5. Gabber's torus trick

We discuss here another criterion to have a large tannakian group that also involves Sidon sets, but in a very different manner from their appearance in the previous sections. This criterion is difficult to apply for an individual object, but it leads to simple specialization results.

We use a version of Gabber's "torus trick" (see [70, Th. 1.0]). The following statement is specialized to the case of $SL_r$ and written in the language of compact Lie groups.

**THEOREM 8.15 (Gabber).** Let $V$ be a finite-dimensional complex vector space of dimension $r \ge 1$, and let $G$ be a connected semisimple compact subgroup of $GL(V)$ that acts irreducibly on $V$. Let $D$ be the subgroup consisting of the elements of $GL(V)$ that are diagonal with respect to some basis, and let $\chi_1, \dots, \chi_r$ be the characters $D \to C^\times$ giving the coefficients of the elements of $D$.

Let $A \subset D$ be a subgroup of the normalizer of $G$ in $GL(V)$. Let $S \subset \tilde{A}$ be the subset of the group of characters of $A$ given by the restrictions to $A$ of the diagonal characters $\chi_i$. If $|S| = r$ and $S$ is a Sidon set in $\tilde{A}$, then $G = SU(V)$.

**REMARK 8.16.** Properly speaking, Gabber's original result implies here that $G$ contains a maximal torus of $SU(V)$, and the fact that $G$ is semisimple and connected then implies that $G$ is $SU(V)$ (see, e.g., [13, p. 36, prop. 13]).

We emphasize that the subgroup $A$ can be arbitrary: it may be finite, and need not be closed.

We can then deduce the following criterion.

**PROPOSITION 8.17.** Let $G$ be a connected commutative algebraic group over the finite field $k$. Let $M$ be a simple perverse sheaf on $G$ which is pure of weight 0 and of tannakian dimension $r \ge 1$. Assume that $M$ is generically unramified.
The geometric tannakian group $G_M^{\text{geo}}$ contains $SL_r$ if and only there exists an unramified character $\chi \in \hat{G}(k_n)$ for some integer $n \ge 1$ such that the eigenvalues of $\Theta_{M,k_n}(\chi)$ are distinct and form a Sidon set in $C^\times$.

PROOF. Suppose that $G_M^{\text{geo}}$ contains $SL_r$. Let $U \subset SU_r(C)$ be the set of matrices whose eigenvalues are distinct and form a Sidon set in $C^\times$. This is an open set (for the Lie group topology), so that equidistribution implies

$$ \liminf_{N \to +\infty} \frac{1}{N} \sum_{1 \le n \le N} \frac{1}{|G(k_n)|} \sum_{\Theta_{M,k_n}(\chi) \in U} 1 > 0, $$

and hence there exists $n \ge 1$ and $\chi \in \hat{G}(k_n)$ such that $\Theta_{M,k_n}(\chi) \in U$.

Conversely, if an unramified character $\chi \in \hat{G}(k_n)$ exists with $\Theta_{M,k_n}(\chi) \in U$, then we can apply Theorem 8.15 to the group A generated by a fixed element in the conjugacy class $\Theta_{M,k_n}(\chi)$, and to the neutral component of the geometric tannakian group of M (which is normalized by $G_M^{\text{ari}}$, since $G_M^{\text{geo}}$ is normal in $G_M^{\text{ari}}$ by Proposition 3.40 and its neutral component is a characteristic subgroup). $\square$

In general, we do not have robust methods to check the existence of a character with the desired properties. However, we may combine this with a specialization argument.

PROPOSITION 8.18. Let $G$ be a connected commutative algebraic group over the finite field $k$. Let $M$ be a simple perverse sheaf on $G$ which is pure of weight 0 and of tannakian dimension $r \ge 1$. Assume that $M$ is generically unramified. Let $f: G \to H$ be a morphism of commutative algebraic groups over $k$.

Suppose that the object $N = \text{Rf}_1 M$ is a geometrically simple perverse sheaf on $H$ that is pure of weight 0, and suppose that $\chi \circ f$ is unramified for $M$ whenever $\chi$ is unramified for $N$.

If the geometric tannakian group $G_N^{\text{geo}}$ contains $SL_r$, then $G_M^{\text{geo}}$ contains $SL_r$.

PROOF. By Proposition 8.17, the assumption implies that there exists a character $\chi \in \hat{H}(k_n)$ unramified for $N$ for which $\Theta_{N,k_n}(\chi)$ has distinct eigenvalues forming a Sidon set. Since $\Theta_{M,k_n}(\chi \circ f)$ has the same characteristic polynomial, the character $\chi \circ f \in \hat{G}(k_n)$ has the same property; by Proposition 8.17 again, it follows that $G_M^{\text{geo}}$ contains $SL_r$. $\square$

## 8.6. Recognition criteria for $E_6$

We include here a criterion of Krämer to recognize the exceptional group $E_6$ in one of its 27-dimensional faithful representations (we always mean by $E_6$ the simply-connected form).

PROPOSITION 8.19 (Krämer). Let $G$ be a connected semisimple linear algebraic group over $\overline{Q}_\ell$ or $C$ and $\varrho$ an irreducible faithful 27-dimensional representation of $G$. If the 729-dimensional representation $\text{End}(\varrho)$ of $G$ contains an irreducible 78-dimensional subrepresentation, then $G$ is isomorphic to the exceptional group $E_6$ and $\varrho$ is one of its two fundamental 27-dimensional representations.

See [92, Lemma 4] for the proof. We will apply this in Section 11.2, although somewhat differently than we use Larsen's Alternative. The following criterion is closer to the spirit of the latter, and might have interesting applications (see again Section 11.2 for an attempt).
PROPOSITION 8.20. Let $\mathbf{G}$ be a connected semisimple linear algebraic group over $\overline{\mathbb{Q}}_\ell$ or $\mathbb{C}$, and
let $\rho$ be a faithful representation of $\mathbf{G}$ of dimension 27. Then $\mathbf{G}$ is isomorphic to the exceptional
group $\mathbf{E}_6$ and $\rho$ to one of the two fundamental 27-dimensional representations of $\mathbf{G}$ if and only if
$M_4(\mathbf{G}, \rho) = 3$ and $\rho$ is not self-dual.

PROOF. Suppose first that $\mathbf{G} = \mathbf{E}_6$ and $\rho$ is one of its fundamental representations of dimension 27. These representations are not self-dual (see [11, Table 1, p. 213]). Using the Weyl dimension formula (see [12, Th. 2, p. 151] and [11, Pl. V, p. 260]), we see that the dimensions of the irreducible representations of $\mathbf{E}_6$ that may possibly occur in the 729-dimensional representation on $\mathrm{End}(\rho)$ are 1, 27, 78, 351, 650. We know that the trivial representation appears once in $\mathrm{End}(\rho)$, and that the 78-dimensional adjoint representation $\mathrm{Ad}$ appears at least once. But the equation

$$729 - 79 = 650 = 27a + 78b + 351c + 650d$$

has the unique non-negative integral solution $(a, b, c, d) = (0, 0, 0, 1)$ (looking modulo 3, it becomes $d \equiv 1 \pmod{3}$). So we must have an isomorphism

$$\mathrm{End}(\rho) \simeq \mathbf{1} \oplus \mathrm{Ad} \oplus \rho_{650},$$

where $\rho_{650}$ has dimension 650, and hence the fourth moment $M_4(\mathbf{E}_6, \rho)$ is equal to 3. (This is also noted without proof by Katz [72, Rem. 1.2.3].)

We now prove the converse, and assume that $M_4(\mathbf{G}, \rho) = 3$ and $\rho$ is not self-dual. Since the fourth moment is $\le 5$, the representation $\rho$ is irreducible (Theorem 8.5, (1)). Now let

$$\mathbf{G}_1 \times \cdots \times \mathbf{G}_k \to \mathbf{G} \xrightarrow{\rho} \mathbf{G}\mathbf{L}_{27}$$

be the representation obtained from the decomposition of the algebraic universal covering of $\mathbf{G}$ in product of almost simple groups. This composition decomposes as an external tensor product

$$\rho_1 \boxtimes \cdots \boxtimes \rho_k$$

of irreducible representations of $\mathbf{G}_i$. We then have

$$3 = M_4(\mathbf{G}, \rho) = \prod_{i=1}^{k} M_4(\mathbf{G}_i, \rho_i)$$

by (8.3). The condition $M_4(\mathbf{G}_i, \rho_i) = 1$ is impossible (since it implies that $\dim(\rho_i) = 1$, and hence $\rho_i$ would be trivial and this contradicts the faithfulness assumption), so we have a single factor $\mathbf{G}_1$.

The representation $\rho_1$ is not self-dual, which implies that the root system of $\mathbf{G}_1$ (and hence of $\mathbf{G}$) can only be of type $\mathbf{E}_6$, or $A_l$ for $l \ge 2$ or $D_l$ with $l \ge 3$ odd (see, e.g. [12, p. 132, prop. 12], combined with the fact that the longest element of the Weyl group acts by $-\mathrm{Id}$ for the other simple root systems).

The groups of type $A_l$ with $l \ge 2$ which have a 27-dimensional irreducible representation are of type $A_2$ (the representation with highest weight $2\varpi_1 + 2\varpi_2$, in the standard notation of Bourbaki) or $A_{26}$ (the standard representation). In the first case, the representation is actually self-dual, and in the second case, the fourth moment is equal to 2, so these are excluded (in particular, groups of type $D_3 = A_2$ are also excluded).

Let $l \ge 5$ be an odd integer. The representations of groups of type $D_l$ which are not self-dual and have smallest possible dimension are the half-spin representations of dimension $2^{l-1}$ (see [12, p. 210]). Thus only $D_5$ could possibly give rise to a representation of dimension 27; but one can check that there is no representation of this dimension of a group of type $D_5$ (e.g., because of the Weyl Dimension Formula, see [12, Th. 2, p. 151]).
We conclude that the group $G_1$ must be $E_6$; since its 27-dimensional representations are faithful, the projection $G_1 \to G$ is an isomorphism. $\square$

REMARK 8.21. This criterion also shows that it may happen that the fourth moment $M_4(G, V)$ of a representation of a group $G$ is equal to 3, but the representation $V$ is not self-dual.

## 8.7. Finiteness of tannakian groups on abelian varieties

The following result strenghtens Theorem 5.7 in situations when one can apply Larsen's Alternative to the fourth moment on abelian varieties.

PROPOSITION 8.22. Let $M$ be a geometrically simple perverse sheaf of weight zero on a simple abelian variety $A$ over $k$. Let $d$ be the tannakian dimension of $M$. If the group $G_M^{\text{ari}}$ is virtually central, then the object $\text{End}(M)$ in $\mathbf{P}_{\text{int}}(G)$ is punctual and the fourth moment of $G_M^{\text{ari}}$ is equal to $d^2$.

PROOF. We observe that $G_M^{\text{ari}}/(Z \cap G_M^{\text{ari}})$ is the arithmetic tannakian group of the arithmetically semisimple object $\text{End}(M)$, and apply Theorem 5.7 to obtain the first conclusion.

In particular, this implies that $\text{End}(M)$, as a representation of $G_M^{\text{ari}}$, is a direct sum of characters. From (8.4), applied to a decomposition in sum of characters, it follows that

$$M_4(G_M^{\text{ari}}) \geq d^2.$$

On the other hand, let $K$ be a maximal compact subgroup of $G_M^{\text{ari}}(C)$, and $\mu$ its Haar probability measure. By (8.2) and Schur's Lemma, we derive the inequality

$$M_4(G_M^{\text{ari}}) = \int_K |\text{Tr}(g)|^4 d\mu(g) \leq d^2 \int_K |\text{Tr}(g)|^2 d\mu(g) = d^2,$$

which concludes the proof. $\square$

REMARK 8.23. There may exist irreducible subgroups $G$ of $\mathbf{GL}(V)$ with fourth moment equal to $\dim(V)^2$. Indeed, this is the case, for instance, of any group which has the property that all irreducible representations with trivial central character have dimension 1, since only such representations can appear in the decomposition of $\text{End}(V)$. A concrete example is given by finite Heisenberg groups (see, e.g., the paper [53] of Gérardin for the relevant facts).
# The product of the additive and the multiplicative groups

## 9.1. Introduction

In this chapter, we consider what is perhaps the simplest case of our equidistribution results beyond those of the additive group and the multiplicative group, namely the case of $G = G_m \times G_a$. Concretely, this means that we are looking at the distribution of two-parameter exponential sums of the type

$$ (9.1) \qquad \frac{1}{p} \sum_{(x,y) \in \mathbf{F}_p^\times \times \mathbf{F}_p} \chi(x) e\left(\frac{ay}{p}\right) t(x, y), $$

where $p$ is a prime number, $\chi$ a complex-valued multiplicative character of the finite field $\mathbf{F}_p$, and the function $t$ is a trace function on $G_m \times G_a$ over $\mathbf{F}_p$. In practice, we mostly consider the analogues over extensions of $\mathbf{F}_p$ of degree $n \to +\infty$, but we will also discuss an horizontal statement in Corollary 9.16.

Throughout this chapter, we denote by $k$ a finite field with an algebraic closure $\bar{k}$, and by $\ell$ a prime different from the characteristic of $k$. We also fix a non-trivial additive character $\psi: k \to \overline{\mathbb{Q}}_\ell^\times$. For every $n \ge 1$, we define $\psi_n = \psi \circ \text{Tr}_{k_n/k}$, a non-trivial additive character of the extension $k_n$ of $k$ of degree $n$ in $\bar{k}$.

We always denote by $G$ the group $G_m \times G_a$, and we will denote by $p_1$ and $p_2$ the projections $G \to G_m$ and $G \to G_a$. For any $n \ge 1$ and any pair $(\chi, a)$ of an $\ell$-adic character of $k_n^\times$ and an element of $k_n$, we will sometimes denote by $\langle \chi, a \rangle$ the character $(x, y) \mapsto \chi(x)\psi_n(ay)$ of $G(k_n)$, and by $\mathcal{L}_{\chi,a}$ the corresponding $\ell$-adic character sheaf.

We first state the specialization of Theorem 4.8 to this case, showing that there is always some equidistribution statement for the sums (9.1) in the vertical direction.

**THEOREM 9.1.** Let $M$ be an arithmetically semisimple $\ell$-adic perverse sheaf on $G_m \times G_a$ over $k$, with trace function over $k_n$ denoted $t(x, y; k_n)$. Assume that $M$ is pure of weight zero.

There exist an integer $r \ge 0$ and a reductive subgroup $G \subset GL_r$ such that the sums

$$ S_n(a, \chi) = \sum_{(x,y) \in k_n^\times \times k_n} \chi(x) \psi_n(ay) t(x, y; k_n), $$

where $(a, \chi)$ are pairs of an element of $k_n$ and a multiplicative character of $k_n^\times$, become equidistributed on average as $n \to +\infty$, with limit measure the image under the trace of the Haar probability measure on a maximal compact subgroup of $G(C)$.

With $G = G_m \times G_a$, $G = G_M^{\text{ari}}$ and $r$ the tannakian dimension of $M$, this is Theorem 4.8 for the object $M$.

The remainder of this chapter will be dedicated to the exploration of special examples. We consider in particular examples where the object $M$ (and hence the trace function in (9.1)) is supported on the “diagonal” $y = x$. Larsen’s Alternative will allow us to prove, with surprisingly
little computation, that in this case the group $\mathbf{G}$ in Theorem 9.1 is always essentially as large as possible.

More precisely, we first define $\Delta: \mathbf{G}_m \to \mathbf{G}_m \times \mathbf{G}_a$ to be the diagonal embedding $x \mapsto (x, x)$; this is a closed immersion. Define the diagonal in $\mathbf{G}_m \times \mathbf{G}_a$ to be the image of $\Delta$, and $j: \mathbf{G}_m \to \mathbf{G}_a$ to be the open immersion.

For any morphism $\lambda: \mathbf{G}_m \to \mathbf{G}$, for an integer $n \ge 1$ and a pair $(\chi, a) \in \hat{\mathbf{G}}_m(k_n) \times k_n$, we denote by $\mathcal{L}_{\chi, a}^{\lambda}$ the sheaf $\lambda_*(\mathcal{L}_{\chi} \otimes j^*\mathcal{L}_{\psi(ay)})$ on $\hat{\mathbf{G}}_k$.

**THEOREM 9.2.** With notation as in Theorem 9.1, suppose that the input object $M$ is geometrically simple and supported on the diagonal. Suppose that $M$ is not punctual and not geometrically isomorphic to $\mathcal{L}_{\eta, b}^{\Delta}[1]$ for some $\langle \eta, b \rangle \in \hat{\mathbf{G}}(k_n)$. Then the integer $r$ is $\ge 2$ and the group $\mathbf{G}$ contains $\mathbf{SL}_r$.

We will see that we can in fact fairly often show that $\mathbf{G} = \mathbf{GL}_r$, and in that setting the sums

$$S_n(\chi, a) = \sum_{x \in k_n^\times} \chi(x)\psi_n(ax)t_M(x, x; k_n)$$

tend to be distributed like the trace of a random matrix in $\mathbf{U}_r(\mathbf{C})$, almost independently of the input object $M$.

**EXAMPLE 9.3.** Let $a \in k$, and let $\chi$ be a multiplicative character of $k$. The Kloosterman–Salié sums over $k$ are defined by

$$\mathrm{Kl}(\chi, a; k) = \frac{1}{\sqrt{|k|}} \sum_{x \in k^\times} \chi(x)\psi(ax + x^{-1}).$$

These sums have been studied extensively, in particular because of their applications in the analytic theory of modular forms (see the surveys [86] or [59]). We may fix $\chi$, obtaining a family of exponential sums parameterized by $a$: this is the discrete additive Fourier transform of the function which is 0 at $x = 0$ and otherwise maps $x$ to $\chi(x)\psi(x^{-1})$. Alternatively, we may fix $a$, and then we are considering the discrete Mellin transform of the function $x \mapsto \psi(ax + x^{-1})$. These are both well-known examples of their respective theories, and their distribution properties are as follows:

- If $\chi$ is the trivial character, we have Kloosterman sums, which are equidistributed with respect to the Sato–Tate measure, that is, to the image of the Haar probability measure on the space of conjugacy classes of $SU_2(\mathbf{C})$; this reflects the fact that the geometric and arithmetic monodromy groups for the $\ell$-adic Fourier transform of the extension by zero of $\mathcal{L}_{\psi(x^{-1})}$ are both equal to $\mathbf{SL}_2$, by work of Katz [69, Thm. 11.1].

- If the characteristic $p$ of $k$ is odd and $\chi$ is the character of order 2, then we have Salié sums, whose arithmetic monodromy group is a finite subgroup of $\mathbf{SL}_2$, isomorphic to a semi-direct product of $\mathbf{F}_p^\times$ and $\mathbf{Z}/2\mathbf{Z}$ (this can be deduced from [70, Cor. 8.9.2]), which shows that the corresponding sheaf is Kummer-induced). The finiteness of the group reflects the fact that Salié sums can be computed elementarily (see, e.g., [14, p. 288, Exerc. 50]), and is also an analogue of the fact that Bessel functions with half-integral index are elementary functions (see, e.g., [14, p. 269, Exerc. 20]).

- If $p \ge 7$ and $\chi$ is fixed, but $\chi^2$ is non-trivial, then the neutral component of the geometric monodromy group is $\mathbf{SL}_2$, but the determinant of geometric monodromy group is not trivial, and more precisely has order equal to the order of $\chi$; see [70, Th. 8.11.3, Lemma 8.11.6].
- If instead we fix $a \in \mathbf{F}_p^\times$ and vary the multiplicative character $\chi$, then the geometric tannakian group (which coincides with the one associated by Katz's theory in [75], see Appendix B) contains $\mathbf{SL}_2$ for all $a$. Indeed, the sheaf $\mathcal{L}_{\psi(ax+x^{-1})}$ on $\mathbf{G}_m$ is not geometrically isomorphic to any of its non-trivial multiplicative translates by the same argument as in the proof of [75, Th. 14.2], and hence the characterisation in [75, Cor. 8.3] shows that this tannakian group is Lie-irreducible; since it is a subgroup of $\mathbf{GL}_2$, it necessarily contains $\mathbf{SL}_2$. If $a = -1$, then the arithmetic and geometric tannakian groups are both equal to $\mathbf{SL}_2$ (this is the case of the Evans sums in [75, Th. 14.2]). In general, the tannakian determinant is geometrically isomorphic to the skyscraper sheaf at $\alpha = -1/a$ (so its Mellin transform is proportional to $\chi \mapsto \chi(\alpha)$) by [75, Th. 21.1]. Letting $n$ denote the order of $\alpha$ in the finite group $\mathbf{F}_p^\times$, it follows that the geometric tannakian group consists of those matrices whose determinant is an $n$th root of unity.

- If $a = 0$ and we vary $\chi$, we have Gauss sums; the arithmetic and geometric tannakian groups are equal to $\mathbf{GL}_1$.

The relation with Theorem 9.1 is the following: we are considering the finite field $k$, and the perverse sheaf $M$ of weight zero is $M = \Delta_*\mathcal{L}[1](1/2)$, where $\mathcal{L}$ is the lisse sheaf $\mathcal{L} = \mathcal{L}_{\psi(x^{-1})}$ of rank one; it is geometrically simple, and perverse since $\Delta$ is a closed immersion (Corollary A.8). The group $\mathbf{G}$ of Theorem 9.1 is then $\mathbf{GL}_2$ (as follows from Theorem 9.2).

Note that when we specialize to a fixed character $\chi$ or a fixed $a$, we obtain a monodromy group or a tannakian group that is a subgroup of $\mathbf{G}$ (as seems natural), which has the following property: the identity component of the derived group $\mathbf{G}'$ is independent of $\chi$ (resp. $a$), except for a finite exceptional set. In fact, the exceptional set for fixed $\chi$ contains only the Legendre character (if $p$ is odd), and the exceptional set for fixed $a$ contains only $a = 0$.

Note also that when we vary $\chi$ for a fixed, only the neutral component of the identity of the geometric tannakian group is independent of $\chi$, but the tannakian group is usually not connected.

Finally, observe that here none of the “specialized” geometric tannakian groups for either $\mathbf{G}_a$ or $\mathbf{G}_m$ coincides with the geometric tannakian group $\mathbf{G} = \mathbf{GL}_2$. However, in an intuitive sense, the collection of all of them “generate” this group.

We expect these phenomena to be very general, and we will consider such questions in greater generality in later works.

**REMARK 9.4.** (1) Theorem 9.2 applies for instance to one-variable exponential sums of the form

$$ \frac{1}{\sqrt{|k|}} \sum_{x \in k^\times} \chi(x) \eta(g(x)) \psi(ax + f(x)) $$

for suitable polynomials $f$ and $g$ and for a multiplicative character $\eta$.

It is worth noting that, even if we are only interested in the distribution of these one-variable sums (and not in the more general sums allowed by Theorem 9.1 with a two-variable trace function), the proof of Theorem 9.1, passing through the tannakian machinery, requires the consideration of objects supported on all of $\mathbf{G}_m \times \mathbf{G}_a$, simply because the convolution of two objects on $\mathbf{G}_m \times \mathbf{G}_a$ that are supported on the diagonal $\Delta$ will be supported on the product set $\Delta \cdot \Delta = \mathbf{G}_m \times \mathbf{G}_a$.

(2) Remark 8.14 (2), suggests a different interpretation of Theorem 9.2. Indeed, using this remark, we can view $\mathbf{G}_m \times \mathbf{G}_a$ as a generalized jacobian of $C = \mathbf{P}^1$ and the diagonal morphism $\mathbf{G}_m \to \mathbf{G}_m \times \mathbf{G}_a$ as a morphism of the type $x \mapsto (x) - (D)$ for a suitable divisor $D$ on $\mathbf{G}_m$. For a
perverse sheaf of weight zero on $\mathbf{G}_m$, we have the arithmetic Mellin transform

$$ \chi \mapsto \sum_{x \in k_n^\times} t_M(x; k_n) \chi(x) $$

as in the work of Katz, which may have a variety of tannakian groups (see [75, Ch. 14 to 27] for examples involving for instance $\mathbf{SL}_n$, $\mathbf{GL}_n$, $\mathbf{O}_{2n}$, $\mathbf{SO}_n$, $\mathbf{Sp}_{2g}$ and $\mathbf{G}_2$). Then the further operation of twisting by an additive character $\psi$ leads to the sums

$$ (\chi, \psi) \mapsto \sum_{x \in k_n^\times} t_M(x; k_n) \chi(x) \psi(x) $$

which correspond to the diagonal object $\Delta_*M$ on the generalized jacobian $\mathbf{G}_m \times \mathbf{G}_a$. The theorem is then, analytically, an instance of the common situation where twisting an exponential sum by a generic additive character leads to “more random” exponential sums (here, replacing a potentially complicated tannakian group on $\mathbf{G}_m$ by one that in almost all cases contains the special linear group). Note however that the tannakian dimension may change when adding this extra twist.

