# Abelian surfaces over $F_q(t)$ with large Tate-Shavarevich groups

Martin Azon

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Martin Azon. Abelian surfaces over $F_q(t)$ with large Tate-Shavarevich groups. 2025. hal-04910117

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# Abelian surfaces over $\mathbb{F}_q(t)$ with large Tate-Shavarevich groups

Martin AZON

**Abstract** — We produce an explicit sequence $(S_a)_{a \ge 1}$ of abelian surfaces over the rational function field $\mathbb{F}_q(t)$ whose Tate-Shafarevich groups are finite and large. More precisely, we establish the estimate $|\mathrm{III}(S_a)| = H(S_a)^{1+o(1)}$ as $a \to \infty$, where $H(S_a)$ denotes the exponential height of $S_a$. Our method is to prove that each $S_a$ satisfies the BSD conjecture, analyse the geometry and arithmetic of its Néron model and give an explicit expression for its $L$-function in terms of Gauss and Kloosterman sums. By studying the relative distribution of the angles associated to these character sums, we estimate the size of the central value of $L(S_a, T)$, hence the order of $\mathrm{III}(S_a)$.

**Keywords:** Abelian surfaces over function fields, Tate-Shafarevich groups, Explicit computation of $L$-functions, BSD conjecture, Gauss and Kloosterman sums.

**2020 Math. Subj. Classification:** 11G10, 11G40, 11G20, 14G10, 11L05.

## 1 Introduction

Consider a finite field $\mathbb{F}_q$ of characteristic $p \ge 7$ and let $K := \mathbb{F}_q(t)$ be the rational function field over $\mathbb{F}_q$. Let $A$ be an abelian variety defined over $K$, the Tate-Shafarevich group $\mathrm{III}(A)$ is the group of isomorphism classes of principal homogeneous spaces of $A$ over $K$ that become trivial in all the local completions of $K$ (see [Hin07] for more details on this definition). It is an arithmetic object that measures the extent to which the local-global principle fails for $A$. Many questions regarding the Tate-Shafarevich group are still open: its finiteness, while proven in various cases, remains conjectural in general. A natural question to ask is: assuming finiteness of $\mathrm{III}(A)$, what can be said about its size? Can we compare it to other arithmetic invariants of $A$? In [GS95] and [Raj97], Goldfeld and Szpiro and, independently, Rajan, addressed the case of elliptic curves and gave upper bounds on the order of the Tate-Shafarevich group in terms of the degree of the conductor and the exponential height. More recently, in [HP16], Hindry and Pacheco relate the order of $\mathrm{III}(A)$ to the exponential height $H(A)$, which measures the arithmetic complexity of $A$.

**Theorem 1.1** (Hindry-Pacheco [HP16]). Let $d \ge 1$ be an integer and $\varepsilon > 0$ be a real number. There exists a real constant $c_{\varepsilon,d} > 0$ such that for any abelian variety $A/K$ of dimension $d$ with finite Tate-Shafarevich group, the following estimate holds:

$$|\mathrm{III}(A)| \le c_{\varepsilon,d} H(A)^{1+\varepsilon}.$$

Kato and Trihan proved in [KT03] that the finiteness of $\mathrm{III}(A)$ implies the Birch and Swinnerton-Dyer conjecture for $A$ (hereafter abbreviated as BSD). Under the hypothesis of the theorem, Hindry and Pacheco use the BSD formula to relate $\mathrm{III}(A)$ to the special value $L^*(A)$ of the $L$-function of $A$. By bounding various arithmetic invariants of $A$ and employing analytic techniques, they manage to estimate the size of $L^*(A)$, hence deriving their theorem.

The main result of this article is that in dimension 2, this bound can essentially not be improved, meaning that the exponent 1 cannot be replaced by a smaller quantity:

**Theorem 1.2** (= 7.1). There exists a sequence $(S_a)_{a \ge 1}$ of non-isotrivial abelian surfaces over $K$, all having finite Tate-Shafarevich groups, satisfying the following property: for any $\varepsilon > 0$, there exist positive constants $c'_\varepsilon, c''_\varepsilon > 0$ such that for all $a \ge 1$,

$$c'_\varepsilon H(S_a)^{1-\varepsilon} \le |\mathrm{III}(S_a)| \le c''_\varepsilon H(S_a)^{1+\varepsilon}.$$
Our proof of this theorem is constructive and effective: we explicitly construct such a sequence of abelian surfaces. We build $S_a$ as the Jacobian of a hyperelliptic genus 2 curve in a tower of Artin-Schreier extensions. Following a construction by Pries and Ulmer (see [PU16]), we define, for any $a \ge 1$, $C_a/\mathbb{K}$ to be the smooth projective curve described by the hyperelliptic equation

$$C_a : y^2 = (x^3 + \wp_a(t) - 2)(x^3 + \wp_a(t) + 2), \quad \text{where } \wp_a(t) := t^{q^a} - t \in \mathbb{F}_q(t).$$

We let $S_a := \text{Jac}(C_a)$ be the Jacobian of $C_a$. One of the main goals of this article is to describe several arithmetic invariants of $S_a$ and their variation as $a$ grows.

Recently, there has been a growing interest in the explicit arithmetic of Jacobians of hyperelliptic curves. In [DDMM23], the authors introduce the notion of cluster picture, a combinatorial object that helps understand the local arithmetic of a hyperelliptic curve and its Jacobian. In Section 2, we apply this theory in our context to describe various invariants of $S_a$. We explicitly obtain:

**Proposition 1.3** (= 2.13 + 2.14 + 2.15). For any $a \ge 1$:

* The local exponents of the conductor are $n_v = 2$ if $v | \wp_a(t)^2 - 4$, $n_\infty = 4$ and $n_v = 0$ for all other places $v$.

* The Tamagawa numbers of $S_a$ satisfy $c_v(S_a) = 1$ if $v \ne \infty$ and $c_\infty(S_a) \in \{1, 3, 9\}$.

* The differential height of $S_a$ is given by $h(S_a) = q^a + 1$.

Since $S_a$ is a Jacobian, its arithmetic is intimately related to the geometry of the underlying curve $C_a$, which is well-described via its minimal regular model $C_a \to \mathbb{P}_{\mathbb{F}_q}^1$. The family $(S_a)_{a \ge 1}$ constitutes an Artin-Schreier family of abelian varieties (such families have been extensively studied in [PU16]). In our context, one can analyse the geometry of $C_a$ and prove that, as a surface over $\mathbb{F}_q$, it is dominated by a product of two curves. Building upon the work of Shioda and Tate, we deduce from the general framework of [PU16]:

**Theorem 1.4** (= 4.4). For any $a \ge 1$, the abelian surface $S_a$ satisfies the BSD conjecture:

* The algebraic and analytic ranks of $S_a$ coincide, i.e. $\text{rk}(S_a(\mathbb{K})) = \text{ord}_{T=q^{-1}} L(S_a, T)$.

* The group $\text{III}(S_a)$ is finite.

* The BSD formula holds:

$$L^*(S_a) = \frac{|\text{III}(S_a)| \text{Reg}(S_a) q^2 \prod_v c_v(S_a)}{|S_a(\mathbb{K})_{\text{tor}}|^2 H(S_a)},$$

where $\text{Reg}(S_a)$ is the Néron-Tate regulator of $S_a$.

The BSD conjecture paves the way for using analytic techniques to compute the rank of $S_a(\mathbb{K})$ and better understand other arithmetic data of $S_a$, such as $|\text{III}(S_a)|$ and $\text{Reg}(S_a)$. The L-function $L(S_a, T)$ has a cohomological interpretation: it is the characteristic polynomial of the Frobenius on certain étale cohomology groups. Using the explicit construction of our hyperelliptic curves $C_a$, we compute in Section 4 the L-function of $S_a$ as follows. By studying the reduction types of the Néron model of $S_a$, we first establish a connection between $L(S_a, T)$ and the zeta function $Z(C_a, T)$ of the surface $C_a/\mathbb{F}_q$. As we stated above, there exist two smooth projective curves $X_a, Y_a$ defined over $\mathbb{F}_q$ such that $C_a$ is dominated by $X_a \times Y_a$. We study in deeper detail the geometry of $X_a \times Y_a \dashrightarrow C_a$ and describe $Z(C_a, T)$ in terms of the cohomology groups of $X_a$ and $Y_a$. This way we obtain a completely explicit expression for $L(S_a, T)$ which is computable in practice.

**Theorem 1.5** (= 4.5). For any $a \ge 1$, the L-function of $S_a$ is given by the formula

$$L(S_a, T) = \prod_{o \in \mathcal{O}_q(a)} \left( 1 - \gamma(o) \text{Kl}(o) T^{|o|} + \gamma(o)^2 q^{|o|} T^{2^{|o|}} \right) \in \mathbb{Z}[T],$$

where $\mathcal{O}_q(a)$ is a finite set of orbits, and for each $o \in \mathcal{O}_q(a)$, $\gamma(o)$ is a Gauss sums and $\text{Kl}(o)$ is a Kloosterman sum. Precise definitions are given in section 3.2.
Various authors have previously computed L-functions of abelian varieties over function fields. For example, in [GdW21] and [BHP+20], the authors use the Grothendieck–Lefschetz trace formula to compute the local factors of the L-function through point-counting arguments and direct manipulation of character sums, and then recover a global formula. The method employed here is inspired by a theorem of Shioda ([Shi92]), who treated the case of elliptic curves, and [GU20, AGTT23] where the authors provide cohomological computations of L-functions.

Theorem 1.5 describes $L(S_a, T)$ in terms of Gauss and Kloosterman sums, which are rather well-understood: in Section 3.2 we recall several properties of these sums (their size, p-adic valuation, ...) that are classical (see [LN97, Coh07]). Moreover, it is possible to associate angles to Gauss and Kloosterman sums: for any $o \in O_q(a)$, there are $\epsilon_o \in [0, 2\pi)$ and $\theta_o \in [0, \pi]$ such that $\gamma(o) = q^{\lceil o \rceil/2} e^{i\epsilon_o}$ and $\text{Kl}(o) = q^{\lceil o \rceil/2} \cos(\theta_o)$ (see Sections 4.4 and 5.2 for details). Using the explicit expression for $L(S_a, T)$ and manipulating trigonometric formulae we establish an explicit expression for the special value $L^*(S_a)$ in terms of Gauss and Kloosterman angles. Using the properties of the character sums mentioned above, we deduce the special value and the rank of the Mordell-Weil group.

**Proposition 1.6** (= 4.11 + 4.12). For any $a \ge 1$, we have

$$L(S_a, q^{-1}) = \prod_{o \in O_q(a)} 4 \left| \sin\left(\frac{\epsilon_o + \theta_o}{2}\right) \sin\left(\frac{\epsilon_o - \theta_o}{2}\right) \right| \neq 0.$$

We deduce that, $L^*(S_a) = L(S_a, q^{-1})$, hence $\text{rk}(S_a(\mathbb{K})) = 0$ and $\text{Reg}(S_a) = 1$.

We then focus on the archimedean size of the special value $L^*(S_a)$. The regulator of $S_a$ being trivial, we use the BSD formula to estimate the order of $\text{III}(S_a)$. This point of view has revealed to be very fruitful: the interested reader is referred to [HP16, Gri19, GdW21] for some examples where the authors approach the arithmetic of abelian varieties via an analytic study of their L-function.

Estimating the size of the special value involves understanding the distribution of Gauss and Kloosterman angles. First, we use Katz's work [Kat88] to establish an equidistribution result concerning Kloosterman angles. More precisely, if one lets $\nu_a$ be the average of Dirac measures supported at Kloosterman angles, then the sequence $(\nu_a)_{a \ge 1}$ converges weak-* to the Sato-Tate measure $\nu_{\text{ST}} := \frac{2}{\pi} \sin^2 \theta \, d\theta$. Moreover, following [Gri19], this equidistribution result is made effective and we give an upper bound on the error term. After that we study the minimal distance between Gauss and Kloosterman angles, topic that was partly addressed in [Gri19, GdW21]. In Section 5.3 we generalise the results of loc. cit. and establish a Diophantine statement asserting that Kloosterman angles “avoid” Gauss angles and cannot be arbitrarily close to them.

With these results at hand, employing analytic techniques we estimate the size of the special value in terms of the exponential height.

**Theorem 1.7** (= 6.1). For any $a \ge 1$, we have

$$\frac{\log L^*(S_a)}{\log H(S_a)} = O\left(\frac{1}{a}\right).$$

Theorem 1.2 follows from Theorem 1.7 using the BSD formula and the estimates on the arithmetic invariants of $S_a$ obtained in Proposition 1.4. We conclude by deriving an analogue of the Brauer-Siegel theorem for the abelian surfaces $S_a$. Our family provides a new example of non-isotrivial higher-dimensional abelian varieties whose Brauer-Siegel ratio $\mathfrak{B}(S_a) := \log(|\text{III}(S_a)|/\text{Reg}(S_a))/\log H(S_a)$ tends to 1 as $a \to \infty$. This provides some more evidence towards the phenomenon (already observed in [HP16, Ulm19] and related work) that the Brauer-Siegel ratio of an abelian variety over $\mathbb{K}$ should “often” be near 1 when the exponential height is large.

## 2 The arithmetic of the abelian surfaces $S_a$

In this section we introduce the curve $C_a/\mathbb{K}$, its Jacobian $S_a/\mathbb{K}$ and we describe different models of those. In particular, we use the minimal regular model of $C_a$ to compute the conductor of $S_a$, and the minimal regular simple normal crossings model of $C_a$ to describe the differential height $h(S_a)$ and the Tamagawa numbers $c_v(S_a)$.

We fix some notation first: let $p \ge 7$ be a prime number, $\mathbb{F}_q$ be a finite field of characteristic $p$ and let $\mathbb{K} := \mathbb{F}_q(t)$ be the function field of $\mathbb{P}_{\mathbb{F}_q}^1$. We write $\mathbb{K}^{\text{sep}}$ for a separable closure of $\mathbb{K}$ and $M_{\mathbb{K}}$
for the set of places of $\mathbb{K}$. Recall that $M_{\mathbb{K}}$ is in bijection with the set of closed points of $\mathbb{P}_{\mathbb{F}_q}^1$. For any $v \in M_{\mathbb{K}}$, we denote the completion of $\mathbb{K}$ at $v$ by $\mathbb{K}_v$, its ring of integers by $\mathcal{O}_v$, and a uniformizer of $\mathcal{O}_v$ by $\pi_v$. Let $\mathbb{F}_v$ be the residue field at $v$, and $d_v := [\mathbb{F}_v : \mathbb{F}_q]$ be the degree of $v$. Denote by $\mathbb{K}_v^{\text{unr}}$ the maximal unramified extension of $\mathbb{K}_v$ and $\mathcal{O}_v^{\text{unr}}$ for its ring of integers: its residue field is an algebraic closure of $\mathbb{F}_v$. Moreover, we let $\infty$ denote the place at infinity of $\mathbb{K}$, corresponding to the valuation defined on $\mathbb{F}_q[t]$ by $P \mapsto -\deg P$, where $\deg P$ denotes the degree of the polynomial $P$. Equivalently, $\infty$ corresponds to the $t^{-1}$-adic valuation on $\mathbb{K}$.

## 2.1 Constructing the curves $C_a$ and their Jacobians

We construct the abelian surfaces $S_a$ as the Jacobians of certain genus 2 curves, in such a way that they satisfy the BSD conjecture (see [HP16] or Section 4.1 for a precise statement of the BSD conjecture). Building upon the work of Shioda and Tate, the following result concerning Jacobians of curves over $\mathbb{K}$ holds:

**Theorem 2.1.** Consider a smooth, proper, geometrically irreducible surface $\mathcal{Z}$ defined over $\mathbb{F}_q$. Assume that $\mathcal{Z}$ is equipped with a generically smooth morphism $\mathcal{Z} \to \mathbb{P}_{\mathbb{F}_q}^1$ and denote by $C/\mathbb{K}$ its generic fiber. If the surface $\mathcal{Z}/\mathbb{F}_q$ is dominated by a product of curves, then the Jacobian of $C/\mathbb{K}$ satisfies the BSD conjecture.

**Skeeth of the proof.** This follows from the equivalence of the BSD conjecture with Tate's conjecture over finite fields, and the fact that products of curves satisfy Tate's conjecture. We refer the reader to [Ulm14, Theorems 6.3.1, 8.1.2 and Lemma 8.2.1]. $\square$

In Theorem 4.4 we will state a precise result in our context and give more details about its proof. The goal of this section is to exploit this phenomenon to construct the curves $C_a$ and their Jacobians $S_a$ presented in the introduction. The discussion below is a particular case of a more general framework, described by Pries and Ulmer in [PU16]. In that article, the authors give an explicit recipe to construct families of abelian varieties in Artin–Schreier extensions, such that all the terms in the family satisfy the BSD conjecture.

Fix an integer $a \ge 1$, consider the morphism $\wp_a : \mathbb{P}_{\mathbb{F}_q}^1 \to \mathbb{P}_{\mathbb{F}_q}^1$ defined on $\mathbb{A}_{\mathbb{F}_q}^1$ by $t \mapsto t^{q^a} - t$, which we call the Artin–Schreier morphism. Let $X_a$ and $Y_a$ be the smooth projective curves over $\mathbb{F}_q$ described on affine subsets of $\mathbb{P}_{\mathbb{F}_q}^2$ by the equations

$$X_a : u^3 = \wp_a(t_1) \quad \text{and} \quad Y_a : v + \frac{1}{v} = \wp_a(t_2). \qquad (2.1)$$

These are instances of so-called Artin–Schreier curves: we will study in detail their geometry in Section 3.4. Let $\mathcal{Z}_a$ be the projective irreducible surface over $\mathbb{F}_q$ defined on an affine open subset by

$$\mathcal{Z}_a : v + \frac{1}{v} - u^3 = \wp_a(t). \qquad (2.2)$$

The morphism $\mathcal{Z}_a \to \mathbb{P}_{\mathbb{F}_q}^1$ extending $(u, v, t) \mapsto t$ turns $\mathcal{Z}_a$ into a fibered surface over $\mathbb{P}_{\mathbb{F}_q}^1$. Using the Jacobian criterion one checks that the generic fiber of $\mathcal{Z}_a \to \mathbb{P}_{\mathbb{F}_q}^1$ is a smooth curve over $\mathbb{K}$, which we denote by $C_a$. The arithmetic surface $\mathcal{Z}_a$ is a model of the curve $C_a$ over $\mathbb{K}$ and equation (2.2), viewed as an element of $\mathbb{K}(u, v)$, defines a dense open affine subset of $C_a/\mathbb{K}$. The surface $\mathcal{Z}_a$ is not regular, but it is birational to a regular proper irreducible surface fibered over $\mathbb{P}_{\mathbb{F}_q}^1$, called the minimal regular model of $C_a$ (we will construct this and other models of $C_a$ in Section 2.2).

The rational map

$$\vartheta_0 : \begin{cases} X_a \times Y_a & \dashrightarrow \mathcal{Z}_a \\ (u, t_1, v, t_2) & \mapsto (u, v, t_2 - t_1), \end{cases}$$

is well-defined because of the relation $\wp_a(t_2 - t_1) = \wp_a(t_2) - \wp_a(t_1)$. We will show in Proposition 3.8 that the map $\vartheta_0$ is dominant. Since $\mathcal{Z}_a$ is birational to the minimal regular model of $C_a$, Theorem 2.1 states that Jac($C_a$) satisfies the BSD conjecture (see Theorem 4.4 for a precise statement).

In the terminology of [PU16], the curves $C_a$ correspond to the type $(3, 1+1)$, our choice of rational functions being $f(u) = u^3$ (which has a pole of order 3 at infinity) and $g(v) = v + v^{-1}$ (which has two poles of order 1). Using Proposition 3.1.5 of the mentioned article, we see that $C_a$ has genus 2,
and is thus hyperelliptic. The hyperelliptic involution, read off from equation (2.2), sends $(u, v)$ to $(u, 1/v)$ and a hyperelliptic equation of $C_a$ is:

$$C_a : y^2 = (x^3 + \wp_a(t) - 2)(x^3 + \wp_a(t) + 2). \qquad (2.3)$$

The change of variables is explicitly given by $(u, v) \mapsto (x, y) = (u, 2v - u^3 - \wp_a(t))$.

**Definition 2.2.** For any $a \ge 1$, let $C_a$ be the genus 2 hyperelliptic curve defined over $\mathbb{K}$ by equation (2.3), and define $S_a := \text{Jac}(C_a)$ to be its Jacobian, which is an abelian surface over $\mathbb{K}$.

**Remark 2.3.** For any $a \ge 1$, let $K_a := \mathbb{K}[z]/(\wp_a(z) - t)$, then $K_a/\mathbb{K}$ is a field extension of degree $q^a$, which corresponds to the covering $\wp_a : \mathbb{P}_{\mathbb{F}_q}^1 \to \mathbb{P}_{\mathbb{F}_q}^1$. We say that $K_a/\mathbb{K}$ is an Artin–Schreier extension. One can also view $C_a$ as the base change from $\mathbb{K}$ to $K_a$ of the hyperelliptic curve $C_0/\mathbb{K}$ defined by $y^2 = (x^3 + t - 2)(x^3 + t + 2)$. In the same way $S_a$ is isomorphic to the base change $\text{Jac}(C_0) \times_{\mathbb{K}} K_a$. We say that $(S_a)_{a \ge 1}$ is an Artin–Schreier family of abelian surfaces.

## 2.2 Models of $C_a$ and $S_a$

In this section we discuss models of $C_a$ and $S_a$ over $\mathbb{P}_{\mathbb{F}_q}^1$. Since each $C_a$ is a hyperelliptic curve, there are many explicit tools that help us understand the geometry and the arithmetic of its Jacobian $S_a$. We refer to [Liu02, §9 & 10] and to [BLR90, §9] for basics about models of curves and Néron models of Jacobians respectively. From now on, we fix the integer $a \ge 1$ and, unless stated otherwise, we omit, the subscript $a$ and write, for example, $C$ instead of $C_a$.

First, we recall from [Liu23] the definition of a Weierstrass model associated to a hyperelliptic equation of a hyperelliptic curve over a PID.

**Definition 2.4.** Let $A$ be a principal ideal domain, $K := \text{Frac}(A)$ and $C/K$ be a hyperelliptic curve of genus $g$ defined by an integral Weierstrass equation

$$y^2 = F(x), \quad F(x) \in A[x]. \qquad (2.4)$$

We define the Weierstrass model of $C$ over $\text{Spec}(A)$ associated to equation (2.4) to be the glueing of the affine schemes

$$\text{Spec}(A[x, y] / (y^2 - F(x))) \quad \text{and} \quad \text{Spec}(A[z, w] / (w^2 - F(z^{-1})))$$

along the identification $(z, w) = (1/x, y/x^{g+1})$. We define the discriminant of such a model as $\Delta := 2^{4g} \text{disc}(F)$. We say that the hyperelliptic equation (2.4) (or the associated Weierstrass model) is minimal at a place $\mathfrak{P}$ of $A$ if the valuation ord$_{\mathfrak{P}}(\Delta)$ is minimal among all other integral hyperelliptic equations for $C$.

We now construct a Weierstrass model of $C$ over $\mathbb{P}_{\mathbb{F}_q}^1$. Let $\mathcal{U}_0 := \text{Spec}(\mathbb{F}_q[t])$ and $\mathcal{U}_\infty := \text{Spec}(\mathbb{F}_q[s])$ be two affine lines covering $\mathbb{P}_{\mathbb{F}_q}^1$, which we glue along the morphism $t \mapsto s = t^{-1}$. Over $\mathcal{U}_\infty$, replacing $t = s^{-1}$ in (2.3) gives a non-integral equation of $C$. In order to construct an integral model of $C$ over $\mathcal{U}_\infty$, let $b \in \{1, 2\}$ be such that $3|q^a + b$. Setting the change of variables $(x_1, y_1, s) := (t^{-(q^a+b)/3}x, -(q^a+b)y, s^{-1})$, equation (2.3) becomes

$$y_1^2 = (x_1^3 + s^b - s^{q^a+b} - 2s^{q^a+b}) (x_1^3 + s^b - s^{q^a+b} + 2s^{q^a+b}). \qquad (2.5)$$

This is an integral equation of $C$ over $\mathcal{O}_\infty$. As we will see in Lemma 2.6, when $b = 1$, i.e. $q^a \equiv 2 \mod 3$, this equation is minimal. However, when $b = 2$, i.e. $q^a \equiv 1 \mod 3$, a further change of variables is required to obtain a minimal equation. Following Liu's minimization algorithm (see [Liu23]), we let $(x_2, y_2) := (s^{-1}x_1, s^{-2}y_1)$. Then (2.5) becomes

$$y_2^2 = (sx_2^3 + 1 - s^{q^a-1} - 2s^{q^a}) (sx_2^3 + 1 - s^{q^a-1} + 2s^{q^a}). \qquad (2.6)$$

**Definition 2.5.** Fix an integer $a \ge 1$. We let $W_0 \to \mathcal{U}_0$ be the Weierstrass model associated to the hyperelliptic equation (2.3). Consider $b \in \{1, 2\}$ be such that $3|q^a + b$ as above. We let $W_\infty \to \mathcal{U}_\infty$
be the Weierstrass model associated to the hyperelliptic equation (2.5) if $b=1$, and to (2.6) if $b=2$. We define the Weierstrass model of $C$ over $\mathbb{P}_{\mathbb{F}_q}^1$, denoted by $W$, as the glueing of $W_0$ and $W_\infty$ along the morphism

$$ (x, y, t) \mapsto \begin{cases} (x_1, y_1, s) = (t^{-(q^a+1)/3}x, t^{-(q^a+1)y}, t^{-1}) & \text{if } b=1 \text{ i.e. } q^a \equiv 2 \mod 3, \\ (x_2, y_2, s) = (t^{-(q^a-1)/3}x, t^{-q^a}y, t^{-1}) & \text{if } b=2 \text{ i.e. } q^a \equiv 1 \mod 3. \end{cases} $$

The scheme $W$ is an irreducible, proper, normal surface over $\mathbb{F}_q$. The projections $W_0 \to U_0$ and $W_\infty \to U_\infty$ glue into a morphism $W \to \mathbb{P}_{\mathbb{F}_q}^1$, whose generic fiber is the curve $C$. The discriminant of $W$ is an effective divisor on $\mathbb{P}_{\mathbb{F}_q}^1$, which we write in multiplicative notation. Recall that we identify finite places of $\mathbb{K}$ with monic irreducible polynomials.

**Lemma 2.6.** The model $W \to \mathbb{P}_{\mathbb{F}_q}^1$ is minimal at all places of $\mathbb{K}$. The discriminant $\Delta(W)$ is given by

$$ \Delta(W) = \prod_{v | \wp_a(t)^2 - 4} (v)^2 (\infty)^{6q^a+10}. $$

*Proof.* One checks that the discriminant of a sextic polynomial of the form $\alpha x^6 + \beta x^3 + \gamma$ is given by $\mathrm{disc}(\alpha x^6 + \beta x^3 + \gamma) = -3^6(\alpha\gamma)^2(\beta^2 - 4\alpha\gamma)^3$. Therefore, the discriminant of $W_0 \to U_0$ is obtained in terms of the discriminant of the defining polynomial, which we compute now:

$$ \mathrm{disc}((x^3 + \wp_a(t) - 2)(x^3 + \wp_a(t) + 2)) = 2^{12} 3^6 (\wp_a(t)^2 - 4)^2. $$

But $\wp_a(t)^2 - 4$ is a square-free polynomial in $t$, so we deduce that $\mathrm{ord}_v(\Delta(W_0)) = \mathrm{ord}_v(\Delta(W)) = 2$ at any $v | \wp_a(t)^2 - 4$. Now Remark 7.3 in [Liu96] implies that $W_0$ (and thus $W$) is minimal at any place $v$ of $U_0$. Applying the minimization algorithm given in [Liu23, §5.5] one checks that (2.5) (resp. (2.6)) is a minimal equation at $\infty$ when $q^a \equiv 2 \mod 3$ (resp. $q^a \equiv 1 \mod 3$). Again, a straightforward computation shows that the discriminant of both (2.5) and (2.6) is $6q^a + 10$, hence the result. $\square$

**Remark 2.7.** When $b=2$, i.e. $q^a \equiv 1 \mod 3$, one can modify the equation defining $W$ (2.6) to obtain a monic polynomial on the right-hand side. Indeed, letting $\wp := (1 - s^{q^a-1})^2 - 4s^{2q^a}$, $\tilde{x} := 1/x_2$ and $\tilde{y} := y_2/(x_2\sqrt{\wp})$ (notice that $\wp$ is a square in $\mathcal{O}_\infty$ by Hensel's lemma), equation (2.6) becomes

$$ \tilde{y}^2 = \left( \tilde{x}^3 + \frac{s(1 - s^{q^a-1}) - 2s^{q^a+1}}{\wp} \right) \left( \tilde{x}^3 + \frac{s(1 - s^{q^a-1}) + 2s^{q^a+1}}{\wp} \right). $$

Thus, independently of $q^a \mod 3$, the defining equation of $W$ at infinity is of the form

$$ y^2 = (x^3 + su - s^{q^a+1}v)(x^3 + su + s^{q^a+1}v) \quad \text{for some } u, v \in \mathcal{O}_\infty^\times. \qquad (2.7) $$

In the following, we will manipulate regular models of $C$. We denote the proper minimal regular model of $C$ by $C \to \mathbb{P}_{\mathbb{F}_q}^1$ (see [Liu02, §9.3.2 & 9.3.3] for its definition and main properties). This is a regular surface fibered over $\mathbb{P}_{\mathbb{F}_q}^1$ whose generic fiber is isomorphic to $C$, and that contains no exceptional divisors. Moreover, $C$ is minimal, in the sense that any birational map $C' \dasharrow C$ of regular surfaces fibered over $\mathbb{P}_{\mathbb{F}_q}^1$ is a birational morphism. We denote the Simple Normal Crossings (SNC) minimal regular model of $C$ by $K \to \mathbb{P}_{\mathbb{F}_q}^1$ (see [Liu02, §9.3.4] for details). This is a regular surface fibered over $\mathbb{P}_{\mathbb{F}_q}^1$, whose generic fiber is isomorphic to $C$, such that all its singularities are ordinary double points. Finally, we let $S \to \mathbb{P}_{\mathbb{F}_q}^1$ be the Néron model of $S$ (see [BLR90, §9.5] for details).