**EXAMPLE 9.5.** The following case of two-variable equidistribution has been studied “by hand” by Kowalski and Nikeghbali [90, §4.1, Th. 11]. Let $d > 5$ be a fixed integer, and consider the sums

$$ S(\chi, a) = \frac{1}{\sqrt{|k|}} \sum_{t \in k} \chi(t^d - dt - a) $$

where the character $\chi$ is extended by $\chi(0) = 0$ if $\chi$ is non-trivial and $\chi(0) = 1$ if $\chi$ is trivial.

We can express these sums as Mellin transforms, namely

$$ S(\chi, a) = \sum_{x \in k^\times} \sum_{y \in k} \hat{S}(x, y) \chi(x) \psi(ay) $$

where

$$ \hat{S}(x, y) = \frac{1}{|\mathbf{G}(k)|} \sum_{\langle \chi, a \rangle \in \hat{\mathbf{G}}(k)} \overline{\chi(x)} \psi(-ay) S(\chi, a). $$

We compute then

$$ \begin{aligned} \hat{S}(x, y) &= \frac{1}{\sqrt{|k|}} \frac{1}{|\mathbf{G}(k)|} \sum_{\langle \chi, a \rangle \in \hat{\mathbf{G}}(k)} \overline{\chi(x)} \psi(-ay) \sum_{t \in k} \chi(t^d - dt - a) \\ &= \frac{1}{\sqrt{|k|}} \frac{1}{|\mathbf{G}(k)|} \sum_{t \in k} \sum_{a \in k} \psi(-ay) \sum_{\chi} \overline{\chi(x)} \chi(t^d - dt - a) \\ &= \frac{1}{|k|^{3/2}} \sum_{\substack{t \in k, a \in k \\ t^d - dt - a = x}} \psi(-ay) = \frac{1}{|k|^{3/2}} \sum_{t \in k} \psi(-y(t^d - dt - x)). \end{aligned} $$

Note that this trace function is not of diagonal type. It was proved however in [90] that when $|k| \to +\infty$ (including the horizontal case where $k = \mathbf{F}_p$ with $p \to +\infty$), the sums $S(\chi, a)$ become equidistributed like the trace of random matrices in the unitary group $\mathbf{U}_{d-1}(\mathbf{C})$. This was done by applying Deligne’s equidistribution theorem, and the computation of the relevant monodromy group by Katz, for each fixed $\chi$, and then averaging over $\chi$.

It would be interesting to recover this result directly from Theorem 9.1 (with $\mathbf{G} = \mathbf{GL}_{d-1}$), but it is not obvious how to do so: the reader can check that the computation of the fourth moment, for instance, is not at all straightforward.
REMARK 9.6. Finally we remark that since the key tool to compute tannakian groups for objects supported on the diagonal will be Larsen's Alternative combined with the Sidon property of the diagonal, one can prove similar results for objects of the form $[x \mapsto (x, x^3)]_*M$ on $G_a^2$, for $M$ on $G_a$ (in characteristic $\neq 3$), and objects of the form $[x \mapsto (x, 1-x)]_*M$ on $G_m^2$ for $M$ on $G_m - \{1\}$ (see Proposition 8.11, (5) and (6)). The corresponding exponential sums are of the form

$$ \sum_{x \in k_n} t_M(x; k_n) \psi_n(ax + bx^3) $$

and

$$ \sum_{x \in k_n^\times - \{1\}} t_M(x; k_n) \chi_1(x) \chi_2(1-x). $$

respectively.

## 9.2. Tannakian group for diagonal objects

We first compute the tannakian dimension $r$ for a perverse sheaf on $G = G_m \times G_a$ which is supported on the diagonal.

LEMMA 9.7. Let $M = \Delta_*(\mathcal{M})[1]$ for some geometrically irreducible middle extension sheaf $\mathcal{M}$ on $G_m$.

(1) The tannakian dimension $r$ of the object $M$ is given by the formula

$$ (9.2) \qquad r = \sum_{\lambda} \max(0, \lambda - 1) + \sum_{x \in \bar{k}^\times} (\mathrm{swan}_x(\mathcal{M}) + \mathrm{drop}_x(\mathcal{M})) + \mathrm{rank}(\mathcal{M}) + \mathrm{swan}_0(\mathcal{M}), $$

where $\lambda$ runs over the breaks of $\mathcal{M}$ at infinity, in the sense of [69, Ch. 1], counted with multiplicity.

(2) We have $r = 1$ if and only if $M = \mathcal{L}_{\eta,b}^{\Delta}[1]$ for some $\langle \eta, b \rangle \in \hat{G}$.

(3) For all but finitely many $a \in \bar{k}$, the tannakian dimension of $M_a = p_{1,*}M \otimes j^*\mathcal{L}_{\psi(ax)}$ on $G_{m,k(a)}$ is equal to $r$.

PROOF. (1) By Proposition 3.17, it is enough to determine the “generic” value of the dimension of the cohomology space

$$ H_c^0(G_{\bar{k}}, M \otimes p_1^*\mathcal{L}_\chi \otimes p_2^*\mathcal{L}_{\psi(ax)}) $$

as $\chi$ varies in $\hat{G}_m$ and $a$ in $\bar{k}$. We have a canonical isomorphism

$$ H_c^0(G_{\bar{k}}, M \otimes p_1^*\mathcal{L}_\chi \otimes p_2^*\mathcal{L}_{\psi(ax)}) = H_c^1(G_{m,\bar{k}}, \mathcal{M} \otimes \mathcal{L}_\chi \otimes j^*\mathcal{L}_{\psi(ax)}), $$

If $\chi$ is non-trivial, this space is also isomorphic to

$$ H_c^1(A_{\bar{k}}, j!(\mathcal{M} \otimes \mathcal{L}_\chi) \otimes \mathcal{L}_{\psi(ax)}). $$

For all but at most one value of $\chi$, the sheaf $j!(\mathcal{M} \otimes \mathcal{L}_\chi)$ is a Fourier sheaf in the sense of [70, (7.3.5)] (i.e., a middle extension sheaf $\mathcal{F}$ such that Deligne's Fourier transform is also a middle extension sheaf). Hence, the space $H_c^1(A_{\bar{k}}^1, j!(\mathcal{M} \otimes \mathcal{L}_\chi) \otimes \mathcal{L}_{\psi(ax)})$ is the stalk at $a$ of the Fourier transform of $j!(\mathcal{M} \otimes \mathcal{L}_\chi)$, and its generic value $r_\chi$ as $a$ varies in $\bar{k}$ is computed in [70, Lemma 7.3.9, (2)], namely

$$ r_{\chi} = \sum_{\lambda} \max(0, \lambda - 1) + \sum_{x \in \bar{k}} (\mathrm{swan}_x(j!(\mathcal{M} \otimes \mathcal{L}_{\chi})) + \mathrm{drop}_x(j!(\mathcal{M} \otimes \mathcal{L}_{\chi}))), $$
where $\lambda$ runs over the breaks at $\infty$ of $j_!(\mathcal{M} \otimes \mathcal{L}_\chi)$, counted with multiplicity. Since $\mathcal{L}_\chi$ is lisse on $\mathbf{G}_m$, the formulas

$$ \mathrm{swan}_x(j_!(\mathcal{M} \otimes \mathcal{L}_\chi)) = \mathrm{swan}_x(\mathcal{M}) \qquad \mathrm{drop}_x(j_!(\mathcal{M} \otimes \mathcal{L}_\chi)) = \mathrm{drop}_x(\mathcal{M}) $$

hold for any $x \in \bar{k}^\times$. Since $\mathcal{L}_\chi$ is tamely ramified at 0 for $\chi$ non-trivial, we have

$$ \mathrm{swan}_0(j_!(\mathcal{M} \otimes \mathcal{L}_\chi)) = \mathrm{swan}_0(\mathcal{M}) \qquad \mathrm{drop}_0(j_!(\mathcal{M} \otimes \mathcal{L}_\chi)) = \mathrm{rank}(\mathcal{M}) $$

for $\chi$ non-trivial, which leads to (9.2).

(2) Since $\mathrm{rank}(\mathcal{M}) \ge 1$, and all terms in the sum (9.2) are non-negative, we deduce that the condition $r = 1$ may hold only if $\mathcal{M}$ has rank 1 and $\mathcal{M}$ is lisse on $\mathbf{G}_m$, tame at 0, and has (unique) break at most 1 at $\infty$. Twisting by a suitable Kummer sheaf, we may then assume that $\mathcal{M}$ is lisse on $\mathbf{A}^1$, and it must then be geometrically isomorphic to an Artin–Schreier sheaf, which by untwisting implies that $M$ is geometrically isomorphic to some $\mathcal{L}_{\eta,b}^\Delta$.

(3) For the object $M_a = \mathcal{M}[1] \otimes j^*\mathcal{L}_{\psi(ax)}$ on $\mathbf{G}_{m,k(a)}$, the tannakian dimension is its compactly-supported Euler-Poincaré characteristic, which is equal to

$$ (9.3) \quad r_a = \mathrm{swan}_0(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}) + \mathrm{swan}_\infty(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}) \\ \qquad + \sum_{x \in \bar{k}^\times} (\mathrm{swan}_x(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}) + \mathrm{drop}_x(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)})) $$

(see (C.12)). Since $\mathcal{L}_{\psi(ax)}$ is lisse on $\mathbf{G}_a$, the formulas

$$ \mathrm{swan}_x(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}) = \mathrm{swan}_x(\mathcal{M}), \qquad \mathrm{drop}_x(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}) = \mathrm{drop}_x(\mathcal{M}) $$

hold for $x \in \bar{k}$.

Assume that $a \ne 0$. Let $\lambda$ be a break of $\mathcal{M}$ at infinity, and $V_\lambda$ the corresponding break-space. Then $V_\lambda \otimes \mathcal{L}_{\psi(ax)}$ coincides with the $\mu$-break-space $W_\mu$ of $\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}$ where $\mu = \max(1, \lambda)$, except possibly if $\lambda = 1$ and $\mathcal{L}_{\psi(-ax)}$ occurs in $V_\lambda$. Thus, for all but finitely many $a$, we have

$$ \mathrm{swan}_\infty(\mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}) = \sum_\mu \mu \dim W_\mu = \sum_\lambda \dim V_\lambda + \sum_{\lambda>1} (\lambda - 1) \dim V_\lambda \\ = \mathrm{rank}(\mathcal{M}) + \sum_\lambda \max(0, \lambda - 1), $$

which leads to $r_a = r$ by comparing (9.3) with (9.2). $\square$

REMARK 9.8. We will classify all objects of tannakian dimension 1 in Section 9.4, and the diagonal objects of tannakian dimension 2 in Section 9.3.

We continue with a lemma to exclude finite tannakian groups in the diagonal situation. The first step is to exploit the specific shape of $G$ to understand the structure of the set of characters which are not Frobenius unramified for suitable objects (or which are ramified, for objects which are generically unramified).

LEMMA 9.9. Let $M$ be a perverse sheaf on $G = \mathbf{G}_m \times \mathbf{G}_a$ and $N$ an object of $\langle\mathrm{M}\rangle^{\mathrm{ari}}$ which is arithmetically semisimple and pure of weight 0. For all but finitely many $a \in \bar{k}$, the set of $\chi \in \hat{\mathbf{G}}_m$ such that $\langle\chi, a\rangle$ is not Frobenius-unramified for $N$ is finite.

In particular, if $M$ has finite arithmetic tannakian group, then for all but finitely many $a \in \bar{k}$, the set of $\chi \in \hat{\mathbf{G}}_m$ such that $\langle\chi, a\rangle$ is ramified is finite.
PROOF. The first statement follows immediately from the proof of Proposition 3.38 combined with Theorem 2.31.

The last statement follows from the first as in the proof of Corollary 3.39. □

LEMMA 9.10. Let $C \subset G = \mathbf{G}_m \times \mathbf{G}_a$ be a line given by $y = \alpha x$ where $\alpha \in k^\times$.

Let $M$ be a geometrically simple perverse sheaf on $\mathbf{G}_m \times \mathbf{G}_a$ supported on $C$ and of weight zero. Assume that the arithmetic tannakian group $\mathbf{G}$ of $M$ is finite. Then $M$ is punctual.

PROOF. The assumption implies that $M$ is generically unramified by Corollary 3.39.

We assume that $M$ is not punctual to get a contradiction. Then $M$ is, up to twist and shift, the pushforward to $G$ of a middle extension sheaf $\mathcal{M}$ on $\mathbf{G}_m \simeq C$.

For all $a$, we denote $\mathcal{M}_a = \mathcal{M} \otimes j^*\mathcal{L}_{\psi(ax)}$; then $M_a = \mathcal{M}_a[1](1/2)$ is a perverse sheaf on $\mathbf{G}_m$.

By Lemma 9.9, there exists $n \ge 1$ and $a \in k_n$ such that for all but finitely many $\chi \in \hat{\mathbf{G}}_m$, the character $\langle \chi, a \rangle$ is unramified for $M$. The action of the Frobenius automorphism of $k_n$ on the space

$$H_c^0(\mathbf{G}_{\bar{k}}, \mathbf{M} \otimes \mathcal{L}_{\chi, a}) = H_c^0(\mathbf{G}_{m, \bar{k}}, \mathbf{M}_a \otimes \mathcal{L}_{\chi})$$

is then by assumption of finite order bounded independently of $\chi$. The corresponding unitary Frobenius elements $\Theta_{M_a, k_{nm}}(\chi)$, for $m \ge 1$, are then dense in a maximal compact subgroup $K$ of the complex points of the arithmetic tannakian group of the perverse sheaf $M_a$ on $\mathbf{G}_m$ by Corollary 4.14. It follows that $K$, and hence also $\mathbf{G}_{M_a}^{\text{ari}}$, is a finite group since a compact real Lie group has no non-trivial small subgroup. By Katz's results on finite tannakian groups on $\mathbf{G}_m$ (see Theorem B.2), this would imply that the perverse sheaf $M_a$ is punctual, which is a contradiction. □

We will now prove a slightly more general statement than Theorem 9.2.

THEOREM 9.11. Let $\lambda$: $\mathbf{G}_m \to \mathbf{G}_m \times \mathbf{G}_a$ be the closed embedding $\lambda(x) = (x, \alpha x)$ for some $\alpha \in k^\times$ and let $C$ be its image.

Let $M$ be a geometrically simple perverse sheaf on $\mathbf{G}_m \times \mathbf{G}_a$ supported on $C$ and of weight zero. Assume that $M$ is not punctual, and that $M$ is not geometrically isomorphic to $\mathcal{L}_{\eta,b}^{\lambda}[1](1/2)$ for some $\langle \eta, b \rangle \in \hat{\mathbf{G}}(k)$.

Let $r \ge 0$ be the tannakian dimension of $M$ and denote $\mathbf{G} = \mathbf{G}_M^{\text{ari}} \subset \mathbf{G}\mathbf{L}_r$.

We then have $r \ge 2$, the group $\mathbf{G}$ contains $\mathbf{SL}_r$ and the standard representation of $\mathbf{G}$ in $\mathbf{GL}_r$ is not self-dual.

Note that the last item implies in particular that $\mathbf{G}$ cannot be equal to $\mathbf{SL}_2$.

PROOF. We may assume that $\alpha = 1$. We first note that our assumptions and Lemma 9.7 imply that $r \ge 2$ (otherwise, $M$ would be punctual or geometrically isomorphic to some perverse sheaf $\mathcal{L}_{\eta,b}^{\lambda}[1](1/2)$).

We will apply Larsen's Alternative. The closed immersion $\lambda$ is a Sidon morphism (Proposition 8.11 (1)), and therefore we have $M_4(\mathbf{G}) = 2$ by Proposition 8.8 (since $r \ge 2$).

Our assumptions therefore imply that $M_4(\mathbf{G}) = 2$. By Larsen's Alternative (Theorem 8.5 (3)), it follows that either $\mathbf{G}$ contains $\mathbf{SL}_r$, or $\mathbf{G}/\mathbf{G} \cap Z$ is finite, where $Z \subset \mathbf{G}\mathbf{L}_r$ is the group of scalar matrices. We must show that this second case actually does not arise. We proceed by contradiction, assuming therefore that $\mathbf{G}/\mathbf{G} \cap Z$ is finite.

The intersection $\mathbf{G} \cap Z$ is either finite or equal to $Z$. In the first case, the group $\mathbf{G}$ would be finite, so that the object $M$ would be punctual by Lemma 9.10, which contradicts our assumptions.
So we are left with the case $G \cap Z = Z$. The object $\mathrm{End}(M)$ of $\langle M \rangle^{\mathrm{ari}}$ has tannakian group $G/G \cap Z$, which is then finite. In particular, this object is generically unramified (Corollary 3.39).

Let $n \ge 1$. For $a \in k_n$, the complex $M_a = p_{1,*}M \otimes j^*\mathcal{L}_{\psi(ax)}$ on $G_{m,k_n}$ is a perverse sheaf, geometrically simple and of weight 0, since the restriction of $p_1$ to $\mathbb{C}$ is an isomorphism. For all but a bounded number of $a \in k_n$, Lemma 9.9 implies that $M_a$ has the property that

$$S(\mathrm{End}(M_a), \chi) = |S(M_a, \chi)|^2 = |S(M, \langle \chi, a \rangle)|^2$$

take only finitely many values as $\chi \in \hat{G}_{m,k_n}$ varies. By equidistribution, this is only possible if the arithmetic tannakian group of the object $\mathrm{End}(M_a) \in \langle M_a \rangle^{\mathrm{ari}}$ on $G_{m,k_n}$ is finite. By Katz's Theorem B.2, this implies that $\mathrm{End}(M_a)$ is punctual, say

$$\mathrm{End}(M_a) = \bigoplus_{s \in S_a} n(a, s) \gamma_{a,s}^{\mathrm{deg}} \otimes \delta_s$$

for a subset $S_a \subset k_n^\times$, integers $n(a, s) \ge 1$ and unitary scalars $\gamma_{a,s}$. For all but finitely many $a \in \bar{k}$, we know also from Lemma 9.7 (3) that

$$r^2 = \dim \mathrm{End}(M) = \dim \mathrm{End}(M_a) = \sum_{s \in S_a} n(a, s).$$

Since all $\chi \in \hat{G}(k_n)$ are unramified for $\mathrm{End}(M_a)$ and $|\gamma_{a,s}| = 1$, we compute

$$\frac{1}{|\hat{G}_m(k_n)|} \sum_{\chi \in \hat{G}_m(k_n)} |S(M_a, \chi)|^4 = \frac{1}{|\hat{G}_m(k_n)|} \sum_{\chi \in \hat{G}_m(k_n)} |S(\mathrm{End}(M_a), \chi)|^2 \\ = \frac{1}{|\hat{G}_m(k_n)|} \sum_{\chi \in \hat{G}_m(k_n)} \left| \sum_{s \in S_a} n(a, s) \gamma_{a,s}^n \chi(s) \right|^2 = \sum_{s \in S_a} n(a, s)^2 |\gamma_{a,s}^n|^2 \ge r^2.$$

Averaging over $a \in k_n$, then letting $n \to +\infty$, it follows that $M_4(G) \ge r^2 \ge 4$, which is a contradiction.

Finally, we note that the tannakian dual of M is supported on the image of the diagonal under the inversion map of $G_m \times G_a$, namely on the hyperbola

$$\{(x^{-1}, -x) \mid x \in G_m\} \subset G_m \times G_a.$$

Since this is not a translate of the diagonal, the tannakian dual of M cannot be geometrically isomorphic to M. $\square$

EXAMPLE 9.12. (1) Suppose that $M = \mathcal{L}_{\eta,b}^{\Delta}[1](1/2)$ for some $(\eta, b) \in \hat{G}(k)$, which corresponds to the case excluded in Theorem 9.11. For $n \ge 1$, denote by $\eta_n$ the character $\eta \circ N_{k_n/k}$ of $k_n^\times$. Then the sums $S_n(\chi, a)$ are essentially Gauss sums, namely

$$S_n(\chi, a) = \frac{1}{|k|^{n/2}} \sum_{x \in k_n^\times} (\chi \eta_n)(x) \psi_n((a+b)x) = \frac{1}{|k|^{n/2}} \overline{(\chi \eta_n)}(a+b) \tau(\chi \eta_n, \psi_n)$$

(see (3) for the normalization).

The equidistribution properties of the Gauss sums are well-known (see for instance [69, Th. 9.5]), and one deduces easily that the arithmetic tannakian group of M is equal to $GL_1$. The fourth moment of all sums $S_n(\chi, a)$ converges to 2, as we saw in the previous proof, but the single contribution to the fourth moment of the (ramified) character $\langle \eta^{-1}, -b \rangle$ is $(|k_n| - 1)^4 / |k_n|^4 \to 1$. (See Proposition 9.20 for the classification of objects of tannakian dimension 1 in general.)
(2) Let $M_0 = \mathcal{K}\ell_{2,\psi}(1/2)$ be the Kloosterman complex of rank 2 on $\mathbf{G}_m$ (see (B.2)) associated to $\psi$, twisted to be pure of weight 0 (see [70, Th. 8.4.13]). It is of the form $\mathcal{M}_0[1](1/2)$ for some middle extension sheaf $\mathcal{M}_0$, pure of weight 0 as lisse sheaf on $\mathbf{G}_m$.

The object $M_0$ has tannakian dimension 1 and geometric tannakian group equal to $\mathbf{GL}_1$ as a $\mathbf{G}_m$-object (since it is a hypergeometric complex, see Theorem B.4). On the other hand, the object $M = \Delta_*M_0 = \Delta_*\mathcal{M}_0[1](1/2)$ on $\mathcal{G}$ has tannakian dimension 2, and arithmetic tannakian group $\mathbf{GL}_2$ by Lemma 9.7 and Theorem 9.11.

We compute the corresponding exponential sums to see the concrete meaning of the theorem in this case. For $n \ge 1$ and $\langle\chi, a\rangle \in \widehat{\mathcal{G}}(k_n)$, we have the formula

$$ 
\begin{aligned}
S_n(\chi, a) &= \frac{1}{|k_n|} \sum_{x \in k_n^\times} \left( \sum_{y \in k_n^\times} \psi_n(xy + 1/y) \right) \chi(x) \psi_n(ax) \\
&= \frac{1}{|k_n|} \sum_{y \in k_n^\times} \psi_n(1/y) \sum_{x \in k_n^\times} \psi_n((a+y)x) \chi(x).
\end{aligned}
$$

For $\chi$ non-trivial, extended by $\chi(0) = 0$, this is equal to

$$ S_n(\chi, a) = \frac{\tau(\chi, \psi_n)}{|k_n|} \sum_{y \in k_n^\times} \overline{\chi(a+y)} \psi_n(1/y). $$

In order to complete the determination of the tannakian group in the situation of Theorem 9.11, we need to compute the tannakian determinant of $M$. There are various tools to do this:

(1) one can attempt to compare the tannakian determinants for $M$ (supported on a line) with those on $\mathbf{G}_m$, which can often be computed using the results of Katz [75];

(2) one can use the relation between the tannakian determinant at $\langle\chi, a\rangle$ and the determinant of Frobenius acting on the cohomology group

$$ H_c^0(\mathcal{G}_{\bar{k}}, \mathcal{M}_{\langle\chi, a\rangle}) \simeq H_c^0(\mathbf{G}_{m, \bar{k}}, \mathcal{M}_a \otimes \mathcal{L}_{\chi}) $$

(with notation as above). The latter determinant (on a curve) may often be computed using the theory of local epsilon factors of Deligne and Laumon (see Appendix C). We will not give explicit examples here, but we perform a computation of this kind in Chapter 10 (see Proposition 10.10).

As an example of the first approach, we have for instance the following criterion:

**PROPOSITION 9.13.** Let $C \subset \mathcal{G} = \mathbf{G}_m \times \mathbf{G}_a$ be a line defined by $y = \alpha x$ where $\alpha \in k^\times$. Let $M$ be a geometrically simple perverse sheaf on $\mathbf{G}_m \times \mathbf{G}_a$ supported on $C$ and of weight zero. Assume that $M$ is not punctual, and that the restriction of $M$ to $C$ is not geometrically isomorphic to $\mathcal{L}_{\eta,b}[1]$ for some multiplicative character $\eta$ and some $b$. Let $r \ge 0$ be the tannakian dimension of $M$.

Suppose that for all but finitely many $a$, the tannakian determinant of $p_{1,*}M_{\langle 1,a\rangle}$ on $\mathbf{G}_m$ is geometrically of infinite order. Then we have $\mathbf{G} = \mathbf{GL}_r$.

**PROOF.** Since $\mathbf{G}$ contains $\mathbf{SL}_r$, it suffices to prove that the determinant of $\mathbf{G}$ is arithmetically of infinite order.

Since $p_1: C \to \mathbf{G}_m$ is an isomorphism, it follows that for any $a \in \mathbf{G}_a$, the object $N_a = p_{1,*}M_{\langle 1,a\rangle}$ on $\mathbf{G}_m$ is a perverse sheaf, and is arithmetically simple and pure of weight 0.

We claim that the assumption implies that the determinants of $\Theta_{M,k_n}(\langle\chi, a\rangle)$ are equidistributed on average on the unit circle, where $\langle\chi, a\rangle$ vary among Frobenius-unramified classes for the determinant. Indeed, denoting $\mathcal{X}$ this set of characters, we have for any non-zero integer $h \in \mathbb{Z}$ the
relation

$$ \frac{1}{|G(k_n)|} \sum_{\langle\chi, a\rangle \in \mathcal{X}(k_n)} \det(\Theta_{M, k_n}(\langle\chi, a\rangle))^h = \frac{1}{|k_n|} \sum_{a \in k_n} \frac{1}{|k_n^\times|} \sum_{\substack{\chi \in \hat{\mathbf{G}}_m(k_n) \\ \langle\chi, a\rangle \in \mathcal{X}(k_n)}} \det(\Theta_{N_a, k_n}(\chi))^h. $$

The contribution of those finitely many $a$ such that $N_a$ has geometrically finite-order determinant tends to 0. For the other values of $a$, we have

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{n \le N} \frac{1}{|k_n^\times|} \sum_{\substack{\chi \in \hat{\mathbf{G}}_m(k_n) \\ \langle\chi, a\rangle \in \mathcal{X}(k_n)}} \det(\Theta_{N_a, k_n}(\chi))^h = 0 $$

by equidistribution, in fact uniformly with respect to $a$ since the complexity of $\det(N_a)$ is bounded independently of $a$. We deduce that

$$ \lim_{N \to +\infty} \frac{1}{N} \sum_{n \le N} \frac{1}{|k_n|} \sum_{a \in k_n} \frac{1}{|k_n^\times|} \sum_{\substack{\chi \in \hat{\mathbf{G}}_m(k_n) \\ \langle\chi, a\rangle \in \mathcal{X}(k_n)}} \det(\Theta_{N_a, k_n}(\chi))^h = 0, $$

which proves the claim.

But by Theorem 4.4, the determinants of $\Theta_{M, k_n}(\langle\chi, a\rangle)$ are known to be equidistributed on average on the subset of the unit circle corresponding to the determinant of the arithmetic tannakian group of M; if the latter were finite, this would be a finite group of roots of unity. By contraposition, the result follows. $\square$

REMARK 9.14. If $\det(M)$ is known to be generically unramified, then it suffices to assume that the tannakian determinant of $p_{1,*}M$ on $\mathbf{G}_m$ is geometrically of infinite order, since in this case we can apply Proposition 3.45 to some twist $M_{\langle\chi_1, a_1\rangle}$ such that the set of characters $\chi$ for which the character $\langle\chi_1, a_1\rangle\langle\chi, 0\rangle$ is unramified is generic.

EXAMPLE 9.15. Proposition 9.13 applies for instance to objects of the form

$$ M = \mathcal{L}_{\eta(f)}[1](1/2) $$

where $\eta$ is a non-trivial multiplicative character of $k$, and $f \in k[X]$ is a polynomial such that $f(0) \ne 0$ with degree $d \ge 2$ such that $\eta^d$ is non-trivial, as explained by Katz in [75, Th. 17.5]. Indeed, in this case, the assumption of the proposition holds for all $a \ne 0$.

The dimension formula (9.2) shows that the tannakian dimension is $d+1$. Note that [75, Th. 17.5] provides the equidistribution for the subfamily with $a=0$, under the assumption that $f$ is not of the form $g(X^b)$ for some $b \ge 2$, but as traces of matrices in $U_d(C)$, because the corresponding object on $\mathbf{G}_m$ has tannakian dimension $d$. This means that the characters $\langle\chi, 0\rangle$ are examples of weakly-unramified characters for M which are not unramified (since they do not give the “right” dimension).

As explained in Remark 4.20 (2), we expect that we can apply Theorem 4.19 unconditionally to $G$. Thus this proposition should imply the following result:

COROLLARY 9.16. Let $\ell$ be a prime number. Assume that Theorem 4.19 holds for $G$. For all $p \ne \ell$, let $M_p$ be a geometrically simple perverse sheaf of weight zero on $(\mathbf{G}_m \times \mathbf{G}_a)_{F_p}$ supported on the diagonal with $c_u(M_p) \ll 1$, where $u$ is the natural locally-closed immersion $\mathbf{G}_m \times \mathbf{G}_a \hookrightarrow \mathbf{A}^2 \hookrightarrow \mathbf{P}^2$. Suppose that the tannakian dimension $r$ of $M_p$ is independent of $p$ and that $M_p$ satisfies the
assumption of Proposition 9.13 for $k = \mathbf{F}_p$. Then the sums

$$S(\chi, a; p) = \sum_{x \in \mathbf{F}_p^\times} t_{M_p}(x)\chi(x)e\left(\frac{ax}{p}\right),$$

for $\chi$ a multiplicative character of $\mathbf{F}_p$ and $a \in \mathbf{F}_p$, become equidistributed according to the trace of a random unitary matrix in $\mathbf{U}_r(\mathbf{C})$.

## 9.3. Diagonal objects of dimension 2

The computation of Lemma 9.7 allow us, for instance, to classify those sheaves $\mathcal{M}$ which give rise to geometrically simple perverse sheaves on the diagonal with tannakian dimension $r = 2$. Indeed, the (usual) rank of $\mathcal{M}$ must be either 1 or 2.

In the first case, one and only one of the following conditions must be true:

(1) $\mathcal{M}$ is lisse on $\mathbf{G}_m$, tamely ramified at 0 and has (unique) break at $\infty$ equal to 2; if the characteristic of $k$ is not equal to 2, then the only such sheaves are isomorphic to

$$\mathcal{L}_{\psi(ax^2+bx)} \otimes \mathcal{L}_{\eta}$$

where $a \neq 0$ and $\eta$ is a multiplicative character. The corresponding exponential sums are “twisted quadratic Gauss sums”.

(2) $\mathcal{M}$ is lisse on $\mathbf{G}_m$ and has Swan conductor 1 at 0 and unique break $\le 1$ at $\infty$; the only such sheaves are isomorphic to

$$\mathcal{L}_{\psi(a/x+bx)} \otimes \mathcal{L}_{\eta}$$

where $a \neq 0$ and $\eta$ is a multiplicative character (we recover the example of Kloosterman–Salié sums).

(3) there exists a unique $\beta \in \bar{k}^\times$ such that $\mathcal{M}$ is lisse on $\mathbf{A}^1 - \{\beta\}$, it has unique break $\le 1$ at $\infty$ and is tamely ramified at 0 and $\beta$; the only such sheaves are isomorphic to

$$\mathcal{L}_{\eta(x-\beta)} \otimes \mathcal{L}_{\xi(x)} \otimes \mathcal{L}_{\psi(\alpha x)}$$

where $\beta \neq 0$, $\alpha \in \mathbf{G}_a$ and $\eta$ and $\xi$ are multiplicative characters. The corresponding exponential sums are

$$\frac{1}{|k_n|^{1/2}} \sum_{x \in k_n^\times} \eta(x - \beta)(\chi\xi)(x)\psi((a + \alpha)x),$$

which can be seen as twisted Jacobi sums.

On the other hand, if $\mathcal{M}$ has rank 2, then it must be lisse on $\mathbf{G}_m$, tamely ramified at 0 and have breaks $\le 1$ at $\infty$. Up to twist by a multiplicative character, we obtain a sheaf lisse on $\mathbf{A}^1$ with breaks $\le 1$ at $\infty$. Since we assume $\mathcal{M}$ to be geometrically irreducible, the two breaks must be equal, say equal to $\lambda$. Their sum is the Swan conductor at $\infty$, which is also the Euler-Poincaré characteristic (since $\mathcal{M}$ is lisse on $\mathbf{G}_m$ and tame at 0, see (C.12)); thus either $\lambda = 1/2$ or $\lambda = 1$. The first case gives Euler-Poincaré characteristic equal to 1, so we have a hypergeometric sheaf of rank 2 by Katz's classification (see Theorem B.4, e.g., a Kloosterman sheaf of rank 2, with the corresponding sums described in Example 9.12 (2)). In the second case, we may have a pullback of such a sheaf by $x \mapsto x^2$. For the pullback of the Kloosterman sheaf, the exponential sums are then
given by the formulas

$$
\begin{align*}
S_n(\chi, a) &= \frac{1}{|k_n|} \sum_{x \in k_n^\times} \left( \sum_{y \in k_n^\times} \psi_n(xy + xy^{-1}) \right) \chi(x) \psi_n(ax) \\
&= \frac{1}{|k_n|} \sum_{y \in k_n^\times} \sum_{x \in k_n^\times} \chi(x) \psi_n(x(a + y + y^{-1})) \\
&= \frac{\tau(\chi, \psi_n)}{|k_n|} \sum_{\substack{y \in k_n^\times \\ a+y+y^{-1} \neq 0}} \overline{\chi(a + y + y^{-1})},
\end{align*}
$$

for $\chi$ non-trivial.

9.4. Negligible objects and objects of dimension one

We conclude our discussion of the group G = G_m × G_a by classifying the negligible objects as
well as the objects of tannakian dimension 1. This may be helpful for further investigations (e.g., to
compute the determinant of the tannakian group in some cases, or to apply the Goursat–Kolchin–
Ribet criterion, see [70, Prop. 1.8.2]).

We will denote by $\mathrm{FT}_{\psi/G_m}$ the relative Fourier transform functor $\mathrm{D}^b_c(G) \to \mathrm{D}^b_c(G)$, defined by

$$
\mathrm{FT}_{\psi/\mathbf{G}_m}(M) = \mathrm{R}q_{2,1}(q_1^*M \otimes \mathcal{L}_{\psi(ay)})
$$

where $q_1$ and $q_2$ are the two projections $\mathbf{G}_m \times \mathbf{G}_a \times \mathbf{G}_a \to \mathbf{G}_m \times \mathbf{G}_a$, and we use coordinates $(x, y, a)$ on $G \times \mathbf{G}_a = \mathbf{G}_m \times \mathbf{G}_a \times \mathbf{G}_a$. This functor satisfies the same basic properties as the Fourier transform over base fields (see, e.g., [78, § 2]), and in particular $\mathrm{FT}_{\psi/\mathbf{G}_m}(M)[1]$ is perverse if $M$ is perverse.

PROPOSITION 9.17. Let M be a simple perverse sheaf on G over $\bar{k}$.

The *perverse sheaf* M is negligible *if and only if* M is isomorphic to an object of the form

$$
(9.4) \qquad p_1^*(\mathbb{N}) \otimes \mathcal{L}_{\psi(ay)}[1]
$$

for some *perverse sheaf* N on $\mathbf{G}_m$ and some $a$, or to an object of the form

$$
(9.5) \qquad \mathcal{L}_{\chi}[1] \otimes p_2^*(M),
$$

for some *perverse sheaf* M on $\mathbf{G}_a$ and some *multiplicative character* $\chi$.

PROOF. It is elementary that the objects of the two forms in the statement are negligible (see
Example 3.5), so we need to prove the converse.

Let M be a simple negligible perverse sheaf on G. We consider the (shifted) Fourier transform

$$
F = \mathrm{FT}_{\psi/\mathbf{G}_m}(M)[1]
$$

of M relative to $\mathbf{G}_m$; this is a perverse sheaf on G. For $a \in \mathbf{G}_a$, the restriction $F_a$ of this complex to $\mathbf{G}_m \times \{a\}$ is isomorphic to $p_1!(M_{\psi(ay)})$. Hence, for $a$ generic, the object $F_a = (F|_{\mathbf{G}_m} \times \{a\})[-1]$ is a perverse sheaf by Proposition 1.1; moreover, if $a$ is such that $\mathbf{G}_m \times \{a\}$ intersects a dense open subset where M is lisse, the generic rank of $F_a$ is still zero.

We now distinguish cases according to the dimension $d$ of the support of F.

(1) If $d = 0$, then F is supported on finitely many points. Since M is simple and the Fourier transform preserves simple perverse sheaves, F is also simple. This implies that the support of F is irreducible, and hence it is a single point $(x, a)$. The point $a$ correspond to the character $\psi(ay)$ via
inverse Fourier transform. Hence, M is of the form $p_1^*(N) \otimes \mathcal{L}_{\psi(ay)}$, where N is a sheaf with finite support in $G_m$, which is an object of the form (9.4).

(2) If $d=1$, then the support of F is a curve $C \subset G_m \times G_a$. If, for generic $a \in G_a$, the intersection of C with $G_m \times \{a\}$ is non-empty, then the support of $F_a$ is finite and non-empty, contradicting the fact that this sheaf is of generic rank zero. Hence, for generic $a \in G_a$, the intersection of C with $G_m \times \{a\}$ is empty. We then deduce that $F_a = 0$ for generic $a$. Hence, C is a finite union of horizontal lines. As in (1), C is irreducible, and hence is of the form $G_m \times \{a\}$ for some $a \in G_a$. Hence, M is of the form $p_1^*(N) \otimes \mathcal{L}_{\psi(ay)}$ for some perverse sheaf N on $G_m$; this is again of the form (9.4).

(3) Finally, assume that $d=2$. Let $\eta$ be the generic point of $G_a$. Then $F_\eta$ is a perverse sheaf with Euler-Poincaré characteristic zero on $G_m$ over $k(\eta)$. By Proposition B.3, it follows that $F_\eta$, viewed as a perverse sheaf on $G_m$ over $k(\eta)$, is geometrically isomorphic to a Kummer perverse sheaf $\mathcal{L}_\chi[1]$ for some multiplicative character $\chi$. Hence, F is of the form $p_2^*(N') \otimes \mathcal{L}_\chi[1]$ for some perverse sheaf $N'$ on $G_a$. Taking the relative inverse (shifted) Fourier transform, we find that there exists some object N of $\mathrm{D}_c^b(G_a)$ such that M is isomorphic to $p_2^*(N) \otimes \mathcal{L}_\chi[1]$. $\square$

We will now classify the objects of tannakian dimension one.

By Proposition 3.48, the most obvious objects of tannakian dimension one on $G_m \times G_a$ are those of the form $H \times N$, for some simple hypergeometric complex $H$ on $G_m$ and some simple perverse sheaf $N$ on $G_a$ with Fourier transform of rank one (we refer again to Section B.4 for reminders concerning hypergeometric complexes, which are the objects of tannakian rank 1 on the multiplicative group). The next lemma provides another class of such objects.

LEMMA 9.18. Let $f \in \bar{k}(x)^\times$ be a rational function and $U$ a dense open set of $G_a$ where $f$ is defined and non-zero. Let $C \subset V = G_m \times U$, with coordinates $(x, a)$, be the curve with equation $f(a) = x$. Let $\bar{Q}_{\ell,f}$ be the intermediate extension to $G_m \times G_a$ of the constant sheaf on $C$ shifted by 1, and let $M_f$ be the inverse relative Fourier transform of $\bar{Q}_{\ell,f}$, also shifted by 1.

$$ (1) \quad \text{Write } f = f_1/f_2 \text{ with } f_i \in \bar{k}[x] \text{ coprime. Let } \tilde{C} \subset G \text{ be the curve with equation} \\ f_1(a) = f_2(a)x, $$

and let $i: \tilde{C} \to G$ be the closed immersion. We then have isomorphisms
$$ \bar{Q}_{\ell,f} \simeq i_*\bar{Q}_{\ell}[1] \simeq i_!\bar{Q}_{\ell}[1]. $$

$$ (2) \quad \textit{The perverse sheaf } M_f \textit{ on } G \textit{ has tannakian dimension one.} $$

$$ (3) \quad \text{For any } y_0 \in G_a, \text{ the restriction of } M_f \text{ to } G_m \times \{y_0\} \subset G \text{ is of the form } \mathcal{G}_{y_0}[2] \text{ for some sheaf } \mathcal{G}_{y_0} \text{ on } G_m, \text{ identified to a sheaf on } G_m \times \{y_0\}, \text{ of generic rank bounded by } \max(\deg(f_1), \deg(f_2)). $$

$$ (4) \quad \text{If } f \in k(x), \text{ then for } n \ge 1, \text{ the Fourier transform on } G(k_n) \text{ of the trace function of } M_f \\ \text{is given by} \\ (\chi, b) \mapsto |k_n|\chi(f(b)) $$

for $\langle \chi, b \rangle \in \hat{G}(k_n)$ in a generic set.

PROOF. The curve $\tilde{C}$ contains $C$, and the assumption that $f_1$ and $f_2$ are coprime implies that $\tilde{C}$ is smooth (since the partial derivative with respect to $x$ is $f_2(a)$, which is non-zero on $\tilde{C}$). It is irreducible since it is isomorphic to the open subset of $G_a$ where $f_1 f_2$ is non-zero by the projection $(x, a) \mapsto a$ with inverse $a \mapsto f_1(a)/f_2(a)$. Since $i_* = i_!$ for a closed immersion, it follows that $i_* \bar{Q}_\ell[1] = i_! \bar{Q}_\ell[1]$ is a perverse sheaf, and since it restricts to $\bar{Q}_{\ell,f}$ on $V$, these perverse sheaves are isomorphic.
We next show that for generic $\langle\chi, b\rangle \in \hat{G}$, we have

$$ \dim H_c^0(G_{\bar{k}}, (M_f)_{\langle\chi, b\rangle}) = 1, $$

which will prove (2).

This cohomology group can be computed by first taking the relative additive Fourier transform $F$ of $M_f$, restricting it to the line $G_m \times \{b\}$, then taking the cohomology of $F \otimes \mathcal{L}_\chi$ on $G_m \times \{b\}$. Since the Fourier transform $F$ of $M_f$ is $\overline{Q}_{\ell,f}$, there exists a dense open set $U$ of $G_a$ such that for $b \in U$, the restriction of $F$ to $G_m \times \{b\}$ is a rank one skyscraper sheaf supported on $f(b)$. Such a sheaf, tensored with any character $\mathcal{L}_\chi$, has its 0-th cohomology group of dimension 1.

In fact, the same argument shows that if $\langle\chi, b\rangle$ is defined over $k_n$, then the action of Frobenius on the one-dimensional space $H_c^0(G_{\bar{k}}, (M_f)_{\langle\chi, b\rangle})$ is $|k_n|\chi(f(b))$, which proves the last statement.

To prove (3), we observe that, by definition of the Fourier transform, yet another description of $M_f$ is

$$ M_f = R\phi_*\mathcal{L}_{\psi(-ay)}[2], $$

where $\phi$ is the restriction of the projection $(x, y, a) \mapsto (x, y)$ to the subvariety

$$ Z = \{(x, y, a) \in G_m \times G_a \times G_a \mid f_1(a) = xf_2(a)\} $$

of $G_m \times G_a^2$. Since $\phi$ is an affine quasi-finite morphism, we obtain (3) with

$$ \mathcal{G} = \tilde{\phi}_*\mathcal{L}_{\psi(-ay_0)}, $$

where $\tilde{\phi}$ is the restriction of $\phi$ to $Z_{y_0}$. This sheaf has generic rank bounded by the size of the fibers of $\phi$, and is $\le \max(\deg(f_1), \deg(f_2))$. $\square$

**REMARK 9.19.** The last statement amounts to the following computation of Fourier transform on $G(k_n)$: by the first part, writing $f = f_1/f_2$, where $f_i$ are polynomials without common factor, the perverse sheaf $\overline{Q}_{\ell,f}$ is the shifted constant sheaf on the smooth irreducible curve defined by $f_1(a) = xf_2(a)$ in $G$. Therefore, the trace function of $\overline{Q}_{\ell,f}$ at $(x, a) \in G(k_n)$ is equal to 1 if $f_1(a) = xf_2(a)$, and 0 otherwise, so the trace function of $M_f$ has the value

$$ \sum_{\substack{a \in k \\ f_1(a) = xf_2(a)}} \psi_n(-ay) $$

at $(x, y) \in k_n^\times \times k_n$. For $(\chi, b) \in \hat{G}(k_n)$, we get the Fourier transform

$$ \sum_{(x,y) \in G(k_n)} \chi(x) \psi_n(by) \sum_{\substack{a \in k_n \\ f_1(a) = xf_2(a)}} \psi(-ay) = \sum_{\substack{a \in k_n \\ f_1(a) = xf_2(a) \neq 0}} \sum_{y \in k_n} \chi(f(a)) \psi_n((b-a)y) $$

and this is equal to $|k_n|\chi(f(b))$ if $f_1(b)f_2(b) \neq 0$, and 0 otherwise.

The basic classes of objects of tannakian dimension 1 we have just described turn out to be sufficient to obtain all of them.

**PROPOSITION 9.20.** Let $M$ be a simple perverse sheaf on $G$ over $\bar{k}$. Assume that $M$ is in $\mathbf{P}_{\text{int}}(G)$. Then $M$ has tannakian dimension one if and only if there exist a rational function $f$, a simple hypergeometric complex $H$ on $G_m$ and a perverse sheaf $N$ on $G_a$ with Fourier transform of rank one such that $M$ is isomorphic to the convolution

$$ M_f *_{\text{int}} (H \boxtimes N). $$
PROOF. Since the tannakian dimension is multiplicative in convolutions, the “if” assertion follows from Lemma 9.18 and the fact that $H \boxtimes N$ has tannakian dimension 1 by an application of Proposition 3.48.

Conversely, let $M$ be a simple perverse sheaf on $G$ of tannakian dimension one. As in the dimension zero case, we consider the shifted Fourier transform $F = \text{FT}_{\psi/G_m}(M)[1]$ of $M$ relative to $G_m$. For generic $a \in G_a$, the object $F_a = F|_{G_m} \times \{a\}$ on $G_m \times \{a\}$ is perverse of generic rank one, as in the beginning of the proof of Proposition 9.17.

In particular, for the generic point $\eta$ of $G_a$, the object $F_\eta$, viewed as a perverse sheaf on $G_{m,k(\eta)}$, is of tannakian dimension one. By Theorem B.4 (2), it is isomorphic to a hypergeometric complex multiplicatively translated by a non-zero rational function $f(\eta)$ of $\eta$, and tensored by a rank one object on $k(\eta)$. Thus, there exists a dense open subset $W$ of $G_m \times G_a$ and an isomorphism

$$ (9.6) \qquad F|W \simeq p_2^*(N)|W \otimes m_f^*(H), $$

where $N$ is a perverse sheaf on $G_a$ of generic rank one, $H$ is a hypergeometric complex on $G_m$ and $m_f: W \to G_m$ is the morphism

$$ (x, a) \mapsto x/f(a) $$

(in particular, $x/f(a)$ is defined and non-zero for $(x, a) \in W$). Using [70, Th. 8.4.10] and Proposition 9.17, we may assume that $H$ is simple.

Let $\tilde{N}$ be the Fourier transform of $N$. We claim that there are isomorphisms

$$ (9.7) \qquad M \simeq (H \boxtimes 1)_{*\text{int}} M_f *_{\text{int}} (1 \boxtimes \tilde{N}) \simeq M_f *_{\text{int}} (H \boxtimes \tilde{N}), $$

which will conclude the proof. The second isomorphism follows from commutativity and associativity of the convolution combined with the isomorphism of Proposition 3.48, hence we need only check the first.

Let $P = (H \boxtimes 1)_{*\text{int}} M_f$; we need to show that $M$ is isomorphic to $P *_{\text{int}} (1 \boxtimes \tilde{N})$. We will do this by showing that the restriction to $W$ of their relative Fourier transforms are isomorphic; since both objects involved are perverse sheaves, this will give the result.

Precisely, denote

$$ P_! = (H \boxtimes 1)_{*!} M_f, \qquad P_* = (H \boxtimes 1)_{*\!} M_f. $$

We claim that there are isomorphisms

$$ (9.8) \qquad P_! \simeq P \simeq P_*, $$

$$ (9.9) \qquad \text{FT}_{\psi/G_m}(P)|W \simeq m_f^*(H), $$

$$ (9.10) \qquad \text{FT}_{\psi/G_m}(P *_{\text{int}} (1 \boxtimes \tilde{N})) \simeq p_2^*(N) \otimes \text{FT}_{\psi/G_m}(P), $$

where $M_f$ is the complex in Lemma 9.18, and $1$ denotes the unit object on $G_a$.

Assuming these to be true, it follows by combining (9.6), (9.9) and (9.10) that

$$ \text{FT}_{\psi/G_m}(M)|W \simeq p_2^*(N)|W \otimes m_f^*(H) \simeq p_2^*(N)|W \otimes \text{FT}_{\psi/G_m}(P)|W \simeq \text{FT}_{\psi/G_m}(P *_! (1 \boxtimes \tilde{N}))|W, $$
proving the first part of (9.7), and thereby concluding the proof.

We will begin with the rigorous sheaf-theoretic computations, but we include afterwards the (potentially more enlightening) computations of trace functions (assuming all objects to be defined over $k$).