For any $v \in M_\mathbb{K}$, we denote by $C_v$ the base change of $C$ to $\mathrm{Spec}(\mathbb{F}_v)$, and we call this the *special fiber* or *reduction* of $C$ at $v$. We define in the same way the fiber at $v$ of the models $W$, $K$ and $S$ introduced above. Since $\mathrm{Spec}(\mathcal{O}_v^{\mathrm{unr}}) \to \mathrm{Spec}(\mathcal{O}_v)$ is étale, Proposition 10.1.17 from [Liu02] states that $C \times_{\mathbb{P}^1} \mathcal{O}_v^{\mathrm{unr}}$ is the minimal regular model of $C \times_K \mathbb{K}_v^{\mathrm{unr}}$ (the same holds if one replaces $C$ by $K$). Similarly, Néron models commute to étale base change, so $S \times_{\mathbb{P}^1} \mathcal{O}_v^{\mathrm{unr}}$ is the Néron model of $S \times_K \mathbb{K}_v^{\mathrm{unr}}$. We denote by $C_{\bar{v}}$ the special fiber of $C \times_{\mathbb{P}^1} \mathcal{O}_v^{\mathrm{unr}}$, and similarly for $K_{\bar{v}}$ and $S_{\bar{v}}$.
## 2.3 Special fibers of $C$, $K$ and $S$

In this section, we describe the fibers of $C$, $K$ and $S$ at various places of $K$ in order to compute arithmetic invariants of $S$ in Section 2.4. To do so, we make use of the theory of cluster pictures: we refer the reader to [DDMM23] for details, or to [BBB$^{+22}$] for a quick survey.

We begin with the minimal regular model $C$. We identify places of $K$ with the corresponding monic irreducible polynomial in $\mathbb{F}_q[t]$. If $v \nmid \Delta(W)$, then the fiber $W_v$ is smooth and $W_v \simeq C_v$ are isomorphic over $\mathbb{F}_v$. We will refer to the places dividing $\Delta(W)$ as *places of bad reduction* (as we will see later, for our specific models, $C$ has bad reduction at $v \in M_K$ if and only if $S$ has bad reduction at $v$). For simplicity, we will write $\mathbb{P}^1$ instead of $\mathbb{P}_{\mathbb{F}_q}^1$ whenever the context is clear.

**Proposition 2.8.** The singular fibers of the minimal regular model $C$ are described as follows:

* At any finite place of bad reduction $v$, the special fibers of $C$ and $W$ are isomorphic over $\mathbb{F}_v$. On an affine subset, they are described by the hyperelliptic equation:

$$C_v \simeq W_v : y^2 = x^3(x^3 + \rho_v) \quad \text{where } \rho_v := \begin{cases} \wp_a(t) - 2 \mod v & \text{if } v | \wp_a(t) + 2, \\ \wp_a(t) + 2 \mod v & \text{if } v | \wp_a(t) - 2. \end{cases}$$

* At infinity, the special fiber $C_\infty/\mathbb{F}_q$ is a tree of $6q^a + 7$ $\mathbb{P}^1$'s arranged as in Figure 1. All components are defined over $\mathbb{F}_q$ and numbers indicate multiplicities.

<figure><img src="image_2.png" /><figcaption>Figure 1: Fiber $C_\infty$.</figcaption></figure>

*Proof.* Fix a finite place of bad reduction $v$. From Lemma 2.6, we know that $W$ is a minimal Weierstrass model at $v$. Proposition 6 from [Liu96] states that $C \times_{\mathbb{P}^1} O_v$ is the minimal desingularization of $W \times_{\mathbb{P}^1} O_v$. Since $\text{disc}(x^3 + \wp_a(t) \pm 2) = -2^4 3^3 (\wp_a(t) \pm 2)^2$, reducing equation (2.3) mod $v$ gives the following affine equation for the special fiber of $W$ at $v$:

$$W_v : y^2 = x^3(x^3 + \rho_v).$$

The projective curve $W_v/\mathbb{F}_v$ admits a singularity at $(0,0)$ and is smooth at all other points. We claim that $W \times_{\mathbb{P}^1} O_v$ is regular at the point corresponding to the ideal $(x, y, \pi_v) \subset O_v[x, y]$. Indeed, $W \times_{\mathbb{P}^1} O_v$, localized at this ideal is

$$\text{Spec}(O_v[x, y]_{(x, y)}/(y^2 - (x^3 + \wp_a(t) - 2)(x^3 + \wp_a(t) + 2))).$$

The defining polynomial is not in $(x, y, \pi_v)^2$ as the constant term $\wp_a(t)^2 - 4$ is square-free, so it has $v$-adic valuation 1. Therefore, Corollary 4.2.12 from [Liu02] implies that $W \times_{\mathbb{P}^1} O_v$ is regular at the point corresponding to the ideal $(x, y, \pi_v)$, hence everywhere. We deduce the existence of an isomorphism $W \times_{\mathbb{P}^1} O_v \simeq C \times_{\mathbb{P}^1} O_v$: the latter is the minimal desingularization of the former, which is already regular. In particular the special fibers $W_v$ and $C_v$ are isomorphic over $\mathbb{F}_v$.

At infinity, we know from Remark 2.7 that we can take (2.7) as a defining equation for $W$. This corresponds to Namikawa–Ueno type $[III_{6q^a}]$ (see [NU73, p.184]): using their table, we see that the fiber of the minimal regular model $C_\infty$ is as depicted in Figure 1. Notice that all singularities in the fiber $C_\infty$ are ordinary double points: hence $C_\infty$ coincides with the fiber $K_\infty$. Using the theory of cluster pictures, we are going to see that all irreducible components appearing in Figure 1 are actually defined over $\mathbb{F}_q$.

The roots of the defining polynomial in (2.7) are $\eta_i^\pm := \zeta_3^i \sqrt{s(u \pm s^{q^a}v)}$, where $\zeta_3 \in \mathbb{K}^{\text{sep}}$ is a third root of unity. It is easy to check that all roots have $s$-adic valuation 1/3. Computing the Taylor
expansion in $\mathcal{O}_\infty$ of $(1 \pm s^{q^a} v/u)^{1/3}$ we see that $\mathrm{ord}_s(\eta_i^+ - \eta_i^-) = 1/3 + q^a$ for any $0 \le i \le 2$, and $\mathrm{ord}_s(\eta_i^+ - \eta_j^-) = 1/3$ if $i \ne j$. We deduce that the cluster picture of $C$ at $\infty$ is the following:

<figure><img src="image_3.png" /><figcaption>Figure 2: Cluster picture of $C$ at $\infty$.</figcaption></figure>

Using [FN20], one can describe the special fiber $K_{\overline{\infty}} = C_{\overline{\infty}}$ and the action of Frobenius on irreducible components, in terms of the cluster picture at $\infty$ and the action of Gal$(K_{\overline{\infty}}^{\mathrm{sep}}/K_{\infty})$ on it. We freely adopt the notations of [FN20] until the end of the proof. The orbit $X_R := \{\mathcal{R}\}$ contributes two central components with tails of $\mathbb{P}^1$'s. Write $t_i := \{\eta_i^+, \eta_i^-\}$ for $0 \le i \le 2$ and let $X_t := \{t_0, t_1, t_2\}$ be the orbit under Galois of $t_0$. Then $X_t$ contributes with a chain of $\mathbb{P}^1$'s of length $6q^a - 1$ linking the two central components associated to the orbit $X_R$. The quantities $\epsilon_{X_R}, \epsilon_{X_t} \in \{\pm 1\}$ describe the action of the Frobenius on irreducible components of $C_{\overline{\infty}}$. It is easy to check that for any $i \in \{0, 1, 2\}$, we have $\mathcal{R}^* = t_i^* = \mathcal{R}$ which implies that $\epsilon_{X_R} = \epsilon_{X_t} = 1$. We then deduce from Theorem 7.21 of [FN20] that the Frobenius fixes each irreducible component of $C_{\overline{\infty}}$. Hence all these components are defined over $\mathbb{F}_q$, so the picture of $C_{\overline{\infty}}$ “descends” to $\mathbb{F}_q$ and $C_{\infty}$ looks like Figure 1. $\square$

**Remark 2.9.** The author fears that the description of the special fiber of the minimal regular SNC model given in [FN20, Theorem 7.18] includes a typographical error. Indeed, applying the mentioned theorem, one obtains for the fiber of $K_{\overline{\infty}}$ a similar drawing as the one obtained in Figure 1, with a subtle difference. The $\mathbb{P}^1$'s of multiplicity 1 branching the “central” components of multiplicity 3 are replaced by a chain of length 2, with multiplicities 2 and 1 (identical to the ones already appearing in Figure 1). The author has checked by other methods that $K_{\overline{\infty}} = C_{\overline{\infty}}$ is indeed as shown in Figure 1, and that all irreducible components are indeed defined over $\mathbb{F}_q$.

**Remark 2.10.** The strategy from [FN20] can also be applied to describe $K_{\overline{v}}$ when $v$ is a finite place of bad reduction. At any such place, it is easy to check that the cluster picture of $C$ is as shown in Figure 3. The second author of loc. cit. has made an explicit table giving the correspondence between cluster pictures and fibers of SNC minimal regular models for genus 2 curves. Our cluster appears in [Now22, p.241] and gives the fiber $K_{\overline{v}}$ shown in Figure 4. In it, numbers denote multiplicities and the label “g1” points out that the component is a genus 1 curve.

<figure><img src="image_4.png" /><figcaption>Figure 3: Cluster picture at $v|\Delta(\mathcal{W})$, $v \ne \infty$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Fiber $K_{\overline{v}}$ at $v|\Delta(\mathcal{W})$, $v \ne \infty$.</figcaption></figure>

We now turn our attention to describing the fibers of the Néron model $S \to \mathbb{P}_{\mathbb{F}_q}^1$ of $S/\mathbb{K}$. For any smooth group scheme $G$ of finite type over $\mathbb{P}_{\mathbb{F}_q}^1$ with geometrically connected generic fiber (e.g. $S$), denote by $G^0$ the relative identity component. This is the open subscheme of $G$ obtained by removing the closed complement of the identity component in the finitely many disconnected fibers. Combining Chevalley’s structure theorem for connected smooth algebraic groups (see [Ros56, Theorem 16]) and the structure theorem for linear connected groups over perfect fields (see [SGA70, Exposé XVII]) we get the following result. At any place $v \in M_{\mathbb{K}}$, $S_v^0$ fits in an exact sequence of $\mathbb{F}_v$-groups

$$1 \to T \times U \to S_v^0 \to A \to 1,$$

where $T$ is a torus, $U$ is a unipotent group and $A$ is an abelian variety. We refer the reader to [Ros56] and [BLR90, §9.1 - 9.5.] for details. The data of these three groups $A$, $T$ and $U$ is what we call the reduction type of $S^0$ at $v$. We will talk about abelian, toric and unipotent dimension of $S_v^0$ to refer to the respective dimensions of the algebraic groups fitting in the exact sequence above. Whenever the unipotent group $U$ is trivial, i.e. $S_v^0$ is the extension of an abelian variety by a torus, we say that
S has semistable reduction at v.

Let $Pic_{C/\mathbb{P}^1}$ be the Picard functor of $C/\mathbb{P}_{\mathbb{F}_q}^1$ (cf. [BLR90]). Proposition 2.8 implies that for any $v \in M_{\mathbb{K}}$, the gcd of the multiplicities of the irreducible components of $C_v$ is 1. Therefore, at any place $v \in M_{\mathbb{K}}$, we have the following isomorphisms over $\mathbb{F}_v$ (see [BLR90, §9.5, §9.6])

$$ S_v^0 \simeq (\mathrm{Pic}_{C/\mathbb{P}^1})_v \simeq \mathrm{Pic}^0(C_v). \qquad (2.8) $$

Using this, we describe the reductions of $S^0$.

**Theorem 2.11.** The special fibers of the identity component of the Néron model $S^0$ are as follows:

* At any $v \nmid \Delta(W)$, $C_v$ is smooth and we have $S_v = S_v^0 = \mathrm{Jac}(C_v)$.

* At any $v \mid \Delta(W)$, $v \neq \infty$, $S_v^0$ fits in the exact sequence of $\mathbb{F}_v$-groups

$$ 1 \to \mathbb{G}_a \to S_v^0 \to E_v \to 1, $$

where $E_v/\mathbb{F}_v$ is the elliptic curve given by the affine equation $E_v : y^2 = x(x^3 + \rho_v)$, and $\rho_v$ is defined as in Proposition 2.8.

* At $\infty$, $S^0$ has totally unipotent reduction, i.e., $S_\infty^0$ is a unipotent group of dimension 2.

*Proof.* For a finite place of bad reduction $v$, we use Section 7.5 in [Liu02] to describe $\mathrm{Pic}^0(C_v)$ in terms of the normalizations of the irreducible components of $C_v$. Recall that the special fiber $C_v \simeq W_v$ is reduced and irreducible. We compute its normalization by blowing up along the closed singular point $(0,0)$. The result is the elliptic curve $E_v/\mathbb{F}_v$ presented in the statement of the theorem and given by

$$ E_v : y^2 = x(x^3 + \rho_v). \qquad (2.9) $$

Theorem 7.5.19 from [Liu02] states that the “abelian part” of $S_v^0$ is the elliptic curve $E_v$. To compute the toric dimension of $S_v^0$, consider the normalization morphism

$$ \nu_v : \left\{ \begin{array}{ccc} E_v & \to & W_v \\ (x, y) & \mapsto & (x, xy). \end{array} \right. $$

Every point on $W_v$ has a unique preimage by $\nu_v$, even the singular point $(0,0)$ as $\nu_v^{-1}((0,0)) = \{(0,0)\}$. The formula given in [Liu02, Proposition 7.5.18] implies that the toric dimension of $S_v^0$ equals 0. Thus, the “linear part” of $S_v^0$ is a unipotent group of dimension 1 defined over the perfect field $\mathbb{F}_v$, which is therefore split and thus isomorphic to $\mathbb{G}_a$, giving the desired exact sequence.