**Proof of (9.8).** Since $G$ is affine, it suffices to prove that $P_!$ is semiperverse: indeed, it first follows that $P_!$ is perverse by [71, Lemma 2.6.7]; since the dual of $P_*$ is $(D(H) \boxtimes 1)_{*!} D(M_f)$, this first fact (applied to the duals of $H$ and $M_f$, which are of the same type) also implies that $P_*$ is perverse, and (9.8) follows.
To prove that $P_!$ is semiperverse, we need to estimate the dimension of the support of the cohomology sheaves $\mathcal{H}^i(P_!)$ Let $i \in \mathbb{Z}$ and $(x_0, y_0) \in \mathcal{G}$. By the definition of convolution and the proper base change theorem, we have an isomorphism

$$ \mathcal{H}^i(P_!)(x_0, y_0) \simeq \mathrm{H}_c^i(\mathcal{G}_{\bar{k}}; [(x, y) \mapsto (x_0x, y_0 + y)])^* (\mathrm{H} \boxtimes \mathbf{1}) \otimes [(x, y) \mapsto (x^{-1}, -y)]^* \mathrm{M}_f. $$

Let $H_{x_0} = [x \mapsto x_0x]^*\mathrm{H}$. The last cohomology group is isomorphic to

$$ \mathrm{H}_c^i(\mathcal{G}_{\bar{k}}, p_1^*\mathrm{H}_{x_0} \otimes p_2^*[y \mapsto y + y_0]^* \mathbf{1} \otimes [(x, y) \mapsto (x^{-1}, -y)]^* \mathrm{M}_f). $$

If we denote by $\tilde{M}_{f,y_0}$ the complex on $\mathbf{G}_m$ given by $[x \mapsto (x^{-1}, -y_0)]^*\mathrm{M}_f$, then the projection formula shows that the cohomology group is isomorphic to

$$ \mathrm{H}_c^i(\mathbf{G}_{m,\bar{k}}, \mathrm{H}_{x_0} \otimes \tilde{\mathrm{M}}_{f,y_0}). $$

By Lemma 9.18 (3) the object $\tilde{\mathrm{M}}_{f,y_0}$ is of the form $\mathcal{G}_{y_0}[2]$ for some sheaf on $\mathbf{G}_m$.

(1) If $H_{x_0}$ is of the form $\mathcal{F}_{x_0}[1]$ for some simple middle extension hypergeometric sheaf on $\mathbf{G}_m$, then $\mathcal{H}^i(P_!)(x_0, y_0)$ is isomorphic to

$$ \mathrm{H}_c^{3+i}(\mathbf{G}_{m,\bar{k}}, \mathcal{F}_{x_0} \otimes \mathcal{G}_{y_0}). $$

It follows immediately that $\mathcal{H}^i(P_!)$ is zero for $i \ge 0$.

For $i = -1$, since $\mathcal{F}_{x_0}$ and $\mathcal{G}_{y_0}$ are lisse on dense open subsets of $\mathbf{G}_m$ and $\mathcal{F}_{x_0}$ is simple, the stalk at $(x_0, y_0)$ is non-trivial if and only if $\mathcal{F}_{x_0}$ is a Jordan-Hölder factor of the dual of $\mathcal{G}_{y_0}$ (on such an open set).

But given $y_0$, the sheaf $\mathcal{G}_{y_0}$ has generic rank $\le \max(\deg(f_1), \deg(f_2))$ by Lemma 9.18 (3), hence has at most as many Jordan-Hölder factors. The hypergeometric sheaves $\mathcal{F}_{x_0}$ are pairwise non-isomorphic by [70, 8.5.6], since they are all multiplicative translates of a fixed hypergeometric sheaf; for a given $y_0$, there are therefore at most $\max(\deg(f_1), \deg(f_2))$ values of $x_0$ for which the stalk of $\mathcal{H}^1(P_!)$ at $(x_0, y_0)$ is non-zero. This implies that the support of $\mathcal{H}^1(P_!)$ is of dimension $\le 1$, so that $P_!$ is semiperverse.

(2) If $\mathrm{H}$ is isomorphic to $\delta_{x_1}$ for some $x_1 \in \mathbf{G}_m$, then $H_{x_0} \simeq \delta_{x_0x_1}$. Then $\mathcal{H}^i(P_!)(x_0, y_0)$ is isomorphic to

$$ \mathrm{H}_c^{2+i}(\mathbf{G}_{m,\bar{k}}, \delta_{x_0x_1} \otimes \mathcal{G}_{y_0}), $$

which is zero if $i \ne -2$. Thus $P_!$ is also semiperverse in that case.

**Proof of (9.9).** We compute $\mathrm{FT}_{\psi/\mathbf{G}_m}(P_!)$, which is the same as $\mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P})$ by the previous result. Consider

$$ X = \{(x, u, v, y, c, d, a) \in \mathbf{G}_m^3 \times \mathbf{G}_a^4 \mid uv = x, c+d=y\}, $$

and the morphism $\sigma_{x,a}: (x, u, v, y, c, d, a) \mapsto (x, a)$ from $X$ to $G$. Then

$$ \mathrm{FT}_{\psi/\mathbf{G}_m}(P_!) = \mathrm{R}\sigma_{x,a,!}(\mathcal{L}_{\psi(ay)}) \otimes \mathrm{H}(u) \otimes \mathbf{1}(c) \otimes \mathrm{M}_f(v, d), $$

where we use a shorthand notation for pullbacks, where, e.g.

$$ \mathrm{H}(u) = [(x, u, v, c, d, y) \mapsto u]^*\mathrm{H}. $$

Denoting

$$ Y = \{(x, u, v, y, c, d, a) \in X \mid c=0\}, \quad Z = \{(x, u, v, y, a) \in \mathbf{G}_m^3 \times \mathbf{G}_a^2 \mid uv = x\}, $$

and noting the isomorphism $Y \to Z$ given by

$$ (x, u, v, y, c, d, a) \mapsto (x, u, v, y, a) $$
with inverse

$$ (x, u, v, y, a) \mapsto (u, x, v, y, 0, y, a), $$

this becomes

$$ \mathrm{R}\sigma_{x,a,!}(\mathcal{L}_{\psi(ay)} \otimes \mathrm{H}(u) \otimes \mathrm{M}_f(v, y)), $$

with an abuse of notation involving in using the notation $\sigma_{x,a}$ again for the projection from points on $Z$ to $(x, a)$. Factoring $\sigma_{x,a}$ into $(x, u, v, y, a) \mapsto (x, u, v, a) \mapsto (x, a)$, we recognize

$$ \mathrm{R}\sigma_{x,a,!}(\mathrm{H}(u) \otimes \mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{M}_f)(v, a)) = \mathrm{R}\sigma_{x,a,!}(\mathrm{H}(u) \otimes \overline{\mathbf{Q}}_{\ell,f}(v, a)), $$

again with some abuse of notation.

We write $f = f_1/f_2$ as in Lemma 9.18. Let

$$ Z_f = \{(x, u, v, a) \mid uv = x, v f_2(a) = f_1(a)\}. $$

By Lemma 9.18, we have an isomorphism

$$ \mathrm{R}\sigma_{x,a,!}(\mathrm{H}(u) \otimes \overline{\mathbf{Q}}_{\ell,f}(v, a)) \simeq \mathrm{R}\sigma_{x,a,!}(\mathrm{H}(u)), $$

and since $\sigma_{x,a}: \sigma_{x,a}^{-1}(W) \to W \subset \mathbf{G}_m \times \mathbf{G}_a$ is an isomorphism with inverse

$$ (x, a) \mapsto (x, x/f(a), f(a), a), $$

this is simply $H(x/f(a)) = m_f^*(H)$ on $W$. In other words, we have proved that there is an isomorphism

$$ \mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P}_!|W) \simeq m_f^*(H), $$

as desired.

(Formally, we can also show that the trace functions on both sides of (9.9) coincide, disregarding the difference between the various convolutions. For $(x, a) \in k^\times \times k$, the object $\mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P})$ has trace function at $(x, a)$ equal to

$$ \sum_{y \in k} \sum_{u \in k^\times} t_H(u) t_{M_f}(x/u, y) \psi(ay) = \sum_{u \in k^\times} t_H(u) \sum_{y \in k} t_{M_f}(x/u, y) \psi(ay). $$

The inner sum over $y$ is the value at $(x/u, a)$ of the trace function of the relative Fourier transform of $M_f$, hence by definition it is the trace function at $(x/u, a)$ of the object $\overline{\mathbf{Q}}_{\ell,f}$. Writing $f = f_1/f_2$ where $f_1$ and $f_2$ are polynomials without common zeros, this value is 1 if $f_1(a) = xu^{-1}f_2(a)$, and 0 otherwise. Thus, provided $f_1(a)f_2(a) \neq 0$, the above expression is

$$ \sum_{u \in k^\times} t_H(u) \sum_{y \in k} t_{M_f}(x/u, y) \psi(ay) = \sum_{\substack{u \in k^\times \\ f_1(a) = xu^{-1}f_2(a)}} t_H(u) = t_H(x/f(a)). $$

If $(x, a) \in W$, this is the same as the trace function of $m_f^*(H)$, as desired.)

**Proof of (9.10).** The argument is similar. Consider

$$ X = \{(x, u, v, y, c, d, a) \in \mathbf{G}_m^3 \times \mathbf{G}_a^4 \mid uv = x, c+d=y\}, $$

and the morphism $\sigma_{x,a}: (x, u, v, y, c, d, a) \mapsto (x, a)$ from $X$ to $G$. Then

$$ \mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P}_! *! (\mathbf{1} \boxtimes \hat{\mathbb{N}})) \simeq \mathrm{R}\sigma_{x,a,!}(\mathcal{L}_{\psi(ay)} \otimes \mathrm{P}_!(u, c) \otimes \mathbf{1}(v) \otimes \hat{\mathbb{N}}(d)), $$

where we use the shorthand notation for pullbacks. Denoting

$$ Y = \{(x, u, v, y, c, d, a) \in X \mid u = x, v = 1, c+d=y\} \simeq Z = \{(x, c, d, y, a) \mid c+d=y\}, $$

this is

$$ \mathrm{R}\sigma_{x,a,!}(\mathcal{L}_{\psi(ay)} \otimes \mathrm{P}_!(x, c) \otimes \hat{\mathbb{N}}(d)) $$
(with again an abuse of notation involved in the notation $\sigma_{x,a}$). From the isomorphism

$$Z \simeq \{(x, c, d, a) \in \mathbf{G}_m \times \mathbf{G}_a^3\}$$

with inverse $(x, c, d, a) \mapsto (x, c, d, c+d, a)$, we get

$$R\sigma_{x,a,!}(\mathcal{L}_{\psi(ac)} \otimes \mathcal{L}_{\psi(ad)} \otimes \mathrm{P}_{!}(x, c) \otimes \hat{\mathbb{N}}(d)).$$

Factor $\sigma_{x,a}$ as $(x, c, d, a) \mapsto (x, c, a) \mapsto (x, a)$; by Fourier inversion, we obtain

$$(9.11) \qquad \mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P}_{!} *_{!}(\mathbf{1} \boxtimes \hat{\mathbb{N}})) \simeq \mathrm{R}\sigma_{x,a,!}(\mathrm{P}_{!}(x, c) \otimes \mathrm{N}(a) \otimes \mathcal{L}_{\psi(ac)}) = \mathrm{N}(a) \otimes \mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P}_{!}).$$

We next claim that we have an isomorphism

$$(9.12) \qquad \mathrm{FT}_{\psi/\mathbf{G}_{m,*}}(\mathrm{P}_{*} *_{*}(\mathbf{1} \boxtimes \hat{\mathbb{N}})) \simeq \mathrm{N}(a) \otimes \mathrm{FT}_{\psi/\mathbf{G}_{m,*}}(\mathrm{P}_{*}).$$

Indeed, we can compute the dual of both sides using the fact that duality transforms the * convolution in the ! convolution, and that both types of Fourier transforms are isomorphic; and the isomorphisms (9.8) and (9.11) applied to the dual of M (with H and N therefore replaced by their respective duals) leads to (9.12).

Again because $\mathrm{P}_{!} \simeq \mathrm{P}_{*} \simeq \mathrm{P}$ and the two Fourier transforms coincide, we conclude that

$$M \simeq \mathrm{P}_{*!}(\mathbf{1} \boxtimes \hat{\mathbb{N}}) \simeq \mathrm{P}_{*}(\mathbf{1} \boxtimes \hat{\mathbb{N}}) \simeq \mathrm{P}_{*\mathrm{int}}(\mathbf{1} \boxtimes \hat{\mathbb{N}}),$$

which establishes (9.10).

(Here also we illustrate the result by computing trace functions. The value at $(x, a)$ of the trace function of the relative Fourier transform of $\mathrm{P}_{*\mathrm{int}}(\mathbf{1} \boxtimes \hat{\mathbb{N}})$ is equal to

$$\sum_{y \in k} \psi(ay) \left( \sum_{v \in k} t_{\mathrm{P}}(x, v) t_{\hat{\mathrm{N}}}(y - v) \right) = \sum_{v \in k} t_{\mathrm{P}}(x, v) \sum_{w \in k} \psi(a(v + w)) t_{\hat{\mathrm{N}}}(w).$$

By Fourier inversion, this is the same as

$$t_{\mathrm{N}}(a) \sum_{v \in k} t_{\mathrm{P}}(x, v) \psi(av),$$

which is the value at $(x, a)$ of the trace function of $p_2^*(N) \otimes \mathrm{FT}_{\psi/\mathbf{G}_m}(\mathrm{P})$. By (9.9), this coincides on W with the trace function of $p_2^*(N)|W \otimes m_f^*(H).$ $\square$

REMARK 9.21. The trace functions (over $k$) of simple negligible objects are of the form

$$(x, y) \mapsto t(x)\psi(by)$$

for some trace function $t$ on $\mathbf{G}_m$ and some $b \in k$, or

$$(x, y) \mapsto \chi(x)t(y)$$

for some trace function $t$ on $\mathbf{G}_a$ and some multiplicative character $\chi$. The trace functions of simple objects of tannakian dimension one are convolutions of functions of the three types

$$(x, y) \mapsto \sum_{f(z)=x} \psi(-yz), \quad (x, y) \mapsto \mathcal{H}(x), \quad (x, y) \mapsto t(y),$$

where $f$ is a non-zero rational function, $\mathcal{H}$ is the trace function of a hypergeometric sheaf and $t$ is the trace function of an object on $\mathbf{G}_a$ whose Fourier transform has generic rank one. The associated exponential sums are (up to normalization by powers of $|k|$) of the form

$$S(\chi, a) = \chi(f(a)) \mathcal{H}(\chi) \hat{t}(a),$$

where $\hat{t}$ is the trace function of an $\ell$-adic character, and $\mathcal{H}$ is a product of monomials in Gauss sums (see (B.1)).
# Variance of arithmetic functions in arithmetic progressions

## 10.1. Introduction

In this chapter, we will consider some of the first natural concrete applications of our results to problems which, as stated, do not seem to refer to algebraic groups, or equidistribution statements of any kind. These problems are related to one of the most essential questions of modern analytic number theory, namely the study of arithmetic functions in arithmetic progressions to large moduli.

Concretely, this means that we are given an arithmetic function $f$ (i.e., a complex-valued function defined on the set of positive integers), an integer $q \ge 1$ (the “modulus”) and $x \ge 2$, and we seek to understand the quantities

$$ \sum_{\substack{n \le x \\ n \equiv a \pmod{q}}} f(n) $$

for $a$ varying among residue classes modulo $q$, or only for $a$ coprime to $q$. The focus is on these sums in settings where both $x$ and $q$ are large, and the goal is often to obtain asymptotic formulas valid for $q$ as large as possible in comparison with $x$.

The literature on this topic is enormous, and the applications cover almost all of analytic number theory: indeed, this subject encompasses, almost by definition, all of sieve theory and its applications (see [46]), and it is in particular at the source of most of the recent developments in prime number theory, going back to the Bombieri–Vinogradov Theorem (see, e.g., [64, Ch. 17]), and including such celebrated results as the Green–Tao Theorem, or Zhang’s Theorem [128], or the Maynard–Tao method (see, e.g., [88]).

The problems that we consider here are the analogue for polynomials over finite fields, and in the limit when the size of the field tends to infinity, of questions related to the distribution of the quantities above, and especially of their variance, as functions of $a$. In other words, we are interested in

$$ \sum_{a \pmod{q}} \left| \sum_{\substack{n \le x \\ n \equiv a \pmod{q}}} f(n) - \frac{1}{q} \sum_{n \le x} f(n) \right|^2 $$

or (often more naturally for applications) the variant

$$ (10.1) \qquad \sum_{\substack{a \pmod{q} \\ (a,q)=1}} \left| \sum_{\substack{n \le x \\ n \equiv a \pmod{q}}} f(n) - \frac{1}{\varphi(q)} \sum_{n \le x} f(n) \right|^2 $$

where the sum covers only invertible residue classes. (In both cases, the choice of “expected main term” is natural, but might require adjustments, depending on the arithmetic function involved.)

The serious study of these function field analogues has been initiated especially by Keating and Rudnick and a number of collaborators (see for instance [81], in the case where $f$ is the von Mangoldt function, using results of Katz [76], which themselves relied on his work on the Mellin transform over finite fields [75]).
It is quite easy to understand the link between a quantity like (10.1), in the function field case, and equidistribution problems of the type considered in Chapter 4. Indeed, we are then in the situation where $q$ is a polynomial in $k[t]$ for some finite field $k$, and the sum over $n \le x$ is replaced by the sum over monic polynomials $g \in k[t]$ of degree $m$. Then for any complex-valued function $f$ defined for polynomials in $k[t]$, we see (using orthogonality of characters, or the discrete Plancherel formula) that the formula

$$ \sum_{a \in (k[t]/qk[t])^\times} \left| \sum_{\substack{\deg(g)=m \\ g \equiv a \pmod q}} f(g) - \frac{1}{|(k[t]/qk[t])^\times|} \sum_{\deg(g)=m} f(g) \right|^2 = \\ \qquad \qquad \qquad \qquad \qquad \qquad \left| (k[t]/qk[t])^\times \right| \sum_{\chi \ne 1 \atop \deg(g)=m} \left| \sum_{\chi(g)=1} \chi(g) f(g) \right|^2 $$

holds, where $\chi$ runs over non-trivial characters of the group $(k[t]/qk[t])^\times$. These characters can be identified with the characters of $G(k)$ for some commutative algebraic group $G$ (by a simple special case of geometric class-field theory; in the case which we will consider, when $q$ is squarefree, it will be a very explicit torus). Moreover, for many natural arithmetic functions, the inner sum over $g$ monic of degree $m$ in $k[t]$ can be interpreted as the value at $\chi$ of the arithmetic Fourier transform of some object on this group $G$. In the limit where $k$ is replaced by its extensions $k_n$ of degree $n \to +\infty$ (and $m$ is fixed), we can therefore expect to determine the asymptotic behavior of this variance from our equidistribution theorems.

We will now consider in detail the version of this question when $f$ is the von Mangoldt function associated to a higher-degree L-function (the classical von Mangoldt function being related to the Riemann zeta function, which has degree 1), in which case Hall, Keating, and Roddity-Gershon [56] have shown that new phenomena appear (again relying on [75]). These are conjectured to correspond to new behavior also in the (currently inaccessible) situation over number fields. We refer the reader to the introductions of both papers [81] and [56] for extensive discussions of these motivating conjectures, and for additional references to other papers.

We will see that, as suggested by the discussion above, the equidistribution theory for arithmetic Fourier transforms on higher-dimensional tori leads to generalizations, strengthenings, and better understanding, of these previous results. This leads in particular to Theorem 3 in the Introduction, but the method is suitable for the proof of many similar statements.

In the remainder of this chapter, as before, we denote by $k$ a finite field, with an algebraic closure $\bar{k}$, and for each $n \ge 1$ by $k_n$ the extension of degree $n$ of $k$ in $\bar{k}$. We fix a prime $\ell$ distinct from the characteristic of $k$, and all complexes are understood to be $\ell$-adic complexes.

## 10.2. Equidistribution on tori associated to polynomials

In what follows, we fix a square-free monic polynomial $f \in k[t]$ of degree $d \ge 2$. We denote by $B$ the (étale) $k$-algebra $B = k[t]/fk[t]$ of degree $d$ over $k$ (in spite of the notation, $B$ depends on $f$), by $Z$ the zero locus of $f$, and by $A_k^1[1/f]$ the complement of $Z$ in the affine line over $k$.

We begin with a result of Katz [76].

**PROPOSITION 10.1 (Katz).** The functor $A \mapsto (B \otimes_k A)^\times$ on $k$-algebras is represented by a torus $T$ defined over $k$. This torus splits over any extension of $k$ where $f$ splits in linear factors.

Moreover, the map $x \mapsto t - x$ defines a closed immersion

$$ i_f: A_k^1[1/f] \to T, $$
and there exists a morphism of algebraic groups

$p: T \to \mathbf{G}_m$

satisfying $p \circ i_f = (-1)^{\deg(f)} f$, where we view $f$ as defining a morphism $\mathbf{A}^1[1/f] \to \mathbf{G}_m$.

REMARK 10.2. As noted by Katz [76, p.3224], the torus $T$ is isomorphic to a generalized jacobian associated to $\mathbf{P}^1$ with divisor $(\infty) + \mathbb{Z}$ (compare Remark 8.14).

We call the morphism $p$ the norm. If $f$ splits completely over $k$, say

$$f = \prod_{z \in \mathbb{Z}} (t - z),$$

then the torus $T$ is split by the morphism sending $g$ to $(g(z))_{z \in \mathbb{Z}}$. The norm is then given by

$$p(g) = \prod_{z \in \mathbb{Z}} g(z),$$

and in particular one has

$$p(i_f(x)) = \prod_{z \in \mathbb{Z}} (z - x) = (-1)^{\deg(f)} f(x).$$

We denote by $\hat{\mathbf{B}}^\times$ (resp. by $\hat{k}^\times$) the group of $\ell$-adic characters of the finite group $\mathbf{B}^\times$ (resp. of $k^\times$). We extend characters of $\mathbf{B}^\times$ to $k[t]$ by putting $\chi(g) = 0$ if $g$ is not coprime to $f$. Since $\mathbf{B}^\times = \mathbf{T}(k)$, the group $\hat{\mathbf{B}}^\times$ of characters of $\mathbf{B}^\times$ is also equal to the group $\hat{\mathbf{T}}(k)$ of characters of $\mathbf{T}(k)$ (although we will sometimes distinguish them to avoid confusion between characters of $\mathbf{B}$, operating on polynomials, and characters of $\mathbf{T}$).

If $f$ splits over $k$ as above, then the Chinese Remainder Theorem induces an isomorphism $(\hat{k}^\times)^Z \to \hat{\mathbf{B}}^\times$, under which an element $(\chi_z)_{z \in \mathbb{Z}} \in (\hat{k}^\times)^Z$ corresponds to the character $\chi$ of $\mathbf{B}^\times$ that maps $g \in k[t]$ to

$$\chi(g) = \prod_{z \in \mathbb{Z}} \chi_z(g(z)).$$

Let $M$ be a perverse sheaf on $\mathbf{A}_k^1[1/f]$ which is pure of weight zero. We are interested in the distribution properties of families of one-variable exponential sums of the type

$$(10.2) \qquad \sum_{x \in k - \mathbb{Z}} t_M(x) \chi(t - x)$$

for $\chi \in \hat{\mathbf{B}}^\times$, or of the underlying L-functions (recall that $t$ is an indeterminate).

We start by interpreting these sums as Mellin transforms on $T$ in order to apply our general equidistribution results. Let $\chi \in \hat{\mathbf{B}}^\times$. Let $\tilde{\chi}$ be the character of $\mathbf{T}(k)$ corresponding to $\chi$. The sum (10.2) takes the form

$$(10.3) \qquad \sum_{x \in k} t_M(x) \chi(t - x) = \sum_{x \in \mathbf{A}^1[1/f](k)} t_M(x) \tilde{\chi}(i_f(x)) = \sum_{y \in \mathbf{T}(k)} t_{i_f * M}(y) \tilde{\chi}(y).$$

Note also that by adapting the argument of [76, Lem. 1.1], for any $n \ge 1$, we have

$$(10.4) \qquad \sum_{x \in k_n} t_M(x; k_n) \chi(N_{k_n/k}(t-x)) = \sum_{y \in \mathbf{T}(k_n)} t_{i_f * M}(y; k_n) \tilde{\chi}(N_{k_n/k}(y)).$$

The variation with $\chi \in \hat{\mathbf{B}}^\times$ of the sums (10.2) is therefore governed by the tannakian group of the perverse sheaf $i_{f*}M$ on $T$. By Theorem 3.27, this perverse sheaf is generically unramified.

We first compute the tannakian dimension of the object $i_{f*}M$, in the most important cases.
LEMMA 10.3. Let $\mathcal{F}$ be a middle extension sheaf on $\mathbf{A}_k^1[1/f]$ which is pure of weight zero<fnref n="1" />.
Define $\mathrm{M} = \mathcal{F}[1](1/2)$, which is a perverse sheaf of weight zero on $\mathbf{A}_k^1[1/f]$. The tannakian
dimension $r$ of $i_{f_*}\mathrm{M}$ is given by

$$
r = (\mathrm{deg}(f) - 1) \mathrm{rank}(\mathcal{F}) + \sum_{x \in \mathbf{P}^1(\bar{k})} \mathrm{swan}_x(\mathcal{F}) + \sum_{x \in \bar{k}} \mathrm{drop}_x(\mathcal{F}) \ge (\mathrm{deg}(f) - 1) \mathrm{rank}(\mathcal{F}).
$$

PROOF. The object M is a perverse sheaf and so is $i_{f_*}M$ because $i_f$ is a closed immersion (see Corollary A.8). The tannakian dimension is the Euler-Poincaré characteristic $\chi_c(T_{\bar{k}}, (i_{f_*}M)_{\chi})$ for a generic character $\chi \in \hat{T}$ (Proposition 3.17).