At infinity, we compute first $S_\infty^0$ using [Lor90, Corollary 1.4]. It states that the abelian dimension of $S_\infty^0$ equals the sum of the genus of the irreducible components of $C_\infty$, and the toric dimension equals the first Betti number of the dual graph of $C_\infty$. But all these irreducible components are $\mathbb{P}^1$'s, and the dual graph of is a tree, which has trivial homology. We deduce that the abelian and toric dimension are both equal to 0. Hence, $S_\infty^0$ is a unipotent group of dimension 2, which is isomorphic to $S_\infty^0$ over $\mathbb{F}_q$, hence the result. $\square$

Recall that an abelian variety defined over $\mathbb{K}$ is said to be *constant* if it is isomorphic over $\mathbb{K}$ to the base change to $\mathbb{K}$ of an abelian variety defined over $\mathbb{F}_q$. An abelian variety is said to be *isotrivial* if it becomes constant after base change to a finite extension of $\mathbb{K}$. From the description of the cluster picture of $C$ at infinity we deduce:

**Proposition 2.12.** The abelian surface $S$ is non-isotrivial.

*Proof.* Let $F/\mathbb{K}$ be the extension where $S$ attains semistable reduction (see [SGA72, Exposé IX]). The cluster picture of $C$ at $\infty$, shown in Figure 2, is such that every cluster has at most 2 odd children. Theorem 10.3 of [DDMM23] then states that the Néron model of $S_F$ has totally toric reduction at $\infty$. By Corollary 7.4.4 of [BLR90], the same holds for $S_{F'}$, where $F'/F$ is any finite extension. But the base change of a constant variety remains constant, and its Néron model must have good reduction everywhere. Hence we conclude that $S$ cannot be isotrivial. $\square$
## 2.4 Arithmetic invariants of S

We now exploit the results from the previous section to estimate arithmetic invariants of S such as its conductor, Tamagawa numbers and differential height.

Fix, once and for all, a prime number $\ell \neq p$. Let $T_\ell(S)$ be the $\ell$-adic Tate module of S and $V_\ell(S) := T_\ell(S) \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell$: this is a 4-dimensional $\mathbb{Q}_\ell$-vector space endowed with a continuous linear action of $\mathrm{Gal}(\mathbb{K}^{\mathrm{sep}}/\mathbb{K})$. At any place $v \in M_{\mathbb{K}}$, denote the inertia subgroup at $v$ by $I_v$, and let $V_\ell(S)^{I_v}$ be the subspace fixed by $I_v$. Let $N(S) = \prod_{v \in M_{\mathbb{K}}} v^{n_v(S)}$ be the conductor divisor of S, written in multiplicative notation. We refer the reader to [Ser70] for details on the conductor of an abelian variety. By the Néron-Ogg-Shafarevich criterion we have $n_v(S) = 0$ at any place $v$ of good reduction.

**Corollary 2.13.** The conductor divisor of S is given by

$$N(S) = \prod_{\substack{v \mid \Delta(W) \\ v \neq \infty}} (v)^2 (\infty)^4.$$

The degree of the conductor is $\deg N(S) = 4q^a + 4$.

*Proof.* Our assumption $\mathrm{char}(\mathbb{K}) > 5$ guarantees that $V_\ell(S)$ is tamely ramified (see Corollary 2 to Theorem 2 in [ST68, §3]), so at any place $v \in M_{\mathbb{K}}$ we have $n_v(S) := \dim V_\ell(S) - \dim V_\ell(S)^{I_v}$. Now Remark 1 from [ST68, §3] states that $\dim V_\ell(S) - \dim V_\ell(S)^{I_v}$ is twice the unipotent dimension of $S_v^0$. Therefore at any finite place of bad reduction $v$, we have $n_v(S) = 2$ and at the place at infinity, $n_\infty(S) = 4$. Finally, we compute

$$\begin{aligned} \deg N(S) &= \sum_{v \mid \wp_a(t)-2} 2d_v + \sum_{v \mid \wp_a(t)+2} 2d_v + 4 \\ &= 2(\deg(\wp_a(t)-2) + \deg(\wp_a(t)+2)) + 4 = 4q^a + 4. \end{aligned}$$

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We turn now to the Tamagawa numbers of S. For $v \in M_{\mathbb{K}}$, the local Tamagawa number of S at $v$ is defined as $c_v(S) := |(S_v/S_v^0)(\mathbb{F}_v)|$.

**Proposition 2.14.** For any $v \neq \infty$, the local Tamagawa number $c_v(S)$ equals 1. At infinity we have $c_\infty(S) \in \{1, 3, 9\}$.

*Proof.* At any finite place of good reduction $v$, the special fiber $S_v$ is connected, so we have $c_v(S) = 1$. Fix now a place of bad reduction $v$, we first compute $c_{\bar{v}}(S) := |(S_{\bar{v}}/S_{\bar{v}}^0)(\mathbb{F}_v)|$. Since $(S_v/S_v^0)(\mathbb{F}_v)$ is a subgroup of $(S_{\bar{v}}/S_{\bar{v}}^0)(\mathbb{F}_v)$, we have $c_v(S) \mid c_{\bar{v}}(S)$. The advantage of introducing this new quantity is that we can apply a result by Lorenzini to compute $c_{\bar{v}}(S)$ in terms of $K_{\bar{v}}$. We write the latter as a divisor $K_{\bar{v}} = \sum_{i=1}^m r_i \Gamma_i$ and let $d_i := \sum_{j \neq i} \Gamma_i \cdot \Gamma_j$ (where $\Gamma_i \cdot \Gamma_j$ stands for the intersection number of $\Gamma_i$ and $\Gamma_j$). Since $S_{\bar{v}}^0$ has toric dimension 0, Corollary 1.5 from [Lor90] yields

$$c_{\bar{v}}(S) = \prod_{i=1}^{m} r_i^{d_i-2}. \qquad (2.10)$$

Using the description of the fibers of $K$ drawn in Figures 1 and 4, we compute $c_{\bar{v}}(S) = 3^{-1}2^{-1}1^{-1}6 = 1$ for any $v \mid \Delta(W), v \neq \infty$ and

$$c_{\infty}(S) = 1^{-1}2^01^{-1}3 \prod_{i=1}^{6q^a-1} (3^0) \cdot 3 \cdot 1^{-1}2^01^{-1} = 9,$$

hence the result.

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To conclude this section, we compute the differential height of S. Let $z : \mathbb{P}_{\mathbb{F}_q}^1 \to S$ be the identity section of the Néron model $S \to \mathbb{P}_{\mathbb{F}_q}^1$ and $\Omega_{S/\mathbb{P}^1}^2$ be the sheaf of relative 2-differentials of S over $\mathbb{P}_{\mathbb{F}_q}^1$. The pullback $\omega_S := z^*\Omega_{S/\mathbb{P}^1}^2$ is a line bundle on $\mathbb{P}_{\mathbb{F}_q}^1$, and we define the differential height of S as

$$h(S) := \deg(\omega_S).$$

We define also the exponential height of S as $H(S) := q^{h(S)}$.
**Proposition 2.15.** The differential height of S is given by $h(S) = q^a + 1$.

*Proof.* Notice first that the polynomial in the right-hand side of (2.3) is unitary and has even degree. Thus, C has two points at infinity, which we denote by $\infty_1, \infty_2$; both are $\mathbb{K}$-rational. Each of them induces a section of $\xi: \mathcal{K} \to \mathbb{P}_{\mathbb{F}_q}^1$, whose image lies in the smooth locus of $\xi$. Let $\omega_{\mathcal{K}/\mathbb{P}^1}$ be the relative dualizing sheaf of $\mathcal{K} \to \mathbb{P}_{\mathbb{F}_q}^1$, then Proposition 7.4 of [BHP+20] gives the isomorphism of line bundles

$$ \omega_S \simeq \bigwedge^2 \xi_* \omega_{\mathcal{K}/\mathbb{P}^1}. $$

Consider the two differentials $\mu_1 := \frac{dx}{y}$ and $\mu_2 := \frac{xdx}{y}$ in $H^0(\mathcal{K}, \omega_{\mathcal{K}/\mathbb{P}^1})$. The contraction of exceptional divisors $\mathcal{K} \to C$ is a birational morphism of regular surfaces over $\mathbb{P}_{\mathbb{F}_q}^1$. Corollary 9.2.25 in [Liu02] then shows that the subgroups $H^0(\mathcal{K}, \omega_{\mathcal{K}/\mathbb{P}^1})$ and $H^0(C, \omega_{\mathcal{C}/\mathbb{P}^1})$ of $\Omega_{\mathcal{C}/\mathbb{K}}^1$ coincide. We thus may view $\mu_1, \mu_2$ as differentials on $C$ and work with $\omega_{\mathcal{C}/\mathbb{P}^1}$ rather than $\omega_{\mathcal{K}/\mathbb{P}^1}$. Moreover $(\mu_{1|C}, \mu_{2|C})$ forms a $\mathbb{K}$-basis of $H^0(C, \Omega_{\mathcal{C}/\mathbb{K}}^1)$, so $\xi_*(\mu_1 \wedge \mu_2)$ is a generator of $\omega_S$. We deduce

$$ \deg(\omega_S) = \deg(\xi_*(\mu_1 \wedge \mu_2)) = \sum_{v \in M_{\mathcal{K}}} \mathrm{ord}_v(\xi_*(\mu_1 \wedge \mu_2)) \deg v. $$

At any place of good reduction $v$, the base change $\mathcal{C} \times_{\mathbb{P}^1} \mathcal{O}_v$ is smooth. Thus, after restriction, $(\mu_1, \mu_2)$ forms an $\mathcal{O}_v$-basis of $H^0(\mathcal{C} \times_{\mathcal{O}_v} \mathcal{O}_{\mathcal{C}/\mathcal{O}_v})$, implying that $\mathrm{ord}_v(\xi_*(\mu_1 \wedge \mu_2)) = 0$. At a finite place of bad reduction $v$, we know from Lemma 2.6 that equation (2.3) defining $\mathcal{C}$ (2.3) is minimal. Moreover, $\mathcal{C}$ has Namikawa–Ueno type $[I_0 - II - 0]$ at $v$, which corresponds to Ogg’s numerical type [0]. Propositions 4 and 7 in [Liu94] then imply that the restriction of $(\mu_1, \mu_2)$ is a basis of $H^0(\mathcal{C} \times_{\mathcal{O}_v} \mathcal{O}_{\mathcal{C}/\mathcal{O}_v})$, hence $\mathrm{ord}_v(\xi_*(\mu_1 \wedge \mu_2)) = 0$.

It follows that $\deg(\omega_S) = \mathrm{ord}_{\infty}(\xi_*(\mu_1 \wedge \mu_2))$. To compute the contribution at $\infty$, let $b \in \{1, 2\}$ be such that $q^a + b \equiv 0 \pmod 3$. We know from Lemma 2.6 that, at $\infty$, the defining equation (2.5) (resp. (2.6)) if $b=1$ (resp. $b=2$) is minimal. The proof of Proposition 2.8 shows that the Namikawa–Ueno type of $\mathcal{C}_{\infty}$ is $[III_{6q^a}]$, which corresponds to Ogg’s numerical type [43]. Write $(x_b, y_b)$ for the coordinates appearing in (2.5) or (2.6) depending on the value of $b$. Then Propositions 4 and 7 in [Liu94] imply that $(\frac{dx_b}{y_b}, \frac{x_b dx_b}{y_b})$ forms an $\mathcal{O}_{\infty}$-basis of $H^0(\mathcal{C} \times_{\mathcal{O}_{\infty}} \mathcal{O}_{\mathcal{C}_{\infty}})$. Using the change of variables giving $(x, y)$ in terms of $(x_b, y_b)$ given in Definition 2.5, it is easy to check that, for both values of $b \in \{1, 2\}$, we have

$$ \mu_1 \wedge \mu_2 = s^{q^a+1} \left( \frac{dx_b}{y_b} \wedge \frac{x_b dx_b}{y_b} \right), $$

which finally gives the result.

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# 3 Character sums and the geometry of $\mathcal{C}$

As we saw in Section 2.1, the minimal regular model $\mathcal{C}$ of $\mathcal{C}$ is dominated by a product of curves. In this section we describe the geometry of $\mathcal{C}$ in further detail in terms of the two auxiliary curves $X$ and $Y$ defined by (2.1) (to ease notation we drop the subscript $a$). We will study the cohomology of $X$ and $Y$: the eigenvalues of the action of the Frobenius on these spaces may be expressed in terms of certain character sums over finite fields. We begin by recalling some facts about the relevant sums.

## 3.1 Background on Gauss and Kloosterman sums

In this section, we let $p \ge 3$ be a prime number and temporarily drop the assumption $p \ge 7$. We fix a finite field $\mathbb{F}_q$ of characteristic $p$.

We first discuss additive and multiplicative characters on finite fields. Our main reference here is [LN97, Chapter 5, §1]. Fix, once and for all, a non-trivial additive character $\psi_0: \mathbb{F}_p \to \mathbb{Q}(\zeta_p)^\times$ with values in the $p$-th cyclotomic field. Let $\mathbb{F}$ be a finite field of characteristic $p$ and write $\mathrm{Tr}_{\mathbb{F}/\mathbb{F}_p}: \mathbb{F} \to \mathbb{F}_p$ for the relative trace map. We recall that all additive characters on $\mathbb{F}$ are obtained in the following way (see [LN97, Theorem 5.7]): for any $\alpha \in \mathbb{F}$, define the map

$$ \psi_{\mathbb{F}, \alpha}: \mathbb{F} \to \mathbb{Q}(\zeta_p)^\times, \quad x \mapsto \psi_0(\mathrm{Tr}_{\mathbb{F}/\mathbb{F}_p}(\alpha x)). $$
One easily checks that $\psi_{\mathbb{F},\alpha}$ is an additive character on $\mathbb{F}$, and that it is non-trivial if and only if $\alpha \neq 0$. To ease notation, we write $\psi_{\mathbb{F}}$ instead of $\psi_{\mathbb{F},1}$. Recall that the group $\hat{\mathbb{F}}$ of additive characters on $\mathbb{F}$ is isomorphic to $\mathbb{F}$ itself via $\alpha \mapsto \psi_{\mathbb{F},\alpha}$.