For any integer $i$, we have natural isomorphisms

$$
H_c^i(T_{\bar{k}}, (i_{f_*}M)_\chi) \simeq H_c^i(\mathbf{A}^1[1/f]_{\bar{k}}, M \otimes i_f^*\mathcal{L}_\chi) \simeq H_c^i(\mathbf{A}^1[1/f]_{\bar{k}}, \mathcal{F}[1] \otimes i_f^*\mathcal{L}_\chi).
$$

As explained in [76, p. 3227], the pullback $i_f^*\mathcal{L}_\chi$ is geometrically isomorphic to the tensor product

$$
\mathcal{L} = \bigotimes_{z \in Z} \mathcal{L}_{\chi(z-x)}
$$

where $x$ is the coordinate on $\mathbf{A}^1[1/f]$ and $\chi$ corresponds to the tuple $(\chi_z)$ of characters of $k^\times$ as above.

Now using the Euler-Poincaré formula on a curve (see Theorem C.2), we obtain

$$
r = -\chi_c(\mathbf{A}^1[1/f]_{\bar{k}}, \mathcal{F} \otimes \mathcal{L}) = -\mathrm{rank}(\mathcal{F})\chi_c(\mathbf{A}^1[1/f]_{\bar{k}}) \\
\qquad + \sum_{x \in \mathbf{P}^1} \mathrm{swan}_x(\mathcal{F} \otimes \mathcal{L}) + \sum_{x \in \mathbf{A}^1[1/f]} \mathrm{drop}_x(\mathcal{F} \otimes \mathcal{L}).
$$

The first term is equal to $\mathrm{rank}(f)(\mathrm{deg}(f) - 1)$ since $f$ is square-free, and the second is the sum of Swan conductors of $\mathcal{F}$, since the sheaf $\mathcal{L}$ is everywhere tame. The third is the sum of the drops of $\mathcal{F}$ on $\mathbf{A}^1[1/f]$, since $\mathcal{L}$ is lisse on $\mathbf{A}^1[1/f]$. $\square$

We now apply Larsen’s Alternative to compute the tannakian group of such perverse sheaves.

PROPOSITION 10.4. Let $\mathcal{F}$ be a middle extension sheaf on $\mathbf{A}_k^1[1/f]$ which is pure of weight zero and irreducible of rank at least 2. Let $\mathrm{M} = \mathcal{F}[1](1/2)$. Assume that $\mathrm{M}$ is not geometrically isomorphic to $i_f^*\mathcal{L}_\chi[1]$ for some character $\chi$ of $\mathrm{G}$.

Then $i_{f_*}M$ is a geometrically simple perverse sheaf, pure of weight zero and of tannakian dimension at least 2.

Moreover, if $\mathrm{deg}(f) \ge 2$, then the fourth moment of the tannakian group $\mathbf{G}_{i_{f_*}M}^{\mathrm{ari}}$ of $i_{f_*}M$ is equal to 2, and if $\mathrm{deg}(f) \ge 4$, then the eighth moment is equal to 24.

PROOF. The previous lemma implies that $i_{f_*}M$ has tannakian dimension $\ge 2$. It is geometrically
simple since $M$ is.

One argument to obtain the result is to observe that $i_f$ is a Sidon morphism when $\deg(f) \ge 2$, and a 4-Sidon morphism when $\deg(f) \ge 4$ (by Proposition 8.11, (4), since these properties can be checked after a finite extension), so that the result follows from Proposition 8.8 since the tannakian dimension is $\ge 2$, and the assumption on $M$).

<footnote n="1">¹ Recall (see Example A.18 (3)) that this means that the restriction of $\mathcal{F}$ to any dense open set where it is lisse is punctually pure of weight 0.</footnote>
For the sake of concreteness, we show also how to perform the computation of the eighth moment using the interpretation of the sums in terms of Dirichlet characters. The eighth moment of the full family of exponential sums over $k$ is equal to

$$ \frac{1}{|\mathbb{B}^\times|} \sum_{\chi \in \hat{\mathbb{B}}^\times} \left| \sum_{x \in k} t_M(x) \chi(t-x) \right|^8 = \sum_{x_1, \dots, x_8} \prod_{i=1}^4 t_M(x_i) \prod_{i=5}^8 \overline{t_M(x_i)} \times \frac{1}{|\mathbb{B}^\times|} \sum_{\chi \in \hat{\mathbb{B}}^\times} \chi((t-x_1) \cdots (t-x_4)) \overline{\chi((t-x_5) \cdots (t-x_8))}. $$

By orthogonality, the inner sum is 0 unless

$$ (t - x_1) \cdots (t - x_4) \equiv (t - x_5) \cdots (t - x_8) (\text{mod } f), $$

in which case it is equal to $|\mathbb{B}^\times|$. Since the degree of $f$ is at least 4, this congruence can only occur when

$$ (t - x_1) \cdots (t - x_4) = (t - x_5) \cdots (t - x_8) $$

in $k[t]$. We then distinguish according to the size of $\{x_1, \dots, x_4\}$. If this set has four elements, then so does $\{x_5, \dots, x_8\}$, and the two sets are equal. The contribution arising from this case is

$$ \sum_{x_1, \dots, x_4} \sum_{\sigma \in \mathfrak{S}_4} t_M(x_1) \cdots t_M(x_4) \overline{t_M(x_{\sigma(1)})} \cdots \overline{t_M(x_{\sigma(4)})} = 24 \left( \sum_{x \in k} |t_M(x)|^2 \right)^4. $$

On the other hand, if the set $\{x_1, \dots, x_4\}$ has three elements, say $x, y$ and $z$, then so does $\{x_5, x_6, x_7, x_8\}$, and there are an absolutely bounded number of possibilities for $(x_1, x_2, x_3, x_4)$ given $x, y$ and $z$. A similar result holds for two or one elements, and since $t_M(x) \ll |k|^{-1/2}$, one sees that these altogether contribute at most

$$ \frac{1}{|k|^4} \sum_{x, y, z \in k} 1 \ll \frac{1}{|k|}. $$

These computations can be repeated over $k_n$ for $n \ge 1$ using (10.4), and using Proposition 8.1, we deduce by letting $n \to +\infty$ that

$$ \frac{1}{|\mathbb{B}^\times|} \sum_{\chi \in \hat{\mathbb{B}}^\times} \left| \sum_{x \in k} t_M(x) \chi(t-x) \right|^8 \to 24 $$

as $|k| \to +\infty$.

Finally, the usual argument using the definition of generic sets of characters together with (10.3) and the Riemann Hypothesis imply that

$$ \frac{1}{|\mathbb{B}^\times|} \sum_{\substack{\chi \in \hat{\mathbb{B}}^\times \\ \tilde{\chi} \text{ ramified}}} \left| \sum_{x \in k} t_M(x) \chi(t-x) \right|^8 \to 0, $$

so that Proposition 8.4 gives the result. $\square$

**COROLLARY 10.5.** *Under the assumptions of the proposition, the tannakian group of $i_{f_*}M$ contains $\mathbf{SL}_r$, where $r$ is the tannakian dimension of $i_{f_*}M$, if $\deg(f) \ge 4$.*

**PROOF.** By Lemma 10.3, the assumption implies $r \ge 4$, and the result follows from Larsen's Alternative, in the form of the eighth moment theorem of Guralnick and Tiep (see Theorem 8.5 (4)). $\square$

<figure><img src="image_16.png" /></figure>
### 10.3. Application to von Mangoldt functions

Suppose again that $M$ is of the form $\mathcal{F}[1](1/2)$ for some middle extension sheaf $\mathcal{F}$ on $\mathbf{A}_k^1[1/f]$ which is pure of weight zero and geometrically irreducible of rank at least 2.

The statement of equidistribution on average for the object $i_{f*}M$ leads automatically to distribution statements of any “continuous” function of the polynomials in the variable $T$ which are the twisted L-functions of $M$, namely

$$ \det(1 - \mathrm{Fr}_k T \mid \mathrm{H}_c^0(\mathbf{A}^1[1/f]_{\bar{k}}, \mathbf{M} \otimes \mathcal{L}_{\tilde{\chi}}^\times)) = \det(1 - \mathrm{Fr}_k T \mid \mathrm{H}_c^0(\mathrm{T}_{\bar{k}}, (i_{f*}M)_\chi)) $$

as $\chi \in \hat{\mathbf{B}}^\times$ varies, where $\tilde{\chi}$ is now the character of the fundamental group of $\mathbf{A}_k^1[1/f]$ that corresponds to $\chi$ by class-field theory, and $\mathcal{L}_{\tilde{\chi}}$ is the associated rank one sheaf.

For instance, this leads to statements concerning the variance of von Mangoldt functions in arithmetic progressions, as we now explain.

Write

$$ \mathrm{L}(\mathbf{M}, T) = \det(1 - \mathrm{Fr}_k T \mid \mathrm{H}_c^0(\mathbf{A}^1[1/f]_{\bar{k}}, \mathbf{M})) = \prod_x \det(1 - \mathrm{Fr}_{k_{\deg(x)}} T^{\deg(x)} \mid \mathcal{F}_x)^{-1}, $$

where $x$ runs over the set of closed points of $\mathbf{A}_k^1[1/f]$, which may be identified with the set of irreducible monic polynomials in $k[t]$ which are coprime to $f$. Expanding the logarithmic derivative of the local factor at a closed point $x$, corresponding to an irreducible monic polynomial $\pi \in k[t]$, we have

$$ -T d \log(\det(1 - \mathrm{Fr}_{k_{\deg(x)}} T^{\deg(x)} \mid \mathcal{F}_x)^{-1}) = \sum_{\nu \ge 1} \Lambda_M(\pi^\nu) T^{\nu \deg(\pi)}, $$

which defines the von Mangoldt function $\Lambda_M(\pi^\nu)$ for any monic irreducible polynomial $\pi$ coprime to $f$ and any $\nu \ge 1$. We further define $\Lambda_M(g) = 0$ if $g \in k[t]$ is not a power of such an irreducible polynomial. The full logarithmic derivative then has the formal power series expansion

$$ -T \frac{L'(\mathbf{M}, T)}{L(\mathbf{M}, T)} = \sum_g \Lambda_M(g) T^{\deg(g)} $$

over all monic polynomials $g \in k[t]$.

For an integer $m \ge 1$ and a polynomial $a \in k[t]$, we then define

$$ \psi_M(m; f, a) = \sum_{\substack{\deg(g)=m \\ g \equiv a \pmod f}} \Lambda_M(g). $$

We consider the average

$$ \mathbf{A}_M(m; f) = \frac{1}{|\mathbf{B}^\times|} \sum_{a \in \mathbf{B}^\times} \psi_M(m; f, a) $$

and the variance

$$ \mathrm{V}_M(m; f) = \frac{1}{|\mathbf{B}^\times|} \sum_{a \in \mathbf{B}^\times} |\psi_M(m; f, a) - \mathrm{A}_M(m; f)|^2. $$

These are related to exponential sums as follows.

**PROPOSITION 10.6.** With assumptions and notation as above, we have

$$ \mathrm{V}_M(m; f) = \frac{1}{|\mathbf{B}^\times|^2} \sum_{\substack{\chi \in \hat{\mathbf{B}}^\times \\ \chi \ne 1}} \mathrm{V}_M(m; \chi) $$
where

$$V_M(m; \chi) = \left| \sum_{x \in k_m} t_M(x; k_m) \chi(N_{k_m/k}(t-x)) \right|^2.$$

In particular, if $\chi$ is weakly unramified for $i_{f*}M$, then we have $V_M(m; \chi) = |\text{Tr}(\Theta_M(\chi)^m)|^2$.

PROOF. The first part is proved, using the orthogonality of characters, exactly like [56, §6, (6.3.4)]. The second assertion then follows from Lemma 3.35 and (10.4). □

REMARK 10.7. The von Mangoldt function can be replaced by many other arithmetic functions in this argument; we refer to the discussion by Sawin in [113] (which proves analogue equidistribution statements to ours for the case of “short intervals”, which amounts to considering a unipotent group instead of a torus) and to [114] for a discussion of how classical arithmetic functions which are related to “factorization functions” (functions of polynomials $g$ that depend only on the factorization type of $g$) can be interpreted as trace functions using representation theory of the symmetric groups.

We now obtain a formula for the variance, with some additional assumption.

COROLLARY 10.8. In addition to the assumptions of this section, assume that $m \ge 2$, and that the tannakian determinant of $M$ is geometrically of infinite order. Then

$$\lim_{|k| \to +\infty} |B^\times|^2 V_M(m; f) = \min(m, r),$$

where $r$ is the tannakian dimension of $i_{f*}M$.

PROOF. Combined with Corollary 10.5, the assumption implies that the arithmetic and geometric tannakian groups of $i_{f*}M$ are both equal to $GL_r$. Thus the limit exists by Theorem 4.15 and is equal to

$$\int_{U_r(C)} |\text{Tr}(g^m)|^2 d\mu(g)$$

where $\mu$ is the Haar probability measure. This matrix integral is equal to $\min(m, r)$ by work of Diaconis and Evans [33, Th. 2.1]. □

To check the assumption on the tannakian determinant, we have a first general criterion, which is however quite restricted.

PROPOSITION 10.9. With notation and assumptions as above, suppose that there exists $z \in Z$ such that the local monodromy representation of $\mathcal{F}$ at $z$ has a non-zero unipotent tame component while the local monodromy at infinity has no unipotent tame component. Then the tannakian determinant of $i_{f*}M$ is geometrically of infinite order.

PROOF. We apply Corollary 3.47 to the norm morphism $p: T \to G_m$. Indeed, $p \circ i_f$ coincides with the finite morphism $\varepsilon f: A^1[1/f] \to G_m$, where $\varepsilon = (-1)^{\deg(f)}$ (Proposition 10.1), so that the equalities $\text{R}_{p!}(i_{f*}M) = \text{R}_{p!}(i_{f!}M) = (\varepsilon f)_*M = ((\varepsilon f)_!\mathcal{F})[1](1/2)$ hold, and the sheaf $((\varepsilon f)_!\mathcal{F})[1](1/2) = ((\varepsilon f)_*\mathcal{F})[1](1/2)$ has no tame unipotent local monodromy at infinity, but has some non-trivial tame unipotent monodromy at 0 in view of the canonical isomorphism

$$((\varepsilon f)_*\mathcal{F})_0 \simeq \bigoplus_{z \in Z} \mathcal{F}_z.$$

Hence, the tannakian determinant of the object $i_{f*}M$ is geometrically of infinite order. □
We now explain the proof of Theorem 3, where we will also use a different approach to checking that the tannakian determinant has infinite order, which may be useful in other contexts.

Let $\pi: \mathcal{E} \to \mathbf{P}^1$ be the morphism which “is” the Legendre elliptic curve. We start with the sheaf

$$ \mathcal{F} = \mathbb{R}^1 \pi_* \overline{\mathbb{Q}}_\ell(1/2). $$

This is a middle extension sheaf on $\mathbf{A}_k^1$. It is pure of weight zero and geometrically irreducible of rank 2 (in particular, its $H_c^2$ vanishes), and is tamely ramified at 0, 1 and $\infty$, with drop equal to 1 at 0 and 1. Using Lemma 10.3, we compute that the tannakian dimension is $r = 2 \deg(f) - 2 + a$, where $a$ is the degree of the $\gcd$ of $f$ and $t(t-1)$.

Now the pullback of $\mathcal{F}$ to $\mathbf{A}_k^1[1/f]$ is a middle extension sheaf, geometrically irreducible of rank 2 and pure of weight 0, for which we keep the same notation. We can then apply Corollary 10.8 to $\mathcal{F}$, using the following proposition. In order to conclude after doing so, we check that the contribution of the local factors at $z \in \mathbb{Z}$ to the L-functions (which might not be of weight 0) is negligible (compare [56, Prop. 6.5.3]).

**PROPOSITION 10.10.** Let $M = \mathcal{F}[1](1/2)$. The tannakian determinant of $i_{f_*M}$ is geometrically of infinite order.

**PROOF.** If $f$ is not coprime to $t(t-1)$, then we can apply Proposition 10.9, since $\mathcal{F}$ has non-trivial tame unipotent monodromy at 0 and 1, and none at infinity. So we assume that $f$ is coprime with $t(t-1)$.

We may assume that the polynomial $f$ splits in linear factors over $k$ and that $k \neq \mathbb{Z} \cup \{0, 1\}$. Fix a non-trivial additive character $\psi$ of $k$. We will then prove in Proposition 10.11 below, using the theory of local constants, that there exists a generic set of characters $\mathcal{X} \subset \mathcal{X}(M)$ and elements $\xi_z \in \mathbf{A}^1[1/f]$ such that for $n \ge 1$ and $\chi \in \mathcal{X}(k_n)$, the equality

$$ \det(\Theta_{M, k_n}(\chi)) = \gamma^n H_1(\chi)^{-1} H_2 \left( \prod_{z \in \mathbb{Z}} \chi_z^{-1} \right)^{-1} $$

holds, for some number $\gamma$ independent of $\chi$ and $n$, where the functions $H_1$ and $H_2$ are products of Gauss sums described in (10.6) and (10.7) below.

On $\mathbf{G}_m$, the function

$$ \chi_z \mapsto \chi_z(\xi_z) \frac{1}{|k|} \left( \sum_{y \in k^\times} \chi_z(y) \psi(y) \right)^2 $$

coincides for $\chi_z$ non-trivial with the arithmetic Mellin transform of the multiplicative translated hypergeometric complex $\text{Hyp}_{\xi_z}(!, \psi, 1, 1; \emptyset)(1/2)$ (see (B.1) for this; in this case, this is a shifted and translated Kloosterman sheaf). Since the function $\chi \mapsto H_1(\chi)^{-1}$ is the product of these functions over $z \in \mathbb{Z}$, it coincides generically with the Mellin transform on $\mathbb{T}$ of the tensor product

$$ \bigotimes_{z \in \mathbb{Z}} p_z^* \text{Hyp}_{\xi_z^{-1}} (!; \psi, 1, 1; \emptyset)(1/2), $$

where $p_z$ is the projection from $\mathbb{T}$ to the $z$-component in the splitting $g \mapsto (g(z))$ of the torus $\mathbb{T}$. (Indeed, this reflects the formula

$$ \sum_{x \in \mathbb{T}(k)} \chi(x) \prod_{z \in \mathbb{Z}} f_z(p_z(x)) = \sum_{(x_z) \in (k^\times)^2} \prod_{z \in \mathbb{Z}} \chi_z(x_z) \prod_{z \in \mathbb{Z}} f_z(x_z) = \prod_{z \in \mathbb{Z}} \sum_{x \in k^\times} \chi_z(x) f_z(x) $$

for arbitrary functions $f_z$ on $k^\times$.)
Similarly, the function $\chi \mapsto H_2(\prod \chi_z^{-1})^{-1}$, which only depends on the product $\eta$ of the component characters $(\chi_z)$, coincides (for $\eta$ non-trivial) with the arithmetic Mellin transform of the object $\Delta_*L$, where $L = \text{Hyp}(!, \psi, \lambda_2, \lambda_2; \emptyset)(1/2)$ and

$$ \Delta: \mathbf{G}_m \to \mathbf{G}_m^Z \simeq \mathbf{T} $$

is the closed immersion $x \mapsto (x^{-1}, \dots, x^{-1})$. This reflects the fact that $\Delta$ is a morphism of algebraic groups, and that the dual $\hat{\Delta}$ on $\hat{T}(k)$ is given by

$$ (\chi_z)_{z \in \mathbb{Z}} \mapsto \prod_{z \in \mathbb{Z}} \chi_z^{-1}. $$

By Theorem 6.11, the formula (10.5) therefore implies that the tannakian determinant of $M$ is geometrically isomorphic in $\bar{\mathbf{P}}(\mathbb{T})$ to the perverse sheaf

$$ D = (\Delta_*L) * \left( \bigotimes_{z \in \mathbb{Z}} p_z^* \mathrm{Hyp}_{\xi_z}(!; \psi, 1, 1; \emptyset)(1/2) \right). $$

The object $D$ visibly has infinite geometric tannakian group since for any $m \ge 1$, we have

$$ D^{*m} = (\Delta_*L)^{*m} * \left( \bigotimes_{z \in \mathbb{Z}} p_z^* \mathrm{Hyp}_{\xi_z^{-1}}(!; \psi, 1, 1; \emptyset)^{*m}(1/2) \right), $$

in $\bar{\mathbf{P}}(\mathbb{T})$, and the $m$-th convolution powers on $\mathbf{G}_m$ of the hypergeometric complexes that appear are not geometrically trivial (see Theorem B.4). $\square$

We complete this section by proving the formula for the determinant.

**PROPOSITION 10.11.** Suppose that $f$ splits in linear factors over $k$. For $z \in \mathbb{Z}$, define

$$ \xi_z = z(z-1) \prod_{x \in \mathbb{Z} - \{z\}} (z-x)^2 \in k^\times. $$

There exist numbers $\varepsilon_0, \varepsilon_1$ with the following property. For a character $\chi \in \mathcal{X}(M)$ such that all components $\chi_z$ for $z \in \mathbb{Z}$ are non-trivial, and such that the product of the components is not of order at most 2, we have

$$ (10.5) \qquad \det(\mathbf{\Theta}_M(\chi))^{-1} = (-1)^r |k| \varepsilon_0 \varepsilon_1 H_1(\chi) H_2\left(\prod_{z \in \mathbb{Z}} \chi_z^{-1}\right) $$

where

$$ (10.6) \qquad H_1(\chi) = \prod_{z \in \mathbb{Z}} \chi_z (\xi_z^{-1}) |k| \left( \sum_{y \in k^\times} \chi_z(y) \psi(y) \right)^{-2}, $$

$$ (10.7) \qquad H_2(\chi) = |k| \left( \sum_{y \in k^\times} (\lambda_2 \chi)(y) \psi(y) \right)^{-2}. $$

**PROOF.** Let $j: \mathbf{A}^1[1/f] \to \mathbf{P}^1$ be the open immersion. Let $\chi \in \hat{\mathbb{B}}$ be a Dirichlet character and $\mathcal{L}_\chi$ the lisse rank 1 sheaf on $\mathbf{A}^1[1/f]$ that corresponds to it. The $L$-function of $j!(M \otimes \mathcal{L}_\chi)$ satisfies a functional equation of the form

$$ L(j!(M \otimes \mathcal{L}_\chi), \mathbb{T}) = \varepsilon(\chi) T^a L(D(j!(M \otimes \mathcal{L}_\chi)), T^{-1}) $$

where $a = -\chi(j!(M \otimes \mathcal{L}_\chi)) = -r$ is an integer and

$$ \varepsilon(\chi) = \det(-\mathrm{Fr}_k | \mathrm{H}^0(\mathbf{P}_k^1, j!(M \otimes \mathcal{L}_\chi)))^{-1} = \det(-\mathrm{Fr}_k | \mathrm{H}_c^0(\mathbf{A}_k^1[1/f], M \otimes \mathcal{L}_\chi))^{-1} $$
By Lemma 3.35, if $\chi \in \hat{T}$ is unramified for M, then we deduce that

$$ (10.8) \qquad \det(\Theta_M(\tilde{\chi})) = (-1)^r \varepsilon(\chi)^{-1}, $$

where $r$ is the tannakian dimension of $i_{f_*}M$. By a theorem of Laumon,<fnref n="2" /> we can express the constant $\varepsilon(\chi)$ as a product over closed points

$$ \varepsilon(\chi) = |k|^{-2} \prod_{x \in |\mathbf{P}^1|} \varepsilon_x(\chi) $$

of local constants, previously defined by Deligne [26] and characterized by the properties of [98, Th. 3.1.5.4]. Precisely, fixing a non-trivial additive character $\psi$ of $k$ and a non-zero meromorphic differential 1-form $\omega$ on $\mathbf{P}^1$, we can then define

$$ \varepsilon_x(\chi) = \varepsilon(\mathbf{P}^1(x), j_*(M \otimes \mathcal{L}_\chi)|\mathbf{P}^1(x), \omega|\mathbf{P}^1(x)) $$

with the notation of loc. cit. See again Section C.1; in particular the factor $|k|^{-2}$ above is given by (C.1), namely the exponent is obtained by the computation

$$ -2 = 1 \cdot (1 - 0) \cdot (-2), $$

where $-2$ is the generic rank of the object M (a sheaf of rank 2 in degree $-1$).

We take $\omega = dt$, where $t$ is the standard coordinate on $\mathbf{P}^1$. The data of $\psi$ and $\omega$ allows us to define non-trivial additive characters $\psi_x$ of the completed local field at any closed point $x \in |\mathbf{P}^1|$ by the recipe in [98, Th. 3.1.5.4, (v)]. For all closed points $x \in \mathbf{A}^1$, the character $\psi_x$ is of conductor zero since $\omega$ is regular at $x$ (see [98, 3.1.3.6]). For $x = \infty$, we have $c(\psi_\infty) = -2$ since $\omega$ has a double pole at $\infty$.