We now discuss multiplicative characters. Fix an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$, and denote the ring of algebraic integers by $\overline{\mathbb{Z}}$. We fix a prime ideal $\mathfrak{p}$ of $\overline{\mathbb{Z}}$ lying over the rational prime $p$ and let $\mathrm{ord}_{\mathfrak{p}} : \overline{\mathbb{Q}} \to \mathbb{Q}$ be the $\mathfrak{p}$-adic valuation on $\overline{\mathbb{Q}}$, normalised so that $\mathrm{ord}_{\mathfrak{p}}(q) = 1$. The quotient $\overline{\mathbb{Z}}/\mathfrak{p}$ is an algebraic closure of $\mathbb{F}_p$, denoted by $\overline{\mathbb{F}}_p$. The reduction map $\overline{\mathbb{Z}} \to \overline{\mathbb{Z}}/\mathfrak{p}$ induces an isomorphism between the group of roots of unity in $\overline{\mathbb{Q}}$ whose order is prime to $p$ and $\overline{\mathbb{F}}_p^\times$. Let $\chi : \overline{\mathbb{F}}_p^\times \to \overline{\mathbb{Q}}^\times$ denote the inverse of this isomorphism. For any positive integer $n$ dividing $|\mathbb{F}^\times|$, define the map

$$ \chi_{\mathbb{F},n} : \mathbb{F}^{\times} \to \overline{\mathbb{Q}}^{\times}, \quad x \mapsto \chi(x)^{|\mathbb{F}^{\times}|/n}. $$

This map is a multiplicative character on $\mathbb{F}^\times$ of exact order $n$ (see [Coh07, §4.3.2]). If $\mathbb{F}_p \subset \mathbb{F} \subset \mathbb{F}'$ are successive finite extensions and $N_{\mathbb{F}'/\mathbb{F}} : (\mathbb{F}')^\times \to \mathbb{F}^\times$ denotes the relative norm, we have the equality $\chi_{\mathbb{F}',n} = \chi_{\mathbb{F},n} \circ N_{\mathbb{F}'/\mathbb{F}}$.

**Definition 3.1.** Let $\mathbb{F}/\mathbb{F}_p$ be a finite field extension, $\psi : \mathbb{F} \to \mathbb{Q}(\zeta_p)^\times$ be an additive character on $\mathbb{F}$ and $\chi : \mathbb{F}^\times \to \overline{\mathbb{Q}}^\times$ a multiplicative character on $\mathbb{F}^\times$. Define the Gauss sum

$$ G_{\mathbb{F}}(\chi, \psi) := - \sum_{x \in \mathbb{F}^{\times}} \chi(x)\psi(x). $$

If $\chi$ has order $n$, without loss of generality we can assume that $\chi$ takes values in the cyclotomic field $\mathbb{Q}(\zeta_n)$. Therefore the sum $G_{\mathbb{F}}(\chi, \psi)$ is an algebraic integer in $\mathbb{Q}(\zeta_{np})$. For any additive character $\psi$ on $\mathbb{F}$ and any multiplicative character $\chi$ on $\mathbb{F}^\times$, we have the following facts:

(Ga 1) If $\psi$ and $\chi$ are non-trivial then, in any complex embedding of $\mathbb{Q}(\zeta_{np})$, $|G_{\mathbb{F}}(\chi, \psi)| = |\mathbb{F}|^{1/2}$.

(Ga 2) For any $\alpha \in \mathbb{F}^\times$, $G_{\mathbb{F}}(\chi, \psi_{\mathbb{F},\alpha}) = \chi(\alpha)^{-1} G_{\mathbb{F}}(\chi, \psi_{\mathbb{F}})$.

(Ga 3) For any finite extension $\mathbb{F}'/\mathbb{F}$, any $\alpha \in \mathbb{F}^\times$,

$$ G_{\mathbb{F}'}(\chi^{|\mathbb{F}'|} \circ N_{\mathbb{F}'/\mathbb{F}}, \psi_{\mathbb{F}', \alpha^{1/|\mathbb{F}'|}}) = G_{\mathbb{F}'}(\chi \circ N_{\mathbb{F}'/\mathbb{F}}, \psi_{\mathbb{F}', \alpha}). $$

(Ga 4) (Hasse–Davenport) For any finite extension $\mathbb{F}'/\mathbb{F}$,

$$ G_{\mathbb{F}'}(\chi \circ N_{\mathbb{F}'/\mathbb{F}}, \psi \circ \mathrm{Tr}_{\mathbb{F}'/\mathbb{F}}) = G_{\mathbb{F}}(\chi, \psi)^{[\mathbb{F}':\mathbb{F}]}. $$

(Ga 5) If $\chi$ has order $n$,

$$ \frac{1}{n} \le \frac{\mathrm{ord}_{\mathfrak{p}}(G_{\mathbb{F}}(\chi, \psi))}{[\mathbb{F} : \mathbb{F}_q]} \le 1 - \frac{1}{n}. $$

These are well-known facts: the reader can find proofs for (Ga 1), (Ga 2), and (Ga 4) in [LN97, Chap. V, §2], (Ga 3) is [Coh07, Lemma 2.5.8], and (Ga 5) follows from Stickelberger theorem: see [AGTT23, Corollary 6.11] with notation adapted to our setting.

**Definition 3.2.** Let $\mathbb{F}/\mathbb{F}_p$ be a finite field extension, $\psi : \mathbb{F} \to \mathbb{Q}(\zeta_p)^\times$ be an additive character, and $\alpha \in \mathbb{F}$. Define the Kloosterman sum

$$ \mathrm{Kl}_{\mathbb{F}}(\psi, \alpha) := - \sum_{x \in \mathbb{F}^{\times}} \psi(x + \frac{\alpha}{x}). $$

It is well-known that the sum $\mathrm{Kl}_{\mathbb{F}}(\psi, \alpha)$ is a totally real algebraic integer in $\mathbb{Q}(\zeta_p)$. For any non-trivial additive character $\psi$ on $\mathbb{F}$, we have the following facts:

(Kl 1) For any finite extension $\mathbb{F}'/\mathbb{F}$ and any $\alpha \in \mathbb{F}$, $\mathrm{Kl}_{\mathbb{F}'}(\psi \circ \mathrm{Tr}_{\mathbb{F}'/\mathbb{F}}, \alpha^{|\mathbb{F}'|}) = \mathrm{Kl}_{\mathbb{F}'}(\psi \circ \mathrm{Tr}_{\mathbb{F}'/\mathbb{F}}, \alpha)$.

(Kl 2) For any $\alpha \in \mathbb{F}^\times$, there is a unique pair $\{\kappa_{1,\mathbb{F}}(\psi, \alpha), \kappa_{2,\mathbb{F}}(\psi, \alpha)\}$ of algebraic integers, whose product is $|\mathbb{F}|$ and such that, for any finite extension $\mathbb{F}'/\mathbb{F}$,

$$ \mathrm{Kl}_{\mathbb{F}'}(\psi \circ \mathrm{Tr}_{\mathbb{F}'/\mathbb{F}}, \alpha) = \kappa_{1,\mathbb{F}}(\psi, \alpha)^{[\mathbb{F}':\mathbb{F}]} + \kappa_{2,\mathbb{F}}(\psi, \alpha)^{[\mathbb{F}':\mathbb{F}]}. $$
(K13) For any $\alpha \in \mathbb{F}^\times$ one has $|\kappa_{1,\mathbb{F}}(\psi, \alpha)| = |\kappa_{2,\mathbb{F}}(\psi, \alpha)| = |\mathbb{F}|^{1/2}$ in any complex embedding. In particular $|\mathrm{Kl}_{\mathbb{F}}(\psi, \alpha)| \le 2|\mathbb{F}|^{1/2}$ and $\kappa_{1,\mathbb{F}}(\psi, \alpha)$, $\kappa_{2,\mathbb{F}}(\psi, \alpha)$, seen as complex numbers, are conjugate.

(K14) If $\mathbb{F}$ is a finite extension of $\mathbb{F}_q$, then for any $\alpha \in \mathbb{F}^\times$, the $\mathfrak{p}$-adic valuations of $\kappa_{1,\mathbb{F}}(\psi, \alpha)$ and $\kappa_{2,\mathbb{F}}(\psi, \alpha)$ are

$$ \{\mathrm{ord}_{\mathfrak{p}}(\kappa_{1,\mathbb{F}}(\psi, \alpha)), \mathrm{ord}_{\mathfrak{p}}(\kappa_{2,\mathbb{F}}(\psi, \alpha))\} = \{0, [\mathbb{F} : \mathbb{F}_q]\}. $$

The first point is straightforward; (K12) and (K13) are Theorems 5.43 and 5.44 in [LN97]. Finally, a proof of (K14) can be found in [GdW21, Lemma 5.1].

## 3.2 Orbits

Assume once again that $p \ge 7$. Fix an integer $a \ge 1$, consider the action of $\langle q \rangle \subset \mathbb{Q}^\times$ on $\mathbb{F}_{q^a}^\times$ via $q \cdot \alpha = \alpha^{1/q}$ for any $\alpha \in \mathbb{F}_{q^a}$. Write $\mathbf{P}_q(a)$ for the set of orbits of $\mathbb{F}_{q^a}$ under this action. $\mathbf{P}_q(a)$ is in bijection with the set of closed points of the multiplicative group $\mathbb{G}_m$ over $\mathbb{F}_q$ of degree dividing $a$. Indeed, for any $v \in \mathbf{P}_q(a)$, any $\beta \in v$, we have $v = \{\beta, \beta^q, \beta^{q^2}, \dots, \beta^{v|v|-1}\}$, and all elements in this set induce the same closed point of $\mathbb{G}_m$. Under the identification between closed points of $\mathbb{P}_{\mathbb{F}_q}^1$ and places of $\mathbb{K}$, $\mathbf{P}_q(a)$ corresponds to the set of places $\ne 0, \infty$ whose degree divides $a$. Given an orbit $v \in \mathbf{P}_q(a)$, its size is $|v| = d_v = [\mathbb{F}_v : \mathbb{F}_q]$.

From now on, we will only need to manipulate multiplicative characters of order 3. Endow the set $\mathbb{I}_a := (\mathbb{Z}/3\mathbb{Z})^\times \times \mathbb{F}_{q^a}^\times$ with the action of $\langle q \rangle$ defined by $q \cdot (j, \alpha) = (q \cdot j \mod 3, \alpha^{1/q})$ for any $(j, \alpha) \in \mathbb{I}_a$. Define $\mathbf{O}_q(a)$ to be the set of orbits under this action. There is a well-defined map $\Theta : \mathbf{O}_q(a) \to \mathbf{P}_q(a)$ given as follows: for any $o \in \mathbf{O}_q(a)$ and any representative $(j, \alpha) \in o$, we let $\Theta(o)$ be the orbit of $\alpha$ in $\mathbf{P}_q(a)$. Moreover, this map $\Theta$ is surjective. The size of an orbit in $\mathbf{O}_q(a)$ can be expressed as follows. For any $o \in \mathbf{O}_q(a)$, let $v := \Theta(o) \in \mathbf{P}_q(a)$, then we have

$$ |o| = \mathrm{lcm}(\mathrm{ord}^\times(q \mod 3), |v|). \qquad (3.1) $$

Indeed, if $(j, \alpha) \in o$ then $|o|$ is the smallest integer $n$ such that $q^n j \equiv j \mod 3$ and $\mathbb{F}_q(\alpha) = \mathbb{F}_v \subset \mathbb{F}_{q^n}$. We can give a more detailed description of the orbits in $\mathbf{O}_q(a)$.

* When $q \equiv 1 \mod 3$, we have $\mathrm{ord}^\times(q \mod 3) = 1$ and $\langle q \rangle$ acts trivially on $(\mathbb{Z}/3\mathbb{Z})^\times$. Hence, in that case, the orbits in $\mathbf{O}_q(a)$ are vertical, in the sense that for any $o \in \mathbf{O}_q(a)$, the “first coordinate” $j \in (\mathbb{Z}/3\mathbb{Z})^\times$ is the same for all representatives $(j, \alpha) \in o$. There is a map $\mathbf{O}_q(a) \to (\mathbb{Z}/3\mathbb{Z})^\times$, sending the orbit $o \in \mathbf{O}_q(a)$ to $j \in (\mathbb{Z}/3\mathbb{Z})^\times$, where $(j, \alpha) \in o$ is any representative. We choose $1, -1 \in \mathbb{Z}$ as representatives of $(\mathbb{Z}/3\mathbb{Z})^\times$ and lift the map above to $\mathrm{pr}_1 : \mathbf{O}_q(a) \to \{\pm 1\}$. Moreover, $\Theta$ is 2-to-1 and (3.1) implies that $|o| = |\Theta(o)|$ for any $o \in \mathbf{O}_q(a)$.

* When $q \equiv 2 \mod 3$, the orbits in $\mathbf{O}_q(a)$ “zigzag”. This means that, for any $o \in \mathbf{O}_q(a)$ the first coordinate of the representatives of $o$ is iteratively exchanged each time $q$ acts on one of them: if $(j, \alpha) \in o$, then $q \cdot (j, \alpha) = (-j, \alpha^{1/q})$. Letting $v := \Theta(o) \in \mathbf{P}_q(a)$, the size of $o$ depends on the parity of $|v|$. If $\alpha \in v$, then the orbits of $(1, \alpha)$ and $(-1, \alpha)$ are equal if and only if $|o| = |v|$. This implies that

$$ |\Theta^{-1}(v)| = \begin{cases} 2 & \text{if } |o| = 2|v|, \\ 1 & \text{if } |o| = |v|. \end{cases} \qquad (3.2) $$

We let $\mathbb{F}_o$ be the extension of $\mathbb{F}_q$ of degree $|o|$: if $(j, \alpha) \in o$ is any representative, then $\mathbb{F}_o$ is the smallest extension of $\mathbb{F}_q$ containing $\alpha$ and having a non-trivial multiplicative character of order 3. For any $d \ge 1$, define $\Pi_q(d)$ to be the number of orbits $v \in \mathbf{P}_q(a)$ having size $|v| = d$.