The main tool to compute the local constants is the formula (C.7) for twisting by a lisse sheaf: for any closed point $x$, if $K$ is an $\ell$-adic complex on the trait $\mathbf{P}^1(x)$ and $F$ is a lisse $\overline{\mathbb{Q}}_\ell$-sheaf on $\mathbf{P}^1(x)$ of rank $r(F)$, then we have

$$ (10.9) \qquad \varepsilon(\mathbf{P}^1(x), (K \otimes F)|\mathbf{P}^1(x), \omega|\mathbf{P}^1(x)) = \det(Fr_x | F)^{a(\mathbf{P}^1(x), K, \omega|\mathbf{P}^1(x))} \varepsilon(\mathbf{P}^1(x), K, \omega|\mathbf{P}^1(x))^{r(F)}, $$

where the local exponent $a(\mathbf{P}^1(x), K, \omega|\mathbf{P}^1(x))$ is defined in (C.4) and (C.3). Moreover, we will often use the formula

$$ \varepsilon(\mathbf{P}^1(x), K[1], \omega) = \varepsilon(\mathbf{P}^1(x), K, \omega)^{-1} $$

(see (C.10)).

Let $(\chi_z)_{z \in Z}$ be the tuple of characters corresponding to $\chi$. We recall that $\mathcal{L}_\chi$ is isomorphic to $\otimes_{z \in Z} \mathcal{L}_{\chi_z(z-t)}$.

We now compute the local constants, distinguishing between the cases $x \in \mathbf{A}^1 - (\{0, 1\} \cup Z)$, $x \in \{0, 1\}$, $x \in Z$ and $x = \infty$.

**Case 1.** Let $x \in \mathbf{A}^1$ and $x \notin Z \cup \{0, 1\}$. In this case, $M \otimes \mathcal{L}_\chi$ is a lisse sheaf shifted by 1, and since $c(\psi_x) = 0$, we find

$$ (10.10) \qquad \varepsilon_x(\chi) = 1 $$

by (10.9).

**Case 2.** Let $x \in \{0, 1\}$. Then $\mathcal{L}_\chi$ is a lisse sheaf at $x$, since we assumed that $f$ is coprime with $t(t-1)$. We find

$$ \varepsilon_x(\chi) = \varepsilon_x t_{\mathcal{L}_\chi}(x)^{a(\mathbf{P}^1(x), M|\mathbf{P}^1(x), dt)} $$

<footnote n="2">² Which, in the case we use it, goes back to Deligne [26, Th. 9.3]; see [98, 3.2.1.9] for references.</footnote>
by (10.9) with $F = \mathcal{L}_\chi$, where $\varepsilon_x = \varepsilon(\mathbf{P}^1_{(x)}, M, dt)$, which is independent of $\chi$. We further compute that

$$a(\mathbf{P}^1_{(x)}, M|\mathbf{P}^1_{(x)}, dt) = -a(\mathbf{P}^1_{(x)}, \mathcal{F}(1/2)|\mathbf{P}^1_{(x)}, dt) = -(2-1+0) = -1$$

by (C.4) and (C.3), since $\mathcal{F}$ has drop 1 at $x$ (see, e.g., [75, p. 73]) and $dt$ is regular at $x$. Hence,

$$(10.11) \qquad \varepsilon_x(\chi) = \varepsilon_x \prod_{z \in \mathbb{Z}} \chi_z(z-x)^{-1}.$$

**Case 3.** Let $x \in \mathbb{Z}$. Then we can write

$$M \otimes \mathcal{L}_{\chi} = \mathcal{F}[1](1/2) \otimes \mathcal{L}^{(x)} \otimes \mathcal{L}_{\chi_x(t-x)} = (\mathcal{F}(1/2) \otimes \mathcal{L}^{(x)} \otimes \mathcal{L}_{\chi_x(t-x)})[1],$$

where $\mathcal{F}$ and $\mathcal{L}^{(x)}$ are both lisse sheaves at $x$. Applying (10.9) after an inversion due to the shift, we get

$$\begin{aligned} \varepsilon_x(\chi) &= \varepsilon(\mathbf{P}^1_{(x)}, \mathcal{F}[1](1/2) \otimes \mathcal{L}^{(x)} \otimes \mathcal{L}_{\chi_x(t-x)}, dt)^{-1} \\ &= \det(\mathrm{Fr}_x | \mathcal{F}(1/2) \otimes \mathcal{L}^{(x)})^{-a} \varepsilon(\mathbf{P}^1_{(x)}, \mathcal{L}_{\chi_x(t-x)}, dt)^{-2} \end{aligned}$$

where

$$a = a(\mathbf{P}^1_{(x)}, \mathcal{L}_{\chi_x(t-x)}, dt) = 1 + 0 - 0 = 1$$

if $\chi_x$ is non-trivial by (C.4) and (C.3) again.

We have

$$\det(\mathrm{Fr}_x | \mathcal{F}(1/2) \otimes \mathcal{L}^{(x)}) = \frac{1}{|k|} \prod_{\substack{z \in \mathbb{Z} \\ z \neq x}} \chi_z(z-x)^2,$$

and by (C.9), we find that

$$\varepsilon(\mathbf{P}^1_{(x)}, \mathcal{L}_{\chi_x(t-x)}, dt) = \varepsilon_0(\mathbf{P}^1_{(x)}, \mathcal{L}_{\chi_x(t-x)}, dt) = -\chi_x(-1) \sum_{y \in k^\times} \chi(y)\psi(y)$$

if $\chi_x$ is not trivial (here we also use the fact that $x \in k$).

These computations imply that

$$(10.12) \qquad \varepsilon_x(\chi) = \prod_{\substack{z \in \mathbb{Z} \\ z \neq x}} \chi_z(z-x)^{-2} |k| \left( \sum_{y \in k^\times} \chi(y)\psi(y) \right)^{-2},$$

if $\chi_x$ is not trivial.

**Case 4.** Let $x = \infty$. Write $u = 1/t$, a uniformizer at $\infty$, so that $dt = -u^{-2}du$. Then $\mathcal{L}_\chi = \mathcal{L}^{(\infty)} \otimes \mathcal{L}_{\eta(u)}$ where

$$\mathcal{L}^{(\infty)} = \bigotimes_{z \in \mathbb{Z}} \mathcal{L}_{\chi_z(uz-1)}, \qquad \eta = \prod_{z \in \mathbb{Z}} \chi_z^{-1}.$$

The sheaf $\mathcal{L}^{(\infty)}$ is lisse at $\infty$ and the local eigenvalue of Frobenius there is equal to $(-1)^{\deg(f)}$. On the other hand, we have $M = \mathcal{F}[1](1/2)$, and $\mathcal{F}$ is of rank 2, tamely ramified at $\infty$ with local monodromy isomorphic to $\mathcal{L}_{\lambda_2} \otimes \mathrm{Unip}(2)$, where $\lambda_2$ is the Legendre character and $\mathrm{Unip}(2)$ is a unipotent Jordan block of size 2 (see, e.g., [75, p. 73]).

Computing first as in the previous case, we get

$$\begin{aligned} \varepsilon_\infty(\chi) &= \varepsilon(\mathbf{P}^1_{(\infty)}, \mathcal{F}(1/2) \otimes \mathcal{L}^{(\infty)} \otimes \mathcal{L}_{\eta(u)}, -u^{-2}du)^{-1} \\ &= \det(\mathrm{Fr}_\infty | \mathcal{L}^{(\infty)})^{-a} \varepsilon(\mathbf{P}^1_{(\infty)}, \mathcal{F}(1/2) \otimes \mathcal{L}_{\eta(u)}, -u^{-2}du)^{-1} \end{aligned}$$
where

$$ 
\begin{aligned}
a &= a(\mathbf{P}_{(\infty)}^{1}, \mathcal{F}(1/2) \otimes \mathcal{L}_{\eta(u)}, -u^{-2}du) \\
&= a(\mathbf{P}_{(\infty)}^{1}, \mathcal{F}(1/2) \otimes \mathcal{L}_{\eta(u)}) - 2 \times 2 = (2+0-2)-4 = -4
\end{aligned}
$$

if $\eta$ is non-trivial (see again (C.4) and (C.3)). Note then that

$$ \det(\mathrm{Fr}_{\infty} | \mathcal{L}^{(\infty)})^{-a} = \prod_{z \in \mathbb{Z}} \chi_z (-1)^4 = 1. $$

The shape of the local monodromy and the multiplicativity property under extensions shows that if $\lambda_2\eta$ is not trivial, then the formula

$$ \varepsilon(\mathbf{P}_{(\infty)}^{1}, \mathcal{F}(1/2) \otimes \mathcal{L}_{\eta(u)}, -u^{-2}du) = \varepsilon(\mathbf{P}_{(\infty)}^{1}, \mathcal{L}_{(\lambda_2\eta)(u)}, -u^{-2}du)^2 $$

holds. Indeed, in this case, the stalk at $\infty$ of $\mathcal{F} \otimes \mathcal{L}_{\eta}$ and of its semisimplification both vanish, so that

$$ \varepsilon(\mathbf{P}_{(\infty)}^{1}, \mathcal{F}(1/2) \otimes \mathcal{L}_{\eta(u)}, -u^{-2}du) = \varepsilon_0(\mathbf{P}_{(\infty)}^{1}, j_{*}\mathcal{F}(1/2) \otimes \mathcal{L}_{\eta(u)}, -u^{-2}du), $$

where $\varepsilon_0$ is the local factor defined by (C.5), and $j$ is the inclusion of the generic point of $\mathbf{P}_{(\infty)}^{1}$, and one can apply (C.6); compare [26, 8.12].

Let $\beta$ be the character of the local field at infinity associated to $\lambda_2\eta$ by local class field theory. Using (C.8), we derive the formula

$$ \varepsilon(\mathbf{P}_{(\infty)}^{1}, \mathcal{L}_{(\lambda_2\eta)(u)}, -u^{-2}du) = \beta(-u^{-2})|k|^{-2}\varepsilon(\mathbf{P}_{(\infty)}^{1}, \mathcal{L}_{(\lambda_2\eta)(u)}, du). $$

From (C.9), we deduce further that if $\lambda_2\eta$ is non-trivial, then

$$ \varepsilon(\mathbf{P}_{(\infty)}^{1}, \mathcal{L}_{(\lambda_2\eta)(u)}, -u^{-2}du) = |k|^{-2} \sum_{y \in k^{\times}} (\lambda_2\eta)(y)\psi(y). $$

The final outcome is that

$$ (10.13) \qquad \varepsilon_{\infty}(\chi) = |k|^4 \left( \sum_{y \in k^{\times}} (\lambda_2\eta)(y)\psi(y) \right)^{-2}, $$

if $\eta \notin \{1, \lambda_2\}$.

We now simply combine the formulas (10.10), (10.11), (10.12) and (10.13) to conclude the proof, noting that the contribution of all $x \in \mathbb{Z}$ involves the product

$$ \prod_{x \in \mathbb{Z}} \prod_{z \in \mathbb{Z}, z \neq x} \chi_z (z-x)^{-2} = \prod_{z \in \mathbb{Z}} \chi_z \left( \prod_{\substack{x \in \mathbb{Z} \\ x \neq z}} (z-x)^{-2} \right). $$

□

**REMARK 10.12.** It also certainly possible to perform this computation by automorphic methods (using the global case of the $GL_2$-Langlands correspondence over $k(t)$, first proved by Drinfeld). However, more general situations might be easier to handle using these geometric arguments.

Yet another possible approach, which would be well-suited for generalizations, would be to use Loeser's general computation of the tannakian determinant for an arbitrary perverse sheaf on a torus $\mathbf{T}$ (see [103, Th. 3.6.1]), which can be identified with an element of the hypergeometric group $\mathbf{H}_{\mathrm{int}}(\mathbf{T})$ of Gabber and Loeser (see Example 3.52). This group is isomorphic (loc. cit.) to $\mathbf{T}(\bar{k}) \times \mathbf{Z}^S$ for some explicit set $S$ (related to sub-tori of dimension 1 in $\mathbf{T}$ and tame $\ell$-adic characters of $\mathbf{G}_m$). It would then be enough to show that there exists some $s \in S$ such that the $s$-component of $\det(\mathbf{M})$ is non-zero to deduce that $\det(\mathbf{M})$ has infinite order (without computing exactly the determinant).
# Equidistribution on abelian varieties

In this chapter, we consider some aspects of equidistribution on abelian varieties. We denote as before by $k$ a finite field, and by $\bar{k}$ an algebraic closure of $k$. We denote by $k_n$ the extension of degree $n$ in $\bar{k}$. The prime $\ell$ is different from the characteristic of $k$.

## 11.1. Equidistribution in the jacobian of a curve

The main result of this section is a generalization of a theorem announced by Katz during a talk at a workshop held at the University of Zürich in September 2012 [74], answering a question of Tsimerman.

Let $\mathbb{C}$ be a smooth projective geometrically connected curve of genus $g \ge 2$ over $k$, and let $\mathcal{A} = \mathrm{Jac}(\mathbb{C})$ be its jacobian. We recall that $\mathbb{C}$ may not have $k$-rational points but always has a $k$-rational divisor of degree one. We fix such a divisor $\Delta$ and we denote by $s_\Delta: \mathbb{C} \hookrightarrow \mathcal{A}$ the closed immersion obtained by sending a point $x$ to the class of the divisor $(x) - \Delta$. Recall that the functor $s_{\Delta*} = s_\Delta!$ preserves perversity (Corollary A.8).

**THEOREM 11.1 (Katz).** Let $\Delta$ be a divisor of degree one on $\mathbb{C}$. Let $M_0$ be a geometrically simple perverse sheaf on $\mathbb{C}$ of generic rank $r \ge 1$ which is pure of weight zero. Let $\mathcal{M} = s_{\Delta*}M_0$ and let $d$ denote the tannakian dimension of $\mathcal{M}$.

(1) We have $d \ge (2g - 2)r \ge 2$.

(2) Assume that $\mathbb{C}$ is hyperelliptic, that $\Delta = (0_{\mathbb{C}})$ for some $k$-rational point $0_{\mathbb{C}} \in \mathbb{C}(k)$ fixed by the hyperelliptic involution $i$, and that $\mathrm{D}(M_0)$ is geometrically isomorphic to $i^*M_0$. Then, up to conjugacy, there are inclusions

$$ \mathbf{G}_{\mathcal{M}}^{\mathrm{geo}} = \mathbf{G}_{\mathcal{M}}^{\mathrm{ari}} = \mathbf{S}_{\mathbf{P}_d} \quad \text{or} \quad \mathbf{S}\mathbf{O}_d \subset \mathbf{G}_{\mathcal{M}}^{\mathrm{geo}} \subset \mathbf{G}_{\mathcal{M}}^{\mathrm{ari}} \subset \mathbf{O}_d. $$

(3) If $\mathbb{C}$ is not hyperelliptic, or if $\mathbb{C}$ is hyperelliptic but $\mathrm{D}(M_0)$ is not geometrically isomorphic to $i^*M_0$, then there are inclusions

$$ \mathbf{SL}_d \subset \mathbf{G}_{\mathcal{M}}^{\mathrm{geo}} \subset \mathbf{G}_{\mathcal{M}}^{\mathrm{ari}} \subset \mathbf{GL}_d. $$

**PROOF.** We write $s = s_\Delta$ for simplicity. Since $\mathcal{A}$ is an abelian variety, the dimension $d$ is the Euler-Poincaré characteristic of $M_\chi$ for any $\chi \in \hat{\mathbb{A}}$ (see Proposition 3.22), in particular for the trivial character, which means that $d = \chi(A_{\bar{k}}, \mathcal{M}) = \chi(C_{\bar{k}}, M_0)$. Write $M_0 = \mathcal{F}_0[1](1/2)$ for some middle extension sheaf $\mathcal{F}_0$ on $\mathbb{C}$ of generic rank $r$; using the Euler-Poincaré characteristic formula on a curve (see (C.11), for instance), it follows that

$$ (11.1) \quad \chi(C_{\bar{k}}, M_0) = \chi(C_{\bar{k}}, \mathcal{F}_0[1]) = (2g-2)r + \sum_{x \in C(\bar{k})} (\mathrm{swan}_x(\mathcal{F}_0) + \mathrm{drop}_x(\mathcal{F}_0)) \ge (2g-2)r. $$

According to Proposition 8.11 (2), the embedding $s$ is a Sidon morphism if $\mathbb{C}$ is not hyperelliptic, and is an $i$-symmetric Sidon morphism in the hyperelliptic situation of (2).

Suppose first that $\mathbb{C}$ is not hyperelliptic. Using the fact that $d \ge 2$, we deduce from Proposition 8.8 that $M_4(\mathbf{G}_{\mathcal{M}}^{\mathrm{ari}}) = 2$. Thus, by Larsen's Alternative (Theorem 8.5 (3)), either $\mathbf{G}_{\mathcal{M}}^{\mathrm{ari}}$ is virtually
central, i.e., $G_M^{\text{ari}}/G_M^{\text{ari}} \cap Z$ is finite, or $G_M^{\text{ari}}$ contains $SL_d$. Proposition 8.22 shows that the first case is not possible, since $M_4(G_M^{\text{ari}}) = 2$ is not the square of an integer. Then the fact that $G_M^{\text{ari}}$ contains $SL_d$ implies that $G_M^{\text{geo}}$ also contains $SL_d$ (indeed, the intersection $G_M^{\text{geo}} \cap SL_d$ is a normal subgroup of $G_M^{\text{ari}}$ by Proposition 3.40, and hence is a normal subgroup of $SL_d$; it is therefore either equal to $SL_d$, or is contained in the center $\mu_d$; but since $d \ge 2$, the latter would imply that $G_M^{\text{ari}}/G_M^{\text{geo}}$ is not abelian).

We now assume that $C$ is hyperelliptic. First we consider the case when $d \ge 3$.

If $\text{D}(M_0)$ is not geometrically isomorphic to $i^*M_0$, then Proposition 8.9 (2) implies $M_4(G_M^{\text{ari}}) = 2$ since we assume that $d \ge 3$; as previously, we then conclude that $G_M^{\text{ari}}$ contains $SL_d$.

If the conditions of (2) hold, then the constant morphism $(s \circ i) + s$ is given by

$$s(i(x)) + s(x) = (x) + i(x) - 2(0_C) = 0,$$

the identity element of $\text{A}$. Proposition 8.9 (1) implies then that $M$ is self-dual and has $M_4(G_M^{\text{ari}}) = 3$, again from our assumption that $d \ge 3$. We conclude in that case by Larsen's Alternative (Theorem 8.5 (5)), combined with the fact that $G_M^{\text{ari}}$ is infinite by Theorem 5.7.

There remains to consider the case when $d = 2$ (and $C$ hyperelliptic). Since $d = \chi(C_{\bar{k}}, M_0)$, formula (11.1) shows that this situation can only occur if $(g, r) = (2, 1)$ and if the sheaf $\mathcal{F}_0$ is lisse on $C$. Thus the curve $C$ has genus 2, and the sheaf $\mathcal{F}_0$ is a rank 1 sheaf corresponding to a character of the fundamental group of $C$. As we will recall below in general, there exists then a character $\chi_0 \in \hat{\text{A}}(k)$ such that $\mathcal{F}_0$ is geometrically isomorphic to $s^*\mathcal{L}_{\chi_0}$ on $C$. The duality condition $M_0 \simeq i^*\text{D}(M_0)$ is then always satisfied.

We claim that in this situation, the fourth moment $M_4(G_M^{\text{ari}})$ is still equal to 2. Indeed, from the proof of Proposition 8.9, we know that

$$\frac{1}{|\text{A}(k_n)|} \sum_{\chi \in \hat{\text{A}}(k_n)} |\text{S}(\text{M}, \chi)|^4$$

converges to 3 as $n \to +\infty$. The contribution of the character $\chi_0^{-1}$, which is the only ramified character, is

$$\frac{1}{|\text{A}(k_n)|} |\text{S}(\text{M}, \chi_0^{-1})|^4 = \frac{1}{|\text{A}(k_n)|} \left| \sum_{x \in \text{C}(k_n)} t_{M_0}(x; k_n) \overline{\chi_0(x)} \right|^4 = \frac{|\text{C}(k_n)|^4}{|k_n|^2 |\text{A}(k_n)|}$$

which converges to 1 as $n \to +\infty$. We then conclude from Larsen's Alternative that the group $G_M^{\text{ari}}$ contains $SL_2 = \mathbf{Sp}_2$. $\square$

REMARK 11.2. (1) Note that the last case provides a concrete example where the limit

$$\lim_{n \to +\infty} \frac{1}{|\text{A}(k_n)|} \sum_{\chi \in \hat{\text{A}}(k_n)} |\text{S}(\text{M}, \chi)|^4$$

exists, where the sum ranges over all characters, but its value is *not* the fourth moment of the standard representation of the tannakian group (see Remark 8.10).

(2) If the curve $C$ has gonality at least 5, then the inclusions

$$SL_d \subset G_M^{\text{geo}} \subset G_M^{\text{ari}} \subset GL_d$$

can be deduced without appealing to Proposition 8.22. Indeed, the immersion $s_\Delta$ is then a 4-Sidon morphism by Proposition 8.11 (3), so we deduce from Proposition 8.8 (2) that $G_M^{\text{ari}}$ (and hence also $G_M^{\text{geo}}$, as before) contains $SL_d$. (Precisely, we are in the excluded case of this statement, but we can observe that there are only finitely many ramified characters here, and that the assumption
implies that the genus of C is at least five, so that the contribution to the 8-th moment of the ramified characters is

$$ \ll \frac{1}{|k_n|^g} |k_n|^{8/2} \to 0, $$

so that we do obtain the correct 8-th moment.)

**REMARK 11.3.** In characteristic zero, Krämer and Weissauer [93] have obtained closely related results, using more geometric methods in the case of the object $M = s_{\Delta*}\overline{Q}_{\ell}[1]$.

We now explain how Theorem 11.1 answers a question of Tsimerman, which was Katz’s original motivation. Let $\rho: \pi(C)^{ab} \to C^\times$ be a character of finite order. By the Riemann hypothesis for curves over finite fields, the Artin L-function $L_C(\rho, s)$ is a polynomial of degree $2g-2$ in the variable $T = q^{-s}$ all of whose reciprocal roots have absolute value $\sqrt{q}$. We can then write

$$ L(\rho, T/\sqrt{q}) = \det (1 - T\Theta_{C/k,\rho}) $$

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

**QUESTION (Tsimerman).** How are these conjugacy classes distributed as $\rho$ varies?

From now on, we shall normalize the characters as follows: we fix a divisor $\Delta = \sum n_i x_i$ of degree one on C and we only consider those characters $\rho$ satisfying

$$ \prod \rho(\mathrm{Fr}_{\kappa(x_i),x_i})^{n_i} = 1. $$

Through the isomorphism $\pi_1(C)^{ab} \simeq \pi_1(A)$ induced by $s_\Delta: C \hookrightarrow A = \mathrm{Jac}(C)$, such normalized characters correspond to characters $\rho: \pi_1(A) \to C^\times$ satisfying $\rho(\mathrm{Fr}_{k,0_A}) = 1$. Since they are in addition supposed to be of finite order, they arise via the Lang isogeny from the elements of $\hat{A}(k)$. Replacing $k$ with $k_n$, we obtain the corresponding characters in $\hat{A}(k_n)$. Thus the following statement answers Tsimerman’s question when considering conjugacy classes associated to normalized characters over $k_n$ and taking $n \to +\infty$.

**COROLLARY 11.4.** Let C be a smooth projective geometrically connected curve of genus $g \ge 2$ over k with jacobian A.

(1) If C is hyperelliptic, the hyperelliptic involution has a fixed k-point $0 \in C(k)$, and we use this point to define the embedding $C \to A$, then the conjugacy classes $(\Theta_{C/k,\chi})_{\chi \in \hat{G}(A_n), \chi \ne 1}$ are conjugacy classes in $US_{2g-2}(C)$ and become equidistributed with respect to the image of the Haar probability measure on the set of conjugacy classes.

(2) If C is not hyperelliptic and $(2g-2)\Delta$ is a canonical divisor on C, then the conjugacy classes $(\Theta_{C/k,\chi})_{\chi \in \hat{A}(k_n), \chi \ne 1}$ are conjugacy classes in $SU_{2g-2}(C)$ and become equidistributed with respect to the image of the Haar probability measure on the set of conjugacy classes.