**Lemma 3.3.** For any $d, a \ge 1$, we have the following estimates:

$$ i) \quad |\Pi_q(d) - \frac{q^d}{d}| \le \frac{q^{d/2}}{1-q^{-1}}. $$

$$ ii) \quad \frac{q^a}{a} \ll_q |\mathbf{P}_q(a)| \ll_q \frac{q^a}{a}. $$

$$ iii) \quad \frac{q^a}{a} \ll_q |\mathbf{O}_q(a)| \ll_q \frac{q^a}{a}. $$
*Proof.* Item i) is the Prime Number Theorem for $\mathbb{F}_q(t)$ (see [Bru92, Proposition 6.3] for details). We deduce ii) from equality $|\mathbf{P}_q(a)| = \sum_{d|a} \Pi_q(d)$. To prove iii), notice that $\Theta$ is surjective and for any $v \in \mathbf{P}_q(a)$, $|\Theta^{-1}(v)| = 1$ or 2 as stated above. We deduce that $|\mathbf{P}_q(a)| \le |\mathbf{O}_q(a)| \le 2|\mathbf{P}_q(a)|$, which combined with ii) gives the desired result. $\square$

Let $o \in \mathbf{O}_q(a)$, applying (Ga 3) to the finite extension $\mathbb{F}_o/\mathbb{F}_q$ shows that for any $(j, \alpha) \in o$, we have the equality $G_{\mathbb{F}_o}(\chi_{\mathbb{F}_o, 3}^j, \psi_{\mathbb{F}_o, \alpha^{1/q}}) = G_{\mathbb{F}_o}(\chi_{\mathbb{F}_o, 3}^j, \psi_{\mathbb{F}_o, \alpha})$. In the same way, (K|1) shows that for any $v \in \mathbf{P}_q(a)$ and any $\beta \in v$, we have $\text{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta^{1/q}}, 1) = \text{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta}, 1)$. Thus, as $(j, \alpha)$ runs through $o$, the Gauss sums $G_{\mathbb{F}_o}(\chi_{\mathbb{F}_o, 3}^j, \psi_{\mathbb{F}_o, \alpha})$ remains constant. In the same way, as $\beta$ runs through $v$, $\text{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta}, 1)$ remains constant too. Thus, it is natural to define:

**Definition 3.4.** Let $v \in \mathbf{P}_q(a)$, $\beta \in v$. We define $\kappa_1(v) := \kappa_{1, \mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta}, 1)$, $\kappa_2(v) := \kappa_{2, \mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta}, 1)$. Let $o \in \mathbf{O}_q(a)$, fix $(j, \alpha) \in o$. We define $\gamma(o) := G_{\mathbb{F}_o}(\chi_{\mathbb{F}_o, 3}^j, \psi_{\mathbb{F}_o, \alpha})$, $\text{Kl}(o) := \text{Kl}_{\mathbb{F}_o}(\psi_{\mathbb{F}_o, \alpha}, 1)$, $\kappa_1(o) := \kappa_{1, \mathbb{F}_o}(\psi_{\mathbb{F}_o, \alpha}, 1)$, and $\kappa_2(o) := \kappa_{2, \mathbb{F}_o}(\psi_{\mathbb{F}_o, \alpha}, 1)$.

With this notation, (K|4) states that $\{\text{ord}_p(\kappa_1(o)), \text{ord}_p(\kappa_2(o))\} = \{0, |o|\}$. We can also compute the $p$-adic valuations of $\gamma(o)$:

**Lemma 3.5.** If $q \equiv 1 \mod 3$, then for any $o \in \mathbf{O}_q(a)$,

$$ \text{ord}_{\mathbf{p}}(\gamma(o)) = \begin{cases} 2|o|/3 & \text{if } \text{pr}_1(o) = 1, \\ |o|/3 & \text{if } \text{pr}_1(o) = -1. \end{cases} $$

If $q \equiv 2 \mod 3$, then $\text{ord}_{\mathbf{p}}(\gamma(o)) = |o|/2$.

*Proof.* The case $q \equiv 1 \mod 3$ is a consequence of Stickelberger's Theorem (see [AGTT23, Theorem 6.10]). The case $q \equiv 2 \mod 3$ is a consequence of a result of Tate and Shafarevich, which states that, under this hypothesis, $\gamma(o) = \pm \chi_{\mathbb{F}_o, 3}^{-j}(\alpha)(-q)^{|o|/2}$ (see [AGTT23, Lemma 6.7]). $\square$

## 3.3 Cohomology of auxiliary curves

In constructing the curve $C/\mathbb{K}$ (Section 2.1), we introduced two smooth projective curves $X$ and $Y$ defined over $\mathbb{F}_q$ (for simplicity, we drop the subscript $a$). We now describe the étale cohomology groups of these curves (see [Mil80]). After a more refined analysis of the geometry of $C$ (cf. Section 3.4), this will help us compute the $L$-function of the abelian surface $S$.

We fix a prime number $\ell \ne p$ and an isomorphism $\overline{\mathbb{Q}}_\ell \simeq \mathbb{C}$. The main reference for this section is [Mil80]. For any variety $V$ defined over $\mathbb{F}_q$, we denote by $H^i(V) := H_{\ell\ell}^i(V_{\overline{\mathbb{F}}_q}, \mathbb{Q}_\ell) \otimes_{\mathbb{Q}_\ell} \overline{\mathbb{Q}}_\ell$ the $i$-th $\ell$-adic cohomology groups of $V$ (see [Mil80, Chapter III]). These are finite dimensional $\overline{\mathbb{Q}}_\ell$-vector spaces that come with a natural action of the $q$-th power geometric Frobenius $\text{Fr}_q$ (we write in the same way the endomorphism of $V_{\overline{\mathbb{F}}_q}$ and the endomorphism on cohomology induced by functoriality).

Recall that the defining equations of $X$ and $Y$ are

$$ X: u^3 = \wp_a(t_1) \quad \text{and} \quad Y: v + \frac{1}{v} = \wp_a(t_2). \qquad (3.3) $$

Since both $X$ and $Y$ are smooth projective curves, $H^0(X)$ and $H^0(Y)$ are one-dimensional $\overline{\mathbb{Q}}_\ell$-vector spaces on which $\text{Fr}_q$ acts trivially (see [Mil80, Chapter V, Theorem 2.5]). By Poincaré duality, $H^2(X)$ and $H^2(Y)$ are also one-dimensional and the eigenvalue of $\text{Fr}_q$ on these lines is $q$. We are interested in $H^1(X)$ and $H^1(Y)$. Applying the Riemann-Hurwitz formula (see [Liu02, Theorem 7.4.16]) we can check that both $X$ and $Y$ have genus $q^a - 1$, so $H^1(X)$ and $H^1(Y)$ have dimension $2(q^a - 1)$. It is clear that the curve $X$ admits a unique point at infinity, denoted by $\infty_X$, which is $\mathbb{F}_q$-rational. On the other hand, $Y$ admits two points at infinity, which we denote by $\infty_{Y_1}, \infty_{Y_2}$, that are $\mathbb{F}_q$-rational. Indeed, $Y$ is hyperelliptic, described in affine coordinates by the hyperelliptic equation $y^2 = \wp_a(t)^2 - 4$, where the change of variables is explicitly given by $y = 2v - \wp_a(t)$.

We first describe $H^1(Y)$ and will then move on to $H^1(X)$. The group $\mathbb{F}_{q^a}$ acts on $Y$ as follows: on the affine subset of $Y$ defined by equation (3.3) we let $\beta \cdot (v, t_2) := (v, t_2 + \beta)$ for $\beta \in \mathbb{F}_{q^a}$ and $(v, t_2) \in Y$.
This action extends to the whole projective curve $Y$. One can check that the only fixed points of this action on $Y$ are $\infty_{Y_i}$, $i=1,2$. Note, however, that this action is not necessarily defined over $\mathbb{F}_q$ since it does not commute with the action of $\text{Fr}_q$ as soon as $a > 1$: we have $\text{Fr}_q(\beta \cdot (v, t_2)) = \beta^q \cdot \text{Fr}_q(v, t_2)$.

By functoriality, $\text{H}^1(Y)$ also carries an action of $\mathbb{F}_{q^a}$. We can decompose this cohomology space into isotypical components indexed by irreducible representations of $\mathbb{F}_{q^a}$. Since $\mathbb{F}_{q^a}$ is an abelian group, its irreducible representations are its characters. The latter are described by the isomorphism $\mathbb{F}_{q^a} \xrightarrow{\sim} \widehat{\mathbb{F}}_{q^a}$, $\beta \mapsto \psi_{\widehat{\mathbb{F}}_{q^a}, \beta}$. Denoting by $\text{H}^1(Y)^\beta$ the subset of $\text{H}^1(Y)$ where $\mathbb{F}_{q^a}$ acts by multiplication by $\psi_{\widehat{\mathbb{F}}_{q^a}, \beta}$, we obtain the decomposition

$$ \text{H}^1(Y) = \bigoplus_{\beta \in \mathbb{F}_{q^a}} \text{H}^1(Y)^\beta. \qquad (3.4) $$

Let us notice that $\text{H}^1(Y)^0$ is exactly the subspace of fixed points by $\mathbb{F}_{q^a}$. Recall that for any $v \in \mathbf{P}_q(a)$, any representative $\beta \in v$ we have $v = \{\beta, \beta^q, \beta^{q^2}, \dots, \beta^{|v|-1}\}$, and $|v| = d_v = [\mathbb{F}_v : \mathbb{F}_q]$.

**Lemma 3.6.** For any $\beta \neq 0$, $\text{H}^1(Y)^\beta$ is 2-dimensional. When $\beta = 0$, $\text{H}^1(Y)^0 = 0$. Moreover, for any $v \in \mathbf{P}_q(a)$, any $\beta \in v$, $\text{Fr}_q^{|v|}$ stabilizes $\text{H}^1(Y)^\beta$, and its 2 eigenvalues are $\kappa_1(v)$ and $\kappa_2(v)$.

*Proof.* The description of how $\text{Fr}_q$, and the action of $\mathbb{F}_{q^a}$ interact on $Y$ and a short computation show that for any $\beta \in \mathbb{F}_{q^a}$, $\text{Fr}_q(\text{H}^1(Y)^\beta) = \text{H}^1(Y)^{\beta^{1/q}}$. Thus, for any $v \in \mathbf{P}_q(a)$, the Frobenius $\text{Fr}_q$ cyclically permutes the subspaces $\text{H}^1(Y)^\beta$ for $\beta \in v$. Therefore, any $\text{H}^1(Y)^\beta$ is stabilized by $\text{Fr}_q^{|v|}$. Following [Kat81, §II], and using arguments of [vdGvdV91, §2], one can show that, for any $\beta \neq 0$

$$ \det \left( 1 - \text{Fr}_q^{|v|} T \middle| \text{H}^1(Y)^\beta \right) = 1 + \text{Kl}_{\mathbb{F}_v} (\psi_{\mathbb{F}_v, \beta}, 1) T + q^{d_v} T^2. $$

In particular $\dim \text{H}^1(Y)^\beta = 2$. Counting dimensions we deduce that $\dim \text{H}^1(Y)^0 = 0$. The two eigenvalues of $\text{Fr}_q^{|v|}$ on $\text{H}^1(Y)^\beta$ are algebraic integers whose product equals $q^{d_v}$ and their sum equals $\text{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta}, 1)$. Thanks to the Riemann hypothesis (see [Del80]), both of these algebraic integers have absolute value $\sqrt{q^{d_v}}$ in any complex embedding. These eigenvalues clearly satisfy relation (K|2), so they must be equal to $\kappa_1(v)$ and $\kappa_2(v)$ by uniqueness of the latter. $\square$

Using [Ulm07, Lemma 2.2] one computes, for any $v \in \mathbf{P}_q(a)$ and any representative $\beta_v \in v$:

$$ \det \left( 1 - \text{Fr}_q T \middle| \bigoplus_{\beta \in v} \text{H}^1(Y)^\beta \right) = \det \left( 1 - \text{Fr}_q^{|v|} T^{|v|} \middle| \text{H}^1(y)^{\beta_v} \right) = \left( 1 - \kappa_1(v) T^{|v|} \right) \left( 1 - \kappa_2(v) T^{|v|} \right). $$

Taking the product over the orbits we deduce

$$ \det(1 - \text{Fr}_q T | \text{H}^1(Y)) = \prod_{v \in P_q(a)} (1 - \kappa_1(v) T^{|v|}) (1 - \kappa_2(v) T^{|v|}). \qquad (3.5) $$

We turn now to the curve $X/\mathbb{F}_q$. Denote by $\mu_3(\overline{\mathbb{F}}_q)$ the set of third roots of unity in $\overline{\mathbb{F}}_q$. The group $G_{X,a} := \mu_3(\overline{\mathbb{F}}_q) \times \mathbb{F}_{q^a}$ acts on $X$: on the affine open subset of $X$ defined by equation (3.3), we set $(\zeta, \alpha) \cdot (u, t_1) = (\zeta u, t_1 + \alpha)$ for any $(\zeta, \alpha) \in G_{X,a}$, and $(u, t_1) \in X$. This action can be extended to the whole projective curve $X$, but it is not necessarily defined over $\mathbb{F}_q$: one checks that $\text{Fr}_q((\zeta, \alpha) \cdot (u, t_1)) = (\zeta^q, \alpha^q) \cdot \text{Fr}_q(u, t_1)$ for any $(u, t_1) \in X, (\zeta, \alpha) \in \mu_3(\overline{\mathbb{F}}_q) \times \mathbb{F}_{q^a}$. The only fixed point of $X$ under this action is $\infty_X$. By functoriality, $G_{X,a}$ acts on $\text{H}^1(X)$ and, again, we decompose this cohomology space into isotypical components:

$$ \text{H}^1(X) = \bigoplus_{\chi \in \widehat{G_{X,a}}} \text{H}^1(X)^\chi. \qquad (3.6) $$

Here $\chi$ runs through the characters of $G_{X,a}$ and $\text{H}^1(X)^\chi$ is the subspace of $\text{H}^1(X)$ on which $G_{X,a}$ acts by multiplication by $\chi$. The group $\widehat{G_{X,a}}$ is isomorphic to $\mathbb{Z}/3\mathbb{Z} \times \mathbb{F}_{q^a}$. Indeed, fix a non-trivial element $\zeta \in \mu_3(\overline{\mathbb{F}}_q)$, the map that sends $(j, \alpha) \mapsto (\zeta \mapsto (e^{2i\pi/3})^j, \psi_{\widehat{\mathbb{F}}_{q^a}, \alpha})$ establishes the desired isomorphism. We denote by $\text{H}^1(X)^{(j,\alpha)}$ the isotypical component corresponding to the character associated to $(j, \alpha)$. Recall the notation $\mathbb{I}_a = (\mathbb{Z}/3\mathbb{Z})^\times \times \mathbb{F}_{q^a}$ introduced above, and note that, for any $(j, \alpha) \in \mathbb{I}_a$, the character associated to $(j, \alpha)$ is non-trivial.
**Lemma 3.7.** For any $(j, \alpha) \in \mathbb{I}_a$ the subspace $H^1(X)^{(j,\alpha)}$ is one-dimensional, and $H^1(X)^{(j,\alpha)} = 0$ if $(j, \alpha) \in (\mathbb{Z}/3\mathbb{Z} \times \mathbb{F}_{q^a}) \setminus \mathbb{I}_a$. For any $o \in \mathbf{O}_q(a)$, any $(j, \alpha) \in o$, $\mathrm{Fr}_q^{|o|}$ stabilizes the line $H^1(X)^{(j,\alpha)}$ and its eigenvalue is the Gauss sum $\gamma(o)$.