**PROOF.** Consider the weight zero perverse sheaf $M_0 = \overline{Q}_{\ell}(1/2)[1]$ on C and set $M = s_{\Delta*}M_0$. For each rank one $\ell$-adic lisse sheaf $\mathcal{L}$ on A, there are isomorphisms

$$ H^i(A_{\bar{k}}, M \otimes \mathcal{L}) \simeq H^i(A_{\bar{k}}, s_{\Delta^*}(M_0 \otimes s_{\Delta}^*\mathcal{L})) \simeq H^i(C_{\bar{k}}, M_0 \otimes s_{\Delta}^*\mathcal{L}) \simeq H^{i+1}(C_{\bar{k}}, s_{\Delta}^*\mathcal{L}(1/2)) $$

by the projection formula and the exactness of $s_{\Delta^*}$. It follows that M has tannakian dimension

$$ -\chi(C_{\bar{k}}, s_{\Delta}^*\mathcal{L}) = 2g - 2, $$

and moreover that all non-trivial characters are unramified for M (since we are considering an abelian variety). By Theorem 4.11, it suffices therefore to prove that the arithmetic and geometric tannakian groups of M coincide and are equal to $Sp_{2g-2}$ in case (1) and to $SL_{2g-2}$ in case (2).
Assume C is hyperelliptic with hyperelliptic involution $i$. Then the sheaf $M_0$ is geometrically isomorphic to $i^*D(M_0)$ and, because of the shift by 1 in the definition of $M_0$, the corresponding self-duality is symplectic. Therefore, if in addition we assume that $i$ has a fixed $k$-point $0 \in C(k)$, which we use as divisor $\Delta$, Theorem 11.1 (2) gives the equality $G_M^{\text{geo}} = G_M^{\text{ari}} = S\mathbf{p}_{2g-2}$.

If C is not hyperelliptic and $(2g-2)\Delta$ is a canonical divisor on C, in view of Theorem 11.1 (3), it suffices to show that the arithmetic group $G_M^{\text{ari}}$ lies in $\mathbf{SL}_{2g-2}$. For this, we compute the determinant of the action of Frobenius on $\mathbb{H}^1(C_{\bar{k}}, \mathcal{L}(1/2))$. Since this cohomology is even-dimensional, this is also the determinant of $-\text{Fr}_k$, which is the constant in the functional equation for the L-function of $\mathcal{L}(1/2)$. By a classical result of Weil [125],<fnref n="1" /> in the case of $\mathcal{L}$ this constant is given by $q^{1-g}\varrho_{\mathcal{L}}(\text{can})$ for a canonical divisor can, where $\varrho_{\mathcal{L}}$ is the character associated to $\mathcal{L}$, which factors through the jacobian. Taking the half-Tate twist into account, along with the fact that $\varrho_{\mathcal{L}}(\text{can}) = 1$ since $(2g-2)\Delta$ is a canonical divisor and characters are normalized to take the value 1 at $\Delta$, it follows that the determinant is trivial, as claimed. □

We conclude this section by a (partial) generalization of Theorem 11.1 to the setting of generalized jacobians arising in geometric class-field theory. This gives a natural example of an application of our results where the algebraic group $G$ is not restricted to being either a torus, an abelian variety or a unipotent group, but may involve all three of these fundamental building blocks (see Remark 8.14). For simplicity, we will only deal with the case where $C$ is not hyperelliptic.

**THEOREM 11.5.** Assume that the curve $C$ is not hyperelliptic. Let $S$ be an effective divisor on the curve $C$. Let $U$ be the complement of the support of $S$ in $C$. Let $\Delta$ be a divisor of degree one on $U$. Let $J_S$ be the generalized jacobian of $C$ relative to the divisor $S$, and let $s_\Delta: U \to J_S$ be the natural immersion defined by $x \mapsto (x) - \Delta$.

Let $M_0$ be a semiperverse object on $U$, mixed of weights $\le 0$ and put $M = s_\Delta!M_0$. Let $d$ be the tannakian dimension of the semisimplification $\tilde{M}$ of the part of $\mathfrak{p}\mathcal{H}^0(M)$ which is pure of weight 0. Assume that $\tilde{M}$ is non-zero.

Then we have $d \ge 2$, and either the arithmetic tannakian group of $\tilde{M}$ contains $\mathbf{SL}_d$ or $G_{\tilde{M}}^{\text{ari}}$ is virtually central in $\mathbf{GL}_d$.

**PROOF.** We note that $M$ is a semiperverse object on $J_S$ since $s_D$ is quasi-finite, and is mixed of weights $\le 0$ by the Riemann Hypothesis.

To check that $d \ge 2$, we use the general Euler-Poincaré characteristic formula (see Theorem C.2) as in (11.1), to conclude. We then need only observe that $s_D$ is a Sidon morphism by Proposition 8.13, and apply Larsen’s Alternative. □

**REMARK 11.6.** (1) Since we do not know in general if perverse sheaves on the group $J_S$ are generically unramified, the corresponding equidistribution statement is currently restricted to the distribution of the arithmetic Fourier transforms

$$ \sum_{x \in U(k_n)} t_M(x; k_n)\chi(x) $$
for $\chi \in \hat{J}_S(k_n)$.

(2) Again because the group $J_S$ is a priori fairly arbitrary here, we can not exclude the possibility that $G_{\tilde{M}}^{\text{ari}}$ is virtually central (e.g., finite), since we do not have currently a general version of

<footnote n="1">¹ Which can also easily be recovered from the theory of local constants, applying the results of Deligne and Laumon (see Appendix C).</footnote>
**Proposition 8.22.** (In our case, since the jacobian of C is a non-trivial quotient of $J_S$, we can expect that the statement should indeed extend.)

(3) It is possible that $\tilde{M}$ is zero; in this case, we have of course $d = 0$, and the tannakian group is trivial.

## 11.2. The intermediate jacobian of a cubic threefold

Our second application involving abelian varieties is related to a very classical and important construction in algebraic geometry, that of the intermediate jacobian of a smooth cubic threefold, which was used by Clemens and Griffiths to prove that these threefolds, over $C$, are not rational (although they are unirational).

The geometric setting, which over finite fields goes back at least to the work of Bombieri and Swinnerton-Dyer [10] (computing the zeta function of smooth cubic threefolds over finite fields) is the following.

Let $k$ be a field of characteristic different from 2, and let $X \subset \mathbb{P}_k^4$ be a smooth cubic threefold over $k$. We denote by $F(X)$ the Fano scheme of lines in $X$, which is a smooth projective and geometrically connected surface over $k$ (see, e.g., [5, § 4] or [10, Lem. 3] or [2, Cor. 1.12, Th. 1.16]; this uses the fact that the characteristic is different from 2). Let then $A(X)$ be the Albanese variety of $F(X)$, which is known to be isomorphic to the Picard variety of $F(X)$ (see, e.g. [61, Cor. 4.3.3]). It has dimension 5, and if the base field is contained in $C$, then the analytification of $A(X)$ is canonically isomorphic to the intermediate jacobian $J(X)$ of Griffiths, which is defined analytically in terms of Hodge theory (this is due to Murre; see [5, Prop. 9]).

The Albanese morphism $s: F(X) \to A(X)$ is a closed immersion, according to a theorem of Beauville [5, p. 201, cor.]. If we view $A(X)$ as the Picard variety, then the morphism $s$ can be identified geometrically with the map sending a line $l \in F(X)$ to the divisor defined by the curve $C_s$ which is the Zariski-closure in $F(X)$ of the set of lines $l' \neq l$ such that $l' \cap l$ is not empty.

The problem we consider is then the following: if $k$ is a finite field of odd characteristic, what is the arithmetic tannakian group of the perverse sheaf $M = s_*\overline{\mathbb{Q}}_l[2](1)$ on $A(X)$? (It is perverse because $s$ is a closed immersion, as in previous similar examples.) The corresponding exponential sums are then

$$S(M, \chi) = \frac{1}{|k_n|} \sum_{l \in F(X)(k_n)} \chi(s(l))$$

for a character $\chi \in \widehat{A(X)}(k_n)$.

Up to correcting a small oversight, the following answer is the analogue over finite fields of a result of Krämer over $C$ (see [92, Th. 2]).

**PROPOSITION 11.7.** Let $k$ be a finite field of characteristic different from 2. Let $X$ be a smooth cubic threefold over $k$, and denote by $F(X)$ the Fano scheme of lines in $X$, by $A(X)$ the Albanese variety of $F(X)$, and by

$$s: F(X) \to A(X)$$

the natural closed immersion.

Let $l$ be a prime different from the characteristic of $k$, and let $M$ be the object $M = s_*\overline{\mathbb{Q}}_l[2](1)$ on $A(X)$. The connected derived subgroup of the arithmetic tannakian monodromy group of the object $M$ of the category $\overline{\mathbb{P}}^{\text{ari}}(A(X))$ is isomorphic to the exceptional group $E_6$.

For the proof, we will use the following lemma, whose proof was communicated to us by Beauville.
LEMMA 11.8 (Beauville). With notation as above, there is no $x \in A(X)$ such that $-s(F(X)) = x + s(F(X))$, and there is no non-zero $x \in A(X)$ such that $s(F(X)) = x + s(F(X))$.

PROOF. We argue by contradiction.

For the first assertion, if $x$ existed such that $-s(F(X)) = x + s(F(X))$, then the involution $a \mapsto -x - a$ of $A(X)$ would induce an involution $i$ of the variety $F(X)$ with a finite number of fixed points. The quotient variety $F(X)/i$ is then a normal variety with only isolated ordinary double points as singularities. In particular, it is Gorenstein (see, e.g., [35, Cor. 21.19]), so its canonical divisor $K_{F(X)/i}$, defined as the direct image of the canonical divisor of the smooth locus of $F(X)/i$, is a Cartier divisor (see e.g [84, p. 79]). Since the projection $p: F(X) \to F(X)/i$ is étale outside of the set of fixed points, the canonical divisor of $F(X)$ is $K = p^*(K_{F(X)/i})$. This implies that $K^2 = 2(K_{F(X)/i}^2)$ is even. However, it is known that $K^2 = 45$, which is odd (see, e.g., [61, Prop. 4.6]).

For the second assertion, note that $s(F(X)) = x + s(F(X))$ would imply that

$$s(F(X)) - s(F(X)) = x + s(F(X)) - s(F(X)),$$

so that the theta divisor $\Theta(X) = s(F(X)) - s(F(X))$ satisfies $\Theta(X) = x + \Theta(X)$. However, Beauville [5, § 3, Prop. 2] showed that $\Theta(X)$ is smooth except for a single singularity, so this equality can only happen if $x = 0$. $\square$

REMARK 11.9. The cohomological analogue of this proposition is not true: for $\varepsilon \in \{-1, 1\}$, the cohomology class of $\varepsilon s(F(X))$ in $H^6(A(X))$ is $\Theta^3/6$, where $\Theta$ is the cohomology class of the symmetric theta divisor $s(F(X)) - s(F(X))$ (the fact that $s(F(X))$ has the same class as $-s(F(X))$ is due to the fact that $x \mapsto -x$ acts trivially on even-degree cohomology groups; the computation in terms of $\Theta$ is explained, e.g., in [61, Cor. 5.3.12, (i)]).

We now give a proof of Proposition 11.7 adapting Krämer's argument over $\mathbb{C}$, the key point being the recognition criterion of $\mathbb{E}_6$ in Proposition 8.19.

PROOF. Since $F(X)$ is a smooth, projective and geometrically connected surface, the object $M$ is a simple perverse sheaf on $A(X)$. The tannakian dimension of $M$ is equal to the Euler-Poincaré characteristic of $M$ over $\bar{k}$ (Proposition 3.22), which is equal to the Euler-Poincaré characteristic of the Fano surface $F(X)$, which is 27 (a result of Fano, see, e.g., [2, Prop. 1.23]).

Let $\Theta(X)$ be the theta divisor $s(F(X)) - s(F(X))$ in $A(X)$, and $i: \Theta(X) \to A(X)$ the closed immersion. The object $M * M^\vee$ contains the object $N = i_*\bar{Q}_\ell[1]$ by the decomposition theorem (see [92, proof of Th. 2]). This is also a simple perverse sheaf since $\Theta$ is a geometrically irreducible divisor (see, e.g., [5, Prop. 2]). The tannakian dimension of $N$ can be computed as in [92, Cor. 6] (or by lifting to characteristic 0, as can be done as in [10, Proof of Lemma 5]), and is equal to 78.

To conclude using Proposition 8.19, applied to the connected derived subgroup $\mathbb{G}$ of $\mathbb{G}_M^{\text{ari}}$, it suffices therefore to check that $\mathbb{G}$ still acts irreducibly on the 27-dimensional representation corresponding to $M$.

To see this, note that the neutral component $(\mathbb{G}_M^{\text{ari}})^\circ$ acts irreducibly by Corollary 5.9 combined with Lemma 11.8. Then its derived group $\mathbb{G}$ must also act irreducibly since

$$(\mathbb{G}_M^{\text{ari}})^\circ = \mathbb{C} \cdot \mathbb{G}$$

for some torus $\mathbb{C}$, which is central by irreducibility. $\square$

It is natural to ask whether this proposition can also be proved using the fourth moment criterion of Proposition 8.20 instead of Krämer's criterion.
We have not fully succeeded in doing so, but we can show that the question translates to an interesting geometric property of the cubic threefolds. Conversely, this property follows in fact from the previous proof, as we will now explain.

In order to apply Proposition 8.20, we need to check that the object M is not self-dual, that it has tannakian dimension 27 and that its fourth moment is $M_4(\mathbf{G}_M^{\text{ari}}) = 3$.

Lemma 11.8 implies that M is not self-dual. The second property is derived as in the beginning of the previous proof. Now we attempt to compute the fourth moment.

We write $F = F(X)$ and $A = A(X)$. We use the diophantine interpretation of the fourth moment. Summing over all characters, we find as usual using orthogonality that for $n \ge 1$, the formula

$$ \frac{1}{|A(k_n)|} \sum_{\chi \in \hat{A}(k_n)} |S(M, \chi)|^4 = \frac{1}{|k_n|^4} \sum_{\substack{(l_1, \dots, l_4) \in F(k_n)^4 \\ s(l_1) - s(l_2) = s(l_3) - s(l_4)}} 1 $$

holds. We rewrite this in the form

$$ \frac{1}{|k_n|^4} \sum_{(l_1, l_2) \in F(k_n)^2} N(l_1, l_2) $$

where

$$ N(l_1, l_2) = |\{(l_3, l_4) \in F(k_n)^2 \mid s(l_1) - s(l_2) = s(l_3) - s(l_4)\}|. $$

For $s \in F(k_n)$, we have

$$ N(s, s) = |F(k_n)|, $$

and hence

$$ \frac{1}{|k_n|^4} \sum_{(l_1, l_2) \in F(k_n)^2} N(l_1, l_2) = \frac{|F(k_n)|^2}{|k_n|^4} + \frac{1}{|k_n|^4} \sum_{l_1 \neq l_2 \in F(k_n)} N(l_1, l_2) $$

Since $F$ is a geometrically irreducible surface over $k$, the first term converges to 1 as $n \to +\infty$. To handle the second term, consider the morphism given by the first projection

$$ f: \mathbb{F}^2 \times_A \mathbb{F}^2 \to \mathbb{F}^2, $$

where the fiber product is defined by the morphisms $(l_1, l_2) \mapsto s(l_1) - s(l_2)$ and $(l_3, l_4) \mapsto s(l_3) - s(l_4)$.

We then have

$$ N(l_1, l_2) = |f^{-1}(l_1, l_2)(k_n)| = |f^{-1}(l_1, l_2)(\bar{k})^{\text{Fr}_{k_n}}|, $$

the number of fixed points of the Frobenius of $k_n$ acting on the fiber of $f$. The fiber product $\mathbb{F}^2 \times_A \mathbb{F}^2$ contains the diagonal $\Delta = \{(l_1, l_2, l_1, l_2)\}$, and we denote by $\tilde{f}$ the morphism obtained by restriction

$$ \tilde{f}: (\mathbb{F}^2 \times_A \mathbb{F}^2) - \Delta \to \mathbb{F}^2. $$

We then deduce that

$$ N(l_1, l_2) = 1 + \tilde{N}(l_1, l_2) \quad \text{where} \quad \tilde{N}(l_1, l_2) = |\tilde{f}^{-1}(l_1, l_2)(\bar{k})^{\text{Fr}_{k_n}}|, $$

and hence

$$ \frac{1}{|k_n|^4} \sum_{(l_1, l_2) \in F(k_n)^2} N(l_1, l_2) = \frac{|F(k_n)|^2}{|k_n|^4} + \frac{|F(k_n)|(|F(k_n)| - 1)}{|k_n|^4} + \frac{1}{|k_n|^4} \sum_{l_1 \neq l_2 \in F(k_n)} \tilde{N}(l_1, l_2). $$
A theorem of Beauville [5, Prop. 8] implies that $\tilde{f}$ is generically finite of degree 5. By the Chebotarev Density Theorem (see, e.g., [79, Th. 9.7.13]) it follows that

$$ \lim_{n \to +\infty} \frac{1}{|k_n|^4} \sum_{l_1 \neq l_2 \in F(k_n)} \tilde{N}(l_1, l_2) $$

is equal to the number of orbits of the Galois group of the Galois closure of $\tilde{f}$ in its permutation representation on the generic fiber of $\tilde{f}$.

The generic point $\eta$ of $F^2$ is a pair of two disjoint lines $\eta = (\tilde{s}_1, \tilde{s}_2)$. Beauville showed that the points $(\tilde{s}_3, \tilde{s}_4) \in F^2$ such that $(\tilde{s}_1, \tilde{s}_2, \tilde{s}_3, \tilde{s}_4)$ is in the fiber over $\eta$ are such that the lines $\tilde{s}_3$ and $\tilde{s}_4$ are contained in the intersection $S$ of $X$ and of the projective 3-space spanned by $(\tilde{s}_1, \tilde{s}_2)$. Thus $S$ is a smooth cubic surface, and the lines $\tilde{s}_3$ and $\tilde{s}_4$ are elements of the set $\Lambda$ of the five lines in $S$ intersecting both $\tilde{s}_1$ and $\tilde{s}_2$; for these geometric facts, see [5, p. 203, rem. 2] or [61, proof of Cor. 4.3.9].

We claim that the subgroup of the Galois group of the 27 lines which fixes the two lines $\tilde{s}_1$ and $\tilde{s}_2$ is the Galois group of the set of seven lines $\{\tilde{s}_1, \tilde{s}_2\} \cup \Lambda$ (see Lemma 11.11 below). Now it follows from the work of Harris on Galois groups of enumerative problems (see [58, p. 718]) that this permutation representation is indeed transitive, in fact that it has image isomorphic to the whole symmetric group $\mathfrak{S}_5$, if we take the base field to be $\mathbb{C}$ and the cubic threefold to be general.

One may expect this to also be true in our situation:

- the restriction of the base field should not be problematic (indeed, the fact that the "generic" Galois group of the 27 lines on a smooth cubic surface is isomorphic to the Weyl group of $\mathbb{E}_6$, which is the starting point of Harris's work, is known in all odd characteristics, by work of Achter [1, Prop. 4.8]);

- the (four-dimensional) family of hyperplane sections that we consider is dominant over the (also four-dimensional) space of hyperplane sections of the cubic hypersurface $X$ (indeed, for any hyperplane $H$ in $\mathbb{P}^4$ intersecting $X$ in a smooth surface, we can pick two distinct lines $(l_1, l_2)$ in $X \cap H$, and the corresponding section is $H \cap X$);

- and the family of all hyperplane sections of $X$ is probably general enough for the result of Harris to extend. (This is in fact the most delicate point.)

If we assume that this expectation holds for $X$, then we would deduce that

$$ (11.2) \qquad \lim_{n \to +\infty} \frac{1}{|k_n|^4} \sum_{l_1 \neq l_2 \in F(k_n)} \tilde{N}(l_1, l_2) = 1 $$

holds, and hence conclude that

$$ \lim_{n \to +\infty} \frac{1}{|k_n|^4} \sum_{(l_1, l_2) \in F(k_n)^2} N(l_1, l_2) = 3. $$

Under this assumption, we therefore derive from Proposition 8.4 that $M_4(\mathbb{G}_M^{\text{ari}}) \le 3$. Since $M$ is not of tannakian dimension 1, the fourth moment is equal to either 2 or 3. We can at least partially exclude the first possibility as follows:

(1) For "most" cubic threefolds, the abelian variety $A$ is absolutely simple (see Lemma 11.12 below for a precise statement). In this case, there are only finitely many characters $\chi \in \tilde{\mathbb{A}}$
which are not weakly-unramified, and for which

$$|S(M, \chi)| = \left| \frac{1}{|k_n|} \sum_{x \in F(k_n)} \chi(s(x)) \right| \ll |k_n|,$$

so that

$$\frac{1}{|\hat{A}(k_n)|} \sum_{\chi \notin \mathcal{X}_w^c(\Lambda)(k_n)} |S(M, \chi)|^4 \ll \frac{|k_n|^4}{|k_n|^5} \to 0$$

as $n \to +\infty$, and from Proposition 8.4, the computation we have performed actually means that the fourth moment is equal to 3.

(2) We may use the beginning of Krämer's proof to deduce that $M*M^\vee$ contains an irreducible summand of dimension 78, which excludes the possibility that the fourth moment be equal to 2.

So under the above assumptions, we conclude that $M_4(G_M^{\text{ari}}) = 3$ and we can then apply Proposition 8.20 (as in the previous argument, we use Corollary 5.9 to deduce that the neutral component of $G_M^{\text{ari}}$ still acts irreducibly).

Now, going backwards, if we use Proposition 11.7, then we do know that the fourth moment of $G_M^{\text{ari}}$ is equal to 3, since the tannakian group is $E_6$. It follows that, at least in the first of the above two situations, the limit formula (11.2) must be true.

**REMARK 11.10.** In contrast with Theorem 11.1, Proposition 11.7 will not extend to compute the fourth moment for perverse sheaves of the form $s_*M$ for a more general simple perverse sheaf M on $F(X)$. One can expect that, in this case, the fourth moment should be equal to 2, but this seems difficult to prove.

We now state and prove the two lemmas we used above. The first one is certainly a standard fact in the study of the 27 lines.

**LEMMA 11.11.** Let S be a smooth cubic hypersurface in $\mathbb{P}^3$ over an algebraically closed field. Let $l_1$ and $l_2$ be two disjoint lines in S. Let $\Lambda$ be the set of the five lines in S intersecting both $l_1$ and $l_2$. Any Galois-automorphism of the twenty seven lines that fixes the lines in $\{l_1, l_2\} \cup \Lambda$ is the identity.

**PROOF.** The key point in this computation is the fact that no line on S is disjoint from all lines in $\Lambda$. More precisely, we use the classical description of S as a blow-up of $\mathbb{P}^2$ in six points which are in general position (see, e.g., [61, Prop. 3.2.3]), and the resulting partition of the 27 lines in subsets

$$\begin{array}{l} E_1, \dots, E_6 \\ L_{ij}, \quad 1 \le i < j \le 6 \\ L_1, \dots, L_6, \end{array}$$

with incidences described as follows:

$$\begin{array}{l} E_i \cap L_j \neq \emptyset \text{ if and only if } i \neq j, \\ E_i \cap L_{i,j} \neq \emptyset \text{ for any } j, \\ L_i \cap L_{i,j} \neq \emptyset \text{ for any } j, \\ L_{i,j} \cap L_{k,l} \neq \emptyset \text{ for } \{i,j\} \cap \{k,l\} = \emptyset, \end{array}$$

all other pairs of lines being disjoint (see, e.g., [61, Rem. 3.2.4, 3.3.1]).

We choose the blow-up, as we may, so that $l_1 = E_1$ and $l_2 = E_2$ (see [61, 3.3.2]). We then have $\Lambda = \{L_{12}, L_3, L_4, L_5, L_6\}$.
Let $\sigma$ be a Galois automorphism which fixes the seven given lines. Since $\sigma$ respects incidence relations, we see:

(1) For any $i$, we have $\sigma(E_i) = E_i$. Indeed, assume that $i = 3$ for simplicity, since all cases are similar. Then $E_3$ meets $L_4, L_5, L_6$, which implies that $\sigma(E_3)$ also intersects these three lines. But the only lines with this property are $E_1, E_2$ and $E_3$; since $\sigma$ fixes the first two of these, we have $\sigma(E_3) = E_3$.

(2) For any $i$, we have $\sigma(L_i) = L_i$. Indeed, assume that $i = 1$; from the previous point, the lines $L_{12}, E_2, \dots, E_6$, which all meet $L_1$, are fixed by $\sigma$, so that $\sigma(L_1)$ fixes all of them. We see that the only line with this property is $L_1$, so that $\sigma(L_1) = L_1$.

(3) For any $i < j$, we have $\sigma(L_{i,j}) = L_{i,j}$. We consider the example of $L_{1,3}$, the other cases being similar. The lines $E_1, E_3, L_1, L_3$ all meet $L_{1,3}$, and hence (by the first two points) also intersect $\sigma(L_{1,3})$. But this means that $\sigma(L_{1,3})$ must of one of the $L_{i,j}$, and the only one that has the desired property is $L_{1,3}$.

□

The second lemma concerns the “generic” simplicity of the intermediate jacobian. Explicit examples that show that this property is not always valid are given for instance by Debarre, Laface and Roulleau [25, Cor. 4.12]; for the Fermat threefold

$$x_0^3 + \dots + x_4^3 = 0$$

over $F_p$, with $p \ge 5$, the intermediate jacobian is isogenous to $E^5$, where $E$ is the Fermat curve $y_0^3 + y_1^3 + y_2^3 = 0$.