*Proof.* This is a generalisation of [Kat81, Corollary 2.2] where Katz describes, when $q = p$, the cohomology of the curve $X$ and the eigenvalues of the Frobenius. $\square$

Using [Ulm07, Lemma 2.2] again, we have, for any $o \in \mathbf{O}_q(a)$ and any representative $(j_o, \alpha_o) \in o$:

$$ \det \left( 1 - \mathrm{Fr}_q T \left| \bigoplus_{(j,\alpha) \in o} \mathrm{H}^1(X)^{(j,\alpha)} \right. \right) = \det \left( 1 - \mathrm{Fr}_q^{|o|} T^{|o|} \left| \mathrm{H}^1(X)^{(j_o,\alpha_o)} \right. \right) = \left( 1 - \gamma(o) T^{|o|} \right), $$

Therefore, we conclude that

$$ \det(1 - \mathrm{Fr}_q T |\mathrm{H}^1(X)) = \prod_{o \in \mathbf{O}_q(a)} (1 - \gamma(o) T^{|o|}). \qquad (3.7) $$

## 3.4 Geometry of the minimal regular model

In this section we give a more thorough analysis of the geometry of the minimal regular model $\mathcal{C}$. We now show that $\mathcal{C}$ is dominated by the product of curves $X \times Y$, thus proving the claim in Section 2.1. Recall from Section 2.1 that there is a rational map

$$ \vartheta_0: \left\{ \begin{array}{lclcl} X \times Y & \dashrightarrow & \mathcal{Z} & & \\ (u, t_1, v, t_2) & \mapsto & (u, v, t_2 - t_1), & & \end{array} \right. $$

to the surface $\mathcal{Z}/\mathbb{F}_q$ defined by (2.2).

**Proposition 3.8.** The map $\vartheta_0$ induces a birational map $\vartheta_1 : (X \times Y)/\mathbb{F}_{q^a} \xrightarrow{\sim} \mathcal{Z}$. In particular, the minimal regular model $\mathcal{C}$ of $\mathcal{C}$, as a surface over $\mathbb{F}_q$, is dominated by $X \times Y$.

*Proof.* We saw above that the group $\mathbb{F}_{q^a}$ acts on both of the curves $X$ and $Y$, so that the product $X \times Y$ also carries a “diagonal” action of $\mathbb{F}_{q^a}$. This action is given in affine coordinates by $\alpha \cdot (u, t_1, v, t_2) = (u, t_1 + \alpha, v, t_2 + \alpha)$ for any $\alpha \in \mathbb{F}_{q^a}$, any $(u, t_1, v, t_2) \in X \times Y$. Its only fixed points are $(\infty_X, \infty_{Y_i})$, $i = 1, 2$. If $(u, t_1, v, t_2) \in X \times Y$ and $\alpha \in \mathbb{F}_{q^a}$, then $\vartheta_0(\alpha \cdot (u, t_1, v, t_2)) = \vartheta_0(u, t_1, v, t_2)$. Thus, $\vartheta_0$ factors through the quotient $(X \times Y)/\mathbb{F}_{q^a}$ and gives a rational map $\vartheta_1 : (X \times Y)/\mathbb{F}_{q^a} \dashrightarrow \mathcal{Z}$. Counting elements in the fibers, one checks that both $\vartheta_0$ and the quotient map $X \times Y \to (X \times Y)/\mathbb{F}_{q^a}$ are of degree $q^a$. We deduce that $\vartheta_1$ has degree 1, hence is a birational map. To prove the last statement, recall that both $\mathcal{Z}$ and $\mathcal{C}$ are models of $\mathcal{C}$ over $\mathbb{P}_{\mathbb{F}_q}^1$, so there exists a birational map $\mathcal{Z} \dashrightarrow \mathcal{C}$. The composition

$$ X \times Y \dashrightarrow (X \times Y)/\mathbb{F}_{q^a} \xrightarrow{\sim} \mathcal{Z} \xrightarrow{\sim} \mathcal{C} $$

is dominant, finally giving the desired result. $\square$

In order to compute the $L$-function of $\mathcal{S}$, we need to understand the indeterminacy of $\vartheta_1$ better. Following [PU16, Proposition 3.1.5], one constructs by successive blow-ups a surface $\mathcal{X}/\mathbb{F}_q$ and a proper birational morphism $\mathrm{res}_1 : \mathcal{X} \to X \times Y$ resolving the indeterminacy of both $\delta_0 : X \times Y \dashrightarrow \mathbb{P}_{\mathbb{F}_q}^1$, $(u, t_1, v, t_2) \mapsto t_2 - t_1$ and $\vartheta_0 : X \times Y \dashrightarrow \mathcal{Z}$. This construction gives morphisms $\delta : \mathcal{X} \to \mathbb{P}_{\mathbb{F}_q}^1$ and $\vartheta : \mathcal{X} \to \mathcal{Z}$ such that the diagram below commutes:

<figure><img src="image_6.png" /></figure>
The diagonal action of $\mathbb{F}_{q^a}$ on $X \times Y$ lifts uniquely to $\mathcal{X}$ and fixes the exceptional divisor pointwise. Thus the morphism $\vartheta: \mathcal{X} \to \mathcal{Z}$ factors through the quotient $\mathcal{X}/\mathbb{F}_{q^a}$. The latter surface might be singular. Again, resolving singularities (see [Liu02, §8.3.4]), we let $\mathrm{res}_2: \tilde{\mathcal{X}} \to \mathcal{X}/\mathbb{F}_{q^a}$ be a proper birational morphism such that $\tilde{\mathcal{X}}$ is a smooth projective surface and the composition $\mathcal{X}/\mathbb{F}_{q^a} \to \mathcal{Z} \dasharrow C$ induces a morphism $\tilde{\mathcal{X}} \to C$. We summarize the situation in the commutative diagram below:

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

We now describe how exceptional divisors behave under the maps in diagram (3.8). Recall that $C$ has two points at infinity, denoted by $\infty_1, \infty_2$, which are $\mathbb{K}$-rational.

**Lemma 3.9.** Let $s_{\infty_i}: \mathbb{P}_{\mathbb{F}_q}^1 \to C, i \in \{1, 2\}$, denote the sections of $C \to \mathbb{P}_{\mathbb{F}_q}^1$ corresponding to the points at infinity on $C/\mathbb{K}$.

i) The strict transforms of $\infty_X \times Y$ and $X \times \infty_Y$ in $\tilde{\mathcal{X}}$ map to $C_\infty$.

ii) Let $\mathcal{E}$ be the image in $\mathcal{X}/\mathbb{F}_{q^a}$ of the exceptional divisor of $\mathcal{X} \to X \times Y$. Then the strict transforms of the components of $\mathcal{E}$ in $\tilde{\mathcal{X}}$ map either to $C_\infty$ or to the image of the infinity sections $s_{\infty_i}$.

iii) Every component of the exceptional divisor of $\tilde{\mathcal{X}} \to \mathcal{X}/\mathbb{F}_{q^a}$ maps to a fiber of $C \to \mathbb{P}_{\mathbb{F}_q}^1$.

*Proof.* The formula defining $\delta_0$ shows that for any $y \in Y(\overline{\mathbb{F}_q}) \setminus \{\infty_Y\}$, $\delta_0$ maps $(\infty_X, y)$ to the point at infinity in $\mathbb{P}_{\mathbb{F}_q}^1$. In the same way, for $x \in X(\overline{\mathbb{F}_q}) \setminus \{\infty_X\}$, we have $\delta_0(x, \infty_Y) = \infty$. Commutativity of diagram (3.8) gives the desired result. The second statement is simply Remark 3.1.6 in [PU16]. Finally, to prove the third point, notice that any component $\Gamma$ of the exceptional divisor of $\tilde{\mathcal{X}} \to \mathcal{X}/\mathbb{F}_{q^a}$ lies over a unique point $P \in (\mathcal{X}/\mathbb{F}_{q^a})(\overline{\mathbb{F}_q})$. If $t \in \mathbb{P}_{\mathbb{F}_q}^1(\overline{\mathbb{F}_q})$ is the image of $P$ by the composition $\mathcal{X}/\mathbb{F}_{q^a} \to \mathcal{Z} \to \mathbb{P}_{\mathbb{F}_q}^1$, then $\tilde{\mathcal{X}} \to C$ maps $\Gamma$ to the fiber $C_t$. $\square$

# 4 The L-function of S

In this section we use the geometric analysis of $C$ from the previous section to show that $S_a$ satisfies the BSD conjecture. We then provide an explicit formula for the L-function of our abelian surface $S$ and conclude by computing the Mordell-Weil rank and the special value $L^*(S)$.

## 4.1 Definition of $L(S, T)$ and statement of the BSD conjecture

Recall that in Section 3.3 we introduced the $\ell$-adic étale cohomology functors $H_{\ell t}^i$ with coefficients in $\overline{\mathbb{Q}}_\ell$. Let us write also $H^1(S) := H_{\ell t}^1(S \times_\mathbb{K} \mathbb{K}^{\mathrm{sep}}, \overline{\mathbb{Q}}_\ell)$ and $H^1(C) := H_{\ell t}^1(C \times_\mathbb{K} \mathbb{K}^{\mathrm{sep}}, \overline{\mathbb{Q}}_\ell)$. It is well-known that we have an isomorphism $H^1(S) \simeq H^1(C)$ (see for example [Mil86, Corollary 9.6]). This is a $\overline{\mathbb{Q}}_\ell$-vector space of dimension 4 on which the absolute Galois group $\mathrm{Gal}(\mathbb{K}^{\mathrm{sep}}/\mathbb{K})$ acts. For any place $v$ of $\mathbb{K}$ corresponding to a geometric point of $\mathbb{P}_{\mathbb{F}_q}^1$ (also denoted by $v$), recall that $I_v$ denotes the inertia subgroup at $v$ and let $H^1(S)_{I_v}$ be the subspace of $H^1(S)$ fixed by inertia. Moreover, we denote by $\mathrm{Fr}_v$ the geometric Frobenius at $v$, induced by $x \mapsto x^{q^{d_v}}$.

**Definition 4.1.** The Hasse-Weil L-function of $S$ is defined by the Euler-product

$$L(S, T) = \prod_{v \in M_\mathbb{K}} \det(1 - \mathrm{Fr}_v T^{d_v} | \mathrm{H}^1(S)_{I_v})^{-1}.$$
The L-function can be expanded into a formal power series in T. The Hasse-Weil bound on the eigenvalues of Fr_v acting on H¹(S) implies that this power series converges on the complex disc {T ∈ C : |T| < q⁻³/²}. The theory of étale cohomology developed by Grothendieck et. al. (see [Mil80]) and the work of Deligne on the Weil conjectures ([Del80]) led to the following theorem, describing the main properties of the L-function of S.

**Theorem 4.2.** Let S/FFFFFFq(t) be as above and N(S) ∈ Div(FFFFFFq) be its conductor divisor.

i) (Rationality) The L-function L(S, T) is a polynomial in T with integer coefficients. Writing b(S) for its degree, we have the equality b(S) = deg N(S) - 8.

ii) (Functional equation) There exists w(S) ∈ {±1} such that L(S, T) satisfies

$$L(S, T) = w(S) (qT)^{b(S)} L(S, (q^2T)^{-1}).$$

iii) (Riemann Hypothesis) If z ∈ C is such that L(S, z) = 0, then |z| = q⁻¹.

The reader can find a sketch of the proof in [Ulm14, §6.2.1].

**Remark 4.3.** i) In general, the L-function of an abelian variety over $K$ is a rational function in T with integral coefficients. Since S is non-isotrivial, [Ulm14, Lemma 6.2.2] implies that L(S, T) is actually a polynomial in T.

ii) We can define a function of the complex variable s ↦ L(S, q⁻s), which converges in the right half-plane Re(s) > 3/2 because of the Hasse-Weil bound. The functional equation ii) shows that it extends to a holomorphic function on the whole complex plane. The third point in Theorem 4.2 is called the Riemann hypothesis because it states that all the zeros of s ↦ L(S, q⁻s) lie on the vertical line Re(s) = 1.

The special value L*(S) of the L-function of S at q⁻¹ is defined as

$$L^*(S) := \frac{L(S, T)}{(1 - qT)^{\mathrm{rk}_{\mathrm{an}}(S)}} \bigg|_{T = q^{-1}} \quad \text{where } \mathrm{rk}_{\mathrm{an}}(S) = \mathrm{ord}_{T = q^{-1}} L(S, T).$$

The integer rk\_an(S) is called the analytic rank of S. The next statement is called the Birch and Swinnerton-Dyer conjecture (BSD): it is conjectured to be true for any abelian variety over $K$, and remains an open question in general. We refer the reader to [Tat68] for a survey. In [PU16], the authors give a recipe to construct sequences of abelian varieties over function fields satisfying the BSD conjecture. As already mentioned in Section 3.4, the construction of our curve C and its Jacobian S follows that of Pries and Ulmer, to ensure that S has the desired properties.

**Theorem 4.4.** The abelian surface S satisfies the BSD conjecture, i.e.,

* the algebraic and analytic ranks of S coincide, that is, rk S(K) = rk\_an(S).

* The group III(S/K) is finite.

* The BSD formula holds:

$$L^*(S) = \frac{|\mathrm{III}(S/K)| \mathrm{Reg}(S) q^2 \prod_v c_v(S)}{|S(\mathbb{K})_{\mathrm{tor}}|^2 H(S)} \quad (4.1)$$

*Proof.* Kato and Trihan have proved that the three statements are equivalent ([KT03]). Thus it suffices to establish the first point, called the “weak BSD conjecture”. We relate this to Tate’s conjecture for C. Tate’s conjecture for a variety V defined over $FFFFFFq$ relates the rank of the Néron- Severi group NS(V) to the order of the pole of the zeta function of V at s = 1 (see [Ulm14, §6] for precise statements). Now Tate’s conjecture is known to be true for curves, and Tate’s theorem on homomorphisms of abelian varieties implies that products of curves satisfy Tate’s conjecture. Moreover, the latter is preserved by domination (see [Ulm11, §2.11 & §2.12]).

In Proposition 3.8 we saw that the minimal regular model C/FFFFFFq is dominated by the product of curves X × Y, and the discussion above implies that C satisfies Tate’s conjecture. But Shioda-Tate theorem implies that Tate’s conjecture for C is equivalent to the BSD conjecture for S, hence the result (the reader can find detailed proofs in [Ulm14, §4.2, §6.2.2]). □
## 4.2 Explicit expression for $L(S, T)$

Let $a \ge 1$, recall from Section 3.2 that $O_q(a)$ denotes the set of orbits of $(\mathbb{Z}/3\mathbb{Z})^\times \times \mathbb{F}_{q^a}^\times$ under the action of $\langle q \rangle$. Moreover, to any $o \in O_q(a)$, we have attached character sums $\gamma(o)$ and $\text{Kl}(o)$.

**Theorem 4.5.** The L-function of S is given by the formula

$$L(S, T) = \prod_{o \in O_q(a)} \left(1 - \gamma(o) \text{Kl}(o) T^{|o|} + \gamma(o)^2 q^{|o|} T^{2|o|}\right). \quad (4.2)$$

We devote the rest of section 4 to the proof of theorem 4.5.