LEMMA 11.12. Let $k$ be a finite field of characteristic $p > 11$. Let $\mathcal{M}$ be the coarse moduli space of smooth projective cubic threefolds over $k$. For any integer $n \ge 1$, let $\mathcal{M}_s(k_n)$ be the subset of $X \in \mathcal{M}(k_n)$ such that the abelian variety $A(X)$ is simple over $k_n$.

There exists $\delta > 0$ such that the asymptotic formula

$$|\mathcal{M}_s(k_n)| = |\mathcal{M}(k_n)|(1 + O(|k_n|^{-\delta}))$$

holds for $n \ge 1$.

PROOF. Fix a prime $\ell$ invertible in $k$. Let $\mathcal{F}$ be the lisse $\ell$-adic sheaf on $\mathcal{M}$ parameterizing the cohomology group $H^1(A(X)_{\bar{k}}, \mathcal{Q}_{\ell})$. The geometric monodromy group of $\mathcal{F}$ is the symplectic group $\mathcal{S}_{P_{10}}$ by a result of Achter [1, Th. 4.3] (based on semicontinuity of monodromy and the extension to positive odd characteristic of a result of Collino [21] over $\mathcal{C}$, which states that the Zariski-closure of the image of $\mathcal{M}$ in the moduli space $\mathcal{A}_5$ of principally polarized abelian varieties of dimension 5 contains the locus $\mathcal{H}_5$ of jacobians of hyperelliptic curves of genus 5).

Using the method in [85, § 6, § 8], this implies that there exists $\delta > 0$ such that the set $\mathcal{M}_i(k_n)$ of threefold $X \in \mathcal{M}(k_n)$ for which the characteristic polynomial of Frobenius acting on $H^1(A(X)_{\bar{k}}, \mathcal{Q}_{\ell})$ is irreducible in $\mathcal{Q}[X]$ satisfies the asymptotic

$$|\mathcal{M}_i(k_n)| = |\mathcal{M}(k_n)|(1 + O(|k_n|^{-\delta}))$$

for $n \ge 1$, and one deduces the lemma since $\mathcal{M}_i(k_n) \subset \mathcal{M}_s(k_n)$.

□

REMARK 11.13. A qualitative form of the result, namely the equality

$$\lim_{n \to +\infty} \frac{|\mathcal{M}_s(k_n)|}{|\mathcal{M}(k_n)|} = 1,$$

can be proved, *mutatis mutandis*, for finite fields of all odd characteristic. It also possible to improve this estimate to obtain absolute simplicity.
“Much remains to be done”

We conclude this book with a selection of open problems (related to the results of this text)
and questions (concerning potential generalizations and more speculative possibilities).

12.1. Problems

(1) Prove a version of the vanishing theorem where the size of the exceptional sets is controlled by the complexity in all cases, and moreover where those sets have a clear algebraic or geometric structure (similar to that of tacs for tori and abelian varieties).

(2) Prove that any object is generically unramified for a general group, or find a counter-example to this statement.

(3) Establish functoriality properties relating tannakian groups of M on G (resp. N on H) with those of $f_*M$ (resp. of $f^*N$) when we have a morphism $f: G \to H$ of commutative algebraic groups.

(4) Study the situation in families over a base like Spec($\mathbb{Z}$).

(5) Find additional and robust tools to compute the tannakian group, or at least to determine some of its properties, which are applicable when Larsen’s Alternative is not. In particular, find analogues (if they exist) of the local monodromy techniques for the Fourier transform on $G_a$ (i.e. of the local Fourier transform functors of Laumon).

We note that recent work of Lawrence and Sawin [99] and Javanpeykar, Krämer, Lehn and Maculan [66] computes the tannakian group of many objects of the form $i_* \overline{Q}_\ell[d]$ on abelian varieties over $\mathbb{C}$, where $i: X \to \mathbb{A}$ is a closed immersion which is either a hypersurface (in the case of [99]) or has dimension $< (\dim(\mathbb{A}) - 1)/2$ (in the case of [66]). It should be possible to extend their results to the situation over finite fields, and it would be interesting to see if it also leads to more cases with other perverse sheaves with similar support conditions. Moreover, Ji [67] has also performed similar computations for $G_m^n$.

(6) Construct interesting concrete perverse sheaves where the tannakian group is an exceptional group. In this direction, we note that automorphic methods have been used by Heinloth, Ngô and Yun [60] to construct sheaves on $G_a$ with any of the exceptional groups as geometric monodromy groups, hence (taking inverse Fourier transforms of these) also to sheaves with these as tannakian groups. Moreover, Katz has shown the existence of examples involving $G_2$, for $G_a$ [69, Th. 11.1], $G_m$ [75, Ch. 26, 27] and on some elliptic curves [77, Th. 4.1]. In the case of $G_m$, his result is of a “statistic” nature: in a certain family of objects whose trace functions are related to hypergeometric sums evaluated at a fixed $a$, he shows that for “most” values of $a$ in $G_m$, the tannakian group is $G_2$. After the first draft of this book was written, Zurbuchen [129] improved this result by showing that (as expected by Katz) any (non-zero) value of $a$ has the desired property, provided the characteristic of the finite field is large enough.

(7) Find further applications!
## 12.2. Questions

Many of the following questions are rather speculative and much more open-ended that the problems above. They may not have any interesting answer, but we find them intriguing.

(1) For a given $G$, what are the tannakian groups that may arise?

This is motivated in part by the striking difference concerning finite groups between $G_a$ and $G_m$ or abelian varieties. In the former case, the solution of Abhyankar's Conjecture gives a characterization of which finite groups will appear, and a recent series of works of Katz, Rojas-Léon and Tiep has shown that there are many possibilities, even when one restricts attention to Fourier transforms of general hypergeometric sheaves (see for instance [80]) and $G_m$). On the other hand, we have already mentioned that Katz proved that finite cyclic groups are the only possible finite geometric tannakian groups on $G_m$, and Corollary 5.10 is a statement going in a similar direction for abelian varieties (although not yet as precise since it requires a priori that the arithmetic tannakian group be finite).

(2) For a given group $G$, if $M$ is a semisimple perverse sheaf associated to a semisimple lisse sheaf $\mathcal{F}$ on an open dense subset of $G$, what (if any) are the relations between the "ordinary" monodromy group $G$ of $\mathcal{F}$ and its tannakian groups?

In particular, suppose that $\mathcal{F}$ has finite monodromy group; what constraints does that impose on the tannakian group of $M$? We note that there is one "obvious" relation: the tannakian group acts irreducibly on its standard representation if and only if the lisse sheaf $\mathcal{F}$ is irreducible.

Since this last fact amounts to the discrete Plancherel formula, or equivalently to a relation between the second moments of both groups, a more specific question could be: are there non-trivial inequalities between the moments of the monodromy group of $\mathcal{F}$ and those of the tannakian group? For instance, does there exist a constant $c \ge 0$, independent of the size of the finite field $k$, such that

$$M_4(\mathbf{G}_M^{\text{ari}}) \le cM_4(\mathbf{G}), \quad \text{and (or)} \quad M_4(\mathbf{G}) \le cM_4(\mathbf{G}_M^{\text{ari}}) ?$$

One can get trivial bounds, similar to the bounds for the norm of the discrete Fourier transform on $G(k_n)$ when viewed as a map from $L^{2m}(G(k_n))$ to $L^{2m}(G(k_n))$ for $m > 1$ and $n$ varying, but this norm has been determined by Gilbert and Rzeszotnik [54, Th. 2.1] and depends on $n$. (On the other hand, a referee has pointed out that certain heuristic examples indicate that the question is most likely "No", and it will be interesting to confirm this rigorously.)

Of course, the meaning of "relation" between the ordinary and tannakian groups could encompass very different aspects, and it is of interest to note that the papers of Lawrence and Sawin and of Javanpeykar, Krämer, Lehn and Maculan use their computations of tannakian groups of certain objects on abelian varieties to study the ordinary monodromy groups of families of subvarieties of abelian varieties (see [99, Th. 5.6] and [66, Th. 4.10] for details).

(3) Can one construct a "natural" fiber functor $\omega$ on the tannakian category for $G$, similar to Deligne's fiber functor for $G_m$?

This would lead to a definition of Frobenius conjugacy classes for all characters (by considering for any $\chi$ the conjugacy class in $G_M^{\text{ari}}$ corresponding to the fiber functor defined by $M \mapsto \omega(M_\chi)$), and potentially provide useful extra information to help determine the tannakian group. This is not even clear in the case of $G_a$, but C. Ji [67, Th. 2.9] has constructed an analogue of Deligne's fiber functor for $G_m^n$ for all integers $n \ge 1$.
(4) Can one find an a priori characterization of the families $(f_n)_{n \ge 1}$, where $f_n: \hat{G}(k_n) \to \mathbf{C}$ is a function, that arise as the arithmetic Fourier transforms of trace functions of complexes, or of perverse sheaves, on $G$?

More generally, is there a natural “geometric” object, with appropriate notions of sheaves, etc, on the “space” of characters of G? A crucial test for such a geometric interpretation of the discrete Fourier transforms would be the definition of an inverse transform.

This geometric description exists when $G$ is unipotent, since the Serre dual $G^\vee$ is also a commutative algebraic group, and the Fourier transform is defined as a functor from $D_c^b(G)$ to $D_c^b(G^\vee)$, but such a strong “algebraicity” property does not hold for other commutative algebraic groups (see for instance [15, Example 1.8], or Remark 5.3).

There are however some hints in a more positive direction:

(a) Gabber and Loeser [50, Th. 3.4.7] have characterized perverse sheaves on tori in terms of the structure of their (coherent) Mellin transforms (which can also be defined for semiabelian varieties), and Loeser [103, Ch. 4] has defined a variant over finite fields taking the Frobenius automorphism into account.

It would be of considerable interest to understand better the (essential) image of these Mellin transforms, and to obtain a geometric form of Mellin inversion in this context.

(b) Considering the well-established analogy of $\ell$-adic sheaves with $\mathcal{D}$-modules (the basic setup of Katz's work [70]), it is well-understood in the complex setting that the Mellin transform of a $\mathcal{D}$-module is a difference equation (e.g., the Mellin transform $\Gamma(s)$ of the exponential satisfies the difference equation $\Gamma(s+1) = s\Gamma(s)$); see for instance the paper of Loeser and Sabbah [104].

(5) *Is there an analogous theory for non-commutative algebraic groups?*

For instance, let $G$ be a reductive group over a finite field $k$, such as $\mathbf{SL}_d(k)$. Deligne–Lusztig Theory parameterizes the irreducible representations of $G(k_n)$ (or some other more convenient basis of the $\ell$-adic representation ring) in terms of pairs $(\mathbf{T}, \theta)$ of a maximal torus of $G$ over $k$ and a character $\theta$ of $\mathbf{T}(k)$ (see for instance [19, Ch. 7]), and the corresponding series of representations have (essentially) constant dimension as $\theta$ varies, so that the character values in such series are suggestively sums of a fixed number of roots of unity. The theory of character sheaves of Lusztig gives a geometric form of this theory.

Are there equidistribution statements for the Fourier coefficients of suitably algebraic conjugacy-invariant functions on $G(k_n)$? In the case of characteristic functions of conjugacy classes, this might lead to interesting consequences concerning the error term in the Chebotarev Density Theorem for Galois extensions with Galois group of the form $G(k_n)$.

In the case of (possibly non-commutative) unipotent groups, the Serre dual still exists as a unipotent group; a theory of character sheaves, and of the Fourier transform has been studied by Lusztig and Boyarchenko–Drinfeld (see for instance the survey [15]).

(6) *Is there an analogue of automorphic duality for other groups than $G_a$?*

What we mean by this is the following: in the case of a simple middle extension sheaf $\mathcal{F}$ on $G_a$ over a finite field $k$ that is pure of weight zero and not geometrically isomorphic to an Artin–Schreier sheaf, there is (by the Langlands correspondence, due to Lafforgue in this generality) an automorphic representation $\pi$ on some general linear group over the adèle ring of $k(t)$ such that (among other properties) the L-functions of (twists of) $\pi$ coincide (up to normalization) with those of (twists of) the Fourier transform of $\mathcal{F}$. Automorphic methods and results are then available to study the Fourier transform of $\mathcal{F}$.
If $\mathcal{G}$ is a commutative algebraic group which is different from $\mathcal{G}_a$, are there objects of a similar nature as automorphic forms and representations that would “correspond” to the arithmetic Fourier transform of suitable perverse sheaves on $\mathcal{G}$? Such objects would presumably have some kind of $L$-function, which would coincide with the $\tilde{L}$-function that we have defined. In particular, is there such a theory for $\mathcal{G}_m$?
# Review of perverse sheaves

In this appendix, we summarize the basic definitions and facts about $\ell$-adic perverse sheaves. The fundamental reference for this material is the work of Beilinson, Bernstein, Deligne and Gabber [8]. Other useful summaries of perverse sheaves are provided by Katz in [71, §2.1 to 2.3] and in [73, § 1.1, 1.2, 1.5]. For basic material on trace functions in this context, see also [98, § 1.1].

## A.1. Complexes of $\ell$-adic sheaves

In this appendix, we work over a field $k$ of characteristic $p$ and fix a prime $\ell \neq p$. For $X$ a separated scheme of finite type over $k$, one can define the triangulated category of complexes of $\ell$-adic sheaves $\mathcal{D}_c^b(X) = \mathcal{D}_c^b(X, \overline{\mathbb{Q}}_\ell)$.

For $M \in \mathcal{D}_c^b(X)$, we write $\mathcal{H}^n(M)$ for the $n$-th cohomology sheaf of $M$, which is an $\ell$-adic constructible sheaf. We denote by $\tau^{<n}$ and $\tau^{\ge n}$ the truncation functors; for every object $M$, we have canonical maps $\tau^{<n}(M) \to M$ and $M \to \tau^{\ge n}(M)$. The composite functor $\tau^{\ge 0} \circ \tau^{<0}$ is canonically isomorphic to $M \mapsto \mathcal{H}^0(M)$.

For varying $X$, the categories $\mathcal{D}_c^b(X)$ satisfy all the properties of Grothendieck’s formalism of the six functors (see [28, 1.12] or [8, 2.2.18] in the case when $k$ is finite or algebraically closed, which suffices for this book).

More precisely, $\mathcal{D}_c^b(X)$ is endowed with two bifunctors

$$ (M, N) \mapsto \mathrm{RHom}(M, N), \quad (M, N) \mapsto M \otimes N $$

from $\mathcal{D}_c^b(X) \times \mathcal{D}_c^b(X)$ to $\mathcal{D}_c^b(X)$, and for a morphism $f: X \to Y$ of finite type, we have functors

$$ M \mapsto Rf_*M \quad M \mapsto Rf_*!M $$

from $\mathcal{D}_c^b(X)$ to $\mathcal{D}_c^b(Y)$, and functors

$$ M \mapsto f^*M \quad M \mapsto f^!M $$

from $\mathcal{D}_c^b(Y)$ to $\mathcal{D}_c^b(X)$. These functors satisfy the usual compatibilities and adjunctions.

The dualizing complex for $X$ is defined to be $s'\overline{\mathbb{Q}}_\ell$, where $s: X \to \mathrm{Spec}(k)$ is the structure morphism, and the Verdier dual of $M \in \mathcal{D}_c^b(X)$ is $\overline{\mathrm{D}}(M) = \mathrm{RHom}(M, s'\overline{\mathbb{Q}}_\ell)$. When $X$ is smooth of pure dimension $d$, there is a canonical isomorphism

$$ (A.1) \qquad s'\overline{\mathbb{Q}}_\ell \simeq \overline{\mathbb{Q}}_\ell(d)[2d]. $$

Let $s: X \to \mathrm{Spec}(k)$ be the structure morphism. For any object $M$ of $\mathcal{D}_c^b(X)$ and $i \in \mathbb{Z}$, the $i$-th cohomology group of $X$ with coefficients in $M$ (resp. cohomology group with compact support of $X$ with coefficients in $M$) is given by

$$ \mathrm{H}^i(X, M) = \mathcal{H}^i(s_*M), \quad \mathrm{H}_c^i(X, M) = \mathcal{H}^i(s!M), $$

where we identify $\ell$-adic sheaves on $\mathrm{Spec}(k)$ with $\overline{\mathbb{Q}}_\ell$-vector spaces.
When X is a smooth curve, two other important results (the Euler-Poincaré characteristic formula and Laumon's product formula for epsilon factors) which are used in this book will be reviewed in Appendix C.

## A.2. Perverse sheaves

**DEFINITION A.1.** A complex $M \in \mathbf{D}_c^b(X)$ is said to be *semiperverse* if its cohomology sheaves satisfy

$\dim \mathrm{supp}(\mathcal{H}^i(M)) \le -i$ for every $i \in \mathbb{Z}$, and M is said to be *perverse* if both $M$ and $D(M)$ are semiperverse (see [8, (4.0.1')]).

We denote by $\mathbf{Perv}(X)$ the full subcategory of perverse sheaves in $\mathbf{D}_c^b(X)$, by $\mathfrak{p}D^{\le 0}(X)$ the full subcategory of semiperverse sheaves, and by $\mathfrak{p}D^{\ge 0}(X)$ the full subcategory of objects M such that $D(M)$ is semiperverse.

**THEOREM A.2.** *The data of $\mathfrak{p}D^{\le 0}(X)$ and $\mathfrak{p}D^{\ge 0}(X)$ give rise to a t-structure on $\mathbf{D}_c^b(X)$. Its heart $\mathbf{Perv}(X) = \mathfrak{p}D^{\le 0}(X) \cap \mathfrak{p}D^{\ge 0}(X)$ is therefore an abelian category.*

**EXAMPLE A.3.** Suppose that X is smooth of pure dimension $d$, and let $\mathcal{F}$ be a lisse $\ell$-adic sheaf on X. Then the complex $\mathcal{F}[d]$ (i.e., the sheaf $\mathcal{F}$ put in degree $-d$) is a perverse sheaf.

Indeed, $\mathcal{F}[d]$ is clearly semiperverse and by (A.1), we see that $D(\mathcal{F}[d]) = \mathcal{F}^{\vee}(d)[d]$, where $\mathcal{F}^{\vee}$ is the dual lisse sheaf of $\mathcal{F}$, so the dual of $\mathcal{F}[d]$ is also semiperverse.

If M is a complex in $\mathbf{D}_c^b(X)$ with support $Y \subset X$, then there exists an open dense subset U of Y such that the restriction of M to U is lisse, i.e., all of the cohomology sheaves of $M|_U$ are lisse sheaves. We then say that M is *lisse on* U.

One also defines

$$ \mathfrak{p}D^{\le n}(X) = \mathfrak{p}D^{\le 0}(X)[n] \text{ and } \mathfrak{p}D^{\ge n}(X) = \mathfrak{p}D^{\ge 0}(X)[n]. $$

The inclusion functors $\mathfrak{p}D^{\le n}(X) \subset \mathbf{D}_c^b(X)$ and $\mathfrak{p}D^{\ge n}(X) \subset \mathbf{D}_c^b(X)$ admit right and left adjoints, called the *perverse truncation functors*, which are denoted

$$ \mathfrak{p}\tau^{\le n}: \mathbf{D}_c^b(X) \to \mathfrak{p}D^{\le n}(X) \text{ and } \mathfrak{p}\tau^{\ge n}: \mathbf{D}_c^b(X) \to \mathfrak{p}D^{\ge n}(X). $$

**DEFINITION A.4.** The $n$-th *perverse cohomology sheaf* of a complex $M \in \mathbf{D}_c^b(X)$ is the *perverse sheaf*

$$ \mathfrak{p}\mathcal{H}^n(M) = \tau^{\le 0}\tau^{\ge 0}(M[n]) \in \mathbf{Perv}(X). $$

Given a distinguished triangle $M \to N \to L \to \text{in } \mathbf{D}_c^b(X)$, we have a long exact sequence (A.2)
$$ \dots \to \mathfrak{p}\mathcal{H}^i(M) \to \mathfrak{p}\mathcal{H}^i(N) \to \mathfrak{p}\mathcal{H}^i(L) \to \mathfrak{p}\mathcal{H}^{i+1}(M) \to \dots $$
of perverse cohomology sheaves.

Let M be a *perverse sheaf* on X. From general principles, there are convergent spectral sequences (A.3)
$$ E_2^{p,q} = \mathbf{H}^p(X, \mathfrak{p}\mathcal{H}^q(M)) \implies \mathbf{H}^{p+q}(X, M), \quad E_2^{p,q} = \mathbf{H}_c^p(X, \mathfrak{p}\mathcal{H}^q(M)) \implies \mathbf{H}_c^{p+q}(X, M), $$
which are called the *perverse spectral sequences*.

We also have an equality

$$ \text{(A.4)} \qquad M = \sum_{j \in \mathbb{Z}} (-1)^j \mathfrak{p}\mathcal{H}^j(M) $$

in the Grothendieck group $K(X)$ (see, e.g., [98, (0.8)]).

As with the standard $t$-structure, *perverse cohomology sheaves* give a criterion to check whether a complex is semiperverse.
LEMMA A.5. A complex $M \in \mathrm{D}_c^b(X)$ is semiperverse if and only if $\mathfrak{P}\mathcal{H}^i(M) = 0$ for all integers $i \ge 1$.

See [8, Prop. 1.3.7] for the proof.

DEFINITION A.6. An exact<fnref n="1" /> functor from $\mathrm{D}_c^b(X)$ to $\mathrm{D}_c^b(Y)$ is said to be left $t$-exact (resp. right $t$-exact) if it sends $\mathfrak{P}D^{\ge 0}(X)$ to $\mathfrak{P}D^{\ge 0}(Y)$ (resp. $\mathfrak{P}D^{\le 0}(X)$ to $\mathfrak{P}D^{\le 0}(Y)$). It is said to be $t$-exact if it is both left and right $t$-exact.

The following important result is a direct consequence of Artin’s cohomological vanishing theorem (see [8, Th. 4.1.1]).

THEOREM A.7. Let $f: X \to Y$ be an affine morphism, then $Rf_*$ is right $t$-exact and $Rf_!$ is left $t$-exact.

Since a closed immersion $i$ is affine and proper (so that $Ri_* = i_!$), we obtain as corollary:

COROLLARY A.8. If $i$ is a closed immersion, then $i_*$ is $t$-exact.

More generally (see [8, Cor. 4.1.3]), the functors $f_!$ and $f_*$ are $t$-exact if $f$ is quasi-finite and affine.

A central result is the construction of the intermediate extension, see [8, Cor. 1.4.25].

PROPOSITION A.9. Let $j: U \to X$ be a locally closed immersion. Let $M$ be a perverse sheaf on $U$. Then there exists a unique perverse sheaf $j_*|_*(M)$ on $X$, called the middle extension or intermediate extension of $M$, such that

- There exists an isomorphism $j^*j_*(M) \simeq M$.

- The perverse sheaf $j_*|_*(M)$ is supported on the closure $\bar{U}$ of $U$.

- The perverse sheaf $j_*|_*(M)$ has no subobject and no quotient supported on $\bar{U} - U$.

The most important example of this construction is when $j: U \to X$ is a dense open immersion, with $U$ smooth of pure dimension $d$, and $M = \mathcal{F}[d]$ for a lisse sheaf $\mathcal{F}$. Note that the uniqueness implies that $D(j_*\mathcal{F}[d]) = j_*\mathcal{F}^\vee(d)[d]$. When $\mathcal{F} = \bar{\mathbb{Q}}_\ell$ is the constant sheaf on $U$, then $j_*\bar{\mathbb{Q}}_\ell[d]$ is called the intersection complex of $X$.

EXAMPLE A.10. Let $X$ be a curve, $U$ a dense open subset of $X$ contained in the smooth locus of $X$ and $\mathcal{F}$ a lisse sheaf on $U$. Then $j_*\mathcal{F}[1] = R^0j_*\mathcal{F}[1]$, where $j: U \to X$ is the open immersion.

The fundamental result concerning the category of perverse sheaves is the following theorem [8, Th. 4.3.1].

THEOREM A.11. The category $\mathbf{Perv}(X)$ is artinian and noetherian, i.e., all objects are of finite length. Its simple objects are of the form $j_*\mathcal{F}[d]$ where $j: U \to X$ is a locally closed immersion with $U$ smooth irreducible of dimension $d$ and $\mathcal{F}$ is an irreducible lisse sheaf on $U$.

EXAMPLE A.12. Let $X$ be a smooth and geometrically connected curve. Following Katz [70, § 7.3], a constructible sheaf $\mathcal{F}$ on $X$ is called a middle extension sheaf if, for any dense open set $U$ of $X$ such that $\mathcal{F}$ is lisse on $U$, with open immersion $j: U \to X$, the canonical morphism $\mathcal{F} \to j_*j^*\mathcal{F}$ is an isomorphism.

There is a one-to-one correspondence between irreducible middle extension sheaves and simple perverse sheaves on $X$ with support equal to $X$; for a middle extension sheaf $\mathcal{F}$, the corresponding

<footnote n="1">¹ Namely, a functor that commutes with shift and preserves distinguished triangles.</footnote>