## 4.3 Cohomological computation of $L(S, T)$

We begin by relating $L(S, T)$ to the zeta function of $\mathcal{C}/\mathbb{F}_q$. Since we will be manipulating zeta functions, we start by recalling their definition and some of their basic facts (see [Ser70] for details).

**Definition 4.6.** Let $\mathbb{F}/\mathbb{F}_q$ be a finite extension. For a variety $V$ over $\mathbb{F}$, define its zeta function as

$$Z(V, T) = \prod_{P \in V} (1 - T^{\text{deg } P})^{-1},$$

where the product runs over the set of closed points of $V$.

As an example, one can compute $Z(\mathbb{P}_q^1, T) = ((1-T)(1-qT))^{-1}$. When $V$ is smooth and projective, the Grothendieck-Lefschetz trace formula implies the rationality of $Z(V, T)$ (see [Mil80, Chapter VI, Theorem 12.4])

$$Z(V, T) = \prod_{i=0}^{2 \dim V} (-1)^{i+1} \det \left( 1 - \mathrm{Fr}_q^{[\mathbb{F}:\mathbb{F}_q]} T \mid \mathrm{H}^i(V) \right). \quad (4.3)$$

Besides, the Frobenius $\mathrm{Fr}_q$ acts as the identity in $V$, so we know its action on $\mathrm{H}^0(V)$. Using Poincaré duality ([Mil80, Theorem 11.1]) we can also deduce its action on $\mathrm{H}^{2 \dim V}(V)$:

$$\det(1 - \mathrm{Fr}_q^{[\mathbb{F}:\mathbb{F}_q]} T \mid \mathrm{H}^0(V)) = 1 - T \quad \text{and} \quad \det(1 - \mathrm{Fr}_q^{[\mathbb{F}:\mathbb{F}_q]} T \mid \mathrm{H}^{2 \dim V}(V)) = 1 - q^{[\mathbb{F}:\mathbb{F}_q]} \dim V T.$$

**Proposition 4.7.** We have the equality

$$L(S, T) = \frac{\det(1 - \mathrm{Fr}_q T \mid \mathrm{H}^2(\mathcal{C}))}{\det(1 - \mathrm{Fr}_q T \mid \mathrm{H}^1(\mathcal{C})) \det(1 - \mathrm{Fr}_q T \mid \mathrm{H}^3(\mathcal{C})) (1 - qT)^{6q^a+8}} \quad (4.4)$$

*Proof.* Formula (4.3) and the discussion below applied to the smooth surface $\mathcal{C}/\mathbb{F}_q$ give

$$Z(\mathcal{C}, T) = \frac{\det(1 - \mathrm{Fr}_q T \mid \mathrm{H}^1(\mathcal{C})) \det(1 - \mathrm{Fr}_q T \mid \mathrm{H}^3(\mathcal{C}))}{(1 - T) \det(1 - \mathrm{Fr}_q T \mid \mathrm{H}^2(\mathcal{C})) (1 - q^2 T)} \quad (4.5)$$

By construction $\mathcal{C}$, as a model of $\mathcal{C}/\mathbb{K}$, is equipped with a surjective morphism $\mathcal{C} \to \mathbb{P}_{\mathbb{F}_q}^1$. We therefore decompose $\mathcal{C}$ as a disjoint union of fibers of closed points of $\mathbb{P}_{\mathbb{F}_q}^1$ (see [Mil80, §VI, 13.8]). Identifying closed points of $\mathbb{P}_{\mathbb{F}_q}^1$ and places of $\mathbb{K}$, we rewrite $Z(\mathcal{C}, T) = \prod_{v \in M_{\mathbb{K}}} Z(C_v, T)$. Splitting the product into places of good and bad reduction for $\mathcal{C}$, and using (4.3) for the smooth fibers $C_v/\mathbb{F}_v$ when $v$ is a place of good reduction we get:

$$Z(\mathcal{C}, T) = \prod_{v \text{ good}} \frac{\det(1 - \mathrm{Fr}_v T^{d_v} \mid \mathrm{H}^1(C_v))}{(1 - T^{d_v})(1 - (qT)^{d_v})} \prod_{v \text{ bad}} Z(C_v, T). \quad (4.6)$$

Let $v \in M_{\mathbb{K}}$ be a place of good reduction. As stated above, we have $\mathrm{H}^1(S) \simeq \mathrm{H}^1(C)$ and the Néron-Ogg-Shafarevich criterion (cf. [ST68]) gives $\mathrm{H}^1(S)^{I_v} \simeq \mathrm{H}^1(S)$. Moreover, the smooth base change theorem ([Mil80, Chapter VI, Corollary 4.2]) implies the isomorphism $\mathrm{H}^1(C) \simeq \mathrm{H}^1(C_v)$. Therefore we have $\mathrm{H}^1(C_v) \simeq \mathrm{H}^1(S)^{I_v}$ and the local factor of $L(S, T)$ at $v$ is $\det(1 - \mathrm{Fr}_v T^{d_v} \mid \mathrm{H}^1(C_v))^{-1}$.
Let now $v$ be a finite place of bad reduction for $C$, we relate $Z(C_v, T)$ to the zeta function of the normalization of $C_v$. From Proposition 2.8 we know that this is the elliptic curve $E_v/\mathbb{F}_v$ described in affine coordinates by (2.9). [AP96] describes $Z(C_v, T)$ in terms of $Z(E_v, T)$, the number and degree of preimages of singular points of $C_v$ by the normalization morphism $\nu_v : E_v \to C_v$. The only singular point in $C_v$ is $(0,0)$, which is $\mathbb{F}_v$-rational. Moreover, $\nu_v$ is one-to-one as the preimage of the singular point of $C_v$ is $\nu_v^{-1}(0,0) = (0,0)$, which is again $\mathbb{F}_v$-rational. Hence, [AP96, Theorem 2.1] gives

$$Z(C_v, T) = Z(E_v, T) = \frac{\det(1 - \mathrm{Fr}_v T^{d_v} | \mathrm{H}^1(E_v))}{(1 - T^{d_v})(1 - (qT)^{d_v})}.$$

Using the universal property of Néron models, one can show that $\mathrm{H}^1(S)^{I_v} \simeq \mathrm{H}^1(S_v^0)$. From Theorem 2.11 we know that the reduction $S_v^0$ is the extension of $E_v$ by a unipotent group of dimension 1, which has trivial $\ell$-adic cohomology, as $\ell \neq p$. Therefore we have $\mathrm{H}^1(S)^{I_v} \simeq \mathrm{H}^1(E_v)$ and the local factor of $L(S, T)$ at $v$ is $\det(1 - \mathrm{Fr}_v T^{d_v} | \mathrm{H}^1(E_v))^{-1}$.

Plugging all this into (4.6) we have

$$Z(C, T) = \left( \prod_{v \text{ good}} \frac{\det(1 - \mathrm{Fr}_v T^{d_v} | \mathrm{H}^1(S)^{I_v})}{(1 - T^{d_v})(1 - (qT)^{d_v})} \right) \left( \prod_{v \text{ finite bad}} \frac{\det(1 - \mathrm{Fr}_v T^{d_v} | \mathrm{H}^1(S)^{I_v})}{(1 - T^{d_v})(1 - (qT)^{d_v})} \right) Z(C_\infty, T).$$

At infinity, $S^0$ has totally unipotent reduction (cf.2.11), so $\mathrm{H}^1(S)^{I_v}$ is trivial and the local factor of $L(S, T)$ at $\infty$ is 1. Regrouping the numerators of the terms inside the parenthesis above, we recognize the inverse of $L(S, T)$. On the other hand, regrouping the denominators inside the parenthesis, one recognizes two zeta functions of $\mathbb{P}_{\mathbb{F}_q}^1$:

$$\prod_{v \neq \infty} ((1 - T^{d_v})(1 - (qT)^{d_v}))^{-1} = Z(\mathbb{P}_{\mathbb{F}_q}^1, T) Z(\mathbb{P}_{\mathbb{F}_q}^1, qT) (1 - T)(1 - qT).$$

The last two factors correspond to the place at infinity that was missing from the left-hand side. Upon using the explicit expression for $Z(\mathbb{P}_{\mathbb{F}_q}^1, T)$, we can rewrite

$$Z(C, T) = L(S, T)^{-1} \frac{Z(C_\infty, T)}{(1 - qT)(1 - q^2T)}. \qquad (4.7)$$

We now only have to compute $Z(C_\infty, T)$. Recall from Proposition 2.8 (see Figure 1) that the fiber $C_\infty$ is a tree of $\mathbb{P}^1$'s, which has $6q^a + 7$ irreducible components, all $\mathbb{F}_q$-rational. All singularities on $C_\infty$ are nodes, so by a direct point counting argument, we have

$$Z(C_\infty, T) = \frac{Z(\mathbb{P}_{\mathbb{F}_q}^1, T)^{6q^a+7}}{Z(\mathrm{Spec}(\mathbb{F}_q), T)^{6q^a+6}} = \frac{1}{(1-T)(1-qT)^{6q^a+7}}. \qquad (4.8)$$

Plugging (4.8) into (4.7), combining it with (4.5), and rearranging terms gives the desired result. $\square$

Some of the terms in formula (4.4) are easily described:

**Proposition 4.8.** The first and third cohomology groups of $C$ are trivial. Therefore we have

$$\det(1 - \mathrm{Fr}_q T | \mathrm{H}^1(C)) = \det(1 - \mathrm{Fr}_q T | \mathrm{H}^3(C)) = 1.$$

*Proof.* Recall from Section 3.4 that the morphism $\tilde{\chi} \to C$ is dominant, so proving that $\mathrm{H}^1(\tilde{\chi}) = 0$ would imply that $\mathrm{H}^1(C)$ is trivial too. Recall from Section 3.4 that $\tilde{\chi}$ is obtained from the quotient $\chi/\mathbb{F}_{q^a}$ by a succession of blow-ups at closed points of $\chi/\mathbb{F}_{q^a}$ and at the image of the exceptional divisor $\mathcal{E}$ (which has trivial $\mathrm{H}^1$ because it is the image of a union of $\mathbb{P}^1$'s). Thus $\mathrm{H}^1(\tilde{\chi})$ decomposes as the direct sum of $\mathrm{H}^1(\chi/\mathbb{F}_{q^a})$ and a certain number of copies of $\mathrm{H}^1(\mathbb{P}^1)$, all of which are trivial. We deduce the isomorphisms

$$\mathrm{H}^1(\tilde{\chi}) \simeq \mathrm{H}^1(\chi/\mathbb{F}_{q^a}) \simeq \mathrm{H}^1(\chi)^{\mathbb{F}_{q^a}},$$
where the upper-script denotes the $\mathbb{F}_{q^a}$-invariant subspace. As explained in Section 3.4, $\mathcal{X}$ is obtained from $X \times Y$ by a series of blow-ups, so that, as above, $H^1(\mathcal{X}) \simeq H^1(X \times Y)$. Applying the Künneth formula ([Mil80, hapter VI, Theorem 8.5]), we have

$$ H^1(\tilde{\mathcal{X}}) \simeq H^1(X \times Y)^{\mathbb{F}_{q^a}} \simeq (H^0(X) \otimes H^1(Y) \oplus H^1(X) \otimes H^0(Y))^{\mathbb{F}_{q^a}} \\ \simeq H^0(X)^{\mathbb{F}_{q^a}} \otimes H^1(Y)^{\mathbb{F}_{q^a}} \oplus H^1(X)^{\mathbb{F}_{q^a}} \otimes H^0(Y)^{\mathbb{F}_{q^a}}. $$

Recall from Lemmas 3.6, 3.7 that $H^1(X)^{\mathbb{F}_{q^a}}$ and $H^1(Y)^{\mathbb{F}_{q^a}}$ are both trivial, so $H^1(\tilde{\mathcal{X}}) = 0$. Therefore $H^1(C) = 0$, and, by Poincaré duality ([Mil80, Corollary 11.2]), that $H^3(C) = 0$ as well. $\square$

The vanishing of the first and third cohomology groups of $C$ clarifies the formula (4.4) of the $L$-function of $S$. We now further simplify the expression of $L(S, T)$. Recall that the Néron-Severi group $\text{NS}(C)$ is defined as the group of divisors of $C$ modulo algebraic equivalence (see [Mil80, Chapter III]). We will denote by $[\Gamma]$ the class in $\text{NS}(C)$ of a divisor $\Gamma \in \text{Div}(C)$. Following [Shi92], we define the trivial lattice $T \subset \text{NS}(C)$ as the sublattice generated by the class of the infinity section $[s_{\infty1}]$ and the classes of the irreducible components of the fibers of $C \to \mathbb{P}_{\mathbb{F}_q}^1$ (we identify the section $s_{\infty1}$ and its image in $\text{Div}(C)$). We write $\gamma_C : \text{NS}(C) \otimes \mathbb{Z}_\ell \to \text{H}^2(C)$ for the map induced by the Kummer exact sequence, and we define the subspace $\Lambda := \gamma_C(T) \subset \text{H}^2(C)$.

**Lemma 4.9.** We have

$$ L(S, T) = \det(1 - \text{Fr}_q T \mid \text{H}^2(C)/\Lambda). $$

*Proof.* Recall from Lemma (2.8) that all fibers of $C \to \mathbb{P}_{\mathbb{F}_q}^1$ at finite places are irreducible, and they are all algebraically equivalent. Fix a finite place $v$. $T$ is generated by $[s_{\infty1}]$, $[C_v]$, and $([\Gamma_i])_i$ where the $\Gamma_i$'s are the irreducible components of $C_\infty$ not meeting $s_{\infty1}$. As Figure 1 in Proposition 2.8 shows, there are $6q^a + 6$ of these irreducible components, that are all $\mathbb{F}_q$-rational. Using the intersection pairing, one sees that $T$ is not spanned by any subfamily of these $6q^a + 8$ classes of divisors, and so $\Lambda \subset \text{H}^2(C)$ has dimension $6q^a + 8$. The Frobenius acts by multiplication by $q$ on $\gamma_C([s_{\infty1}]), \gamma_C([C_v])$ and each of the $\gamma_C([\Gamma_i])$, so we obtain

$$ \det(1 - \text{Fr}_q T \mid \Lambda) = (1 - qT)^{6q^a + 8}. $$

But we have $\det(1 - \text{Fr}_q T \mid \text{H}^2(C)/\Lambda) = \det(1 - \text{Fr}_q T \mid \text{H}^2(C))/\det(1 - \text{Fr}_q T \mid \Lambda)$, so combining the discussion above with (4.4) and Proposition 4.8 gives the desired result. $\square$

We have now all the necessary tools to prove Theorem 4.5.

*Proof of Theorem 4.5.* Given Lemma 4.9, we need to compute the characteristic polynomial of $\text{Fr}_q$ on $\text{H}^2(C)/\Lambda$. We start by relating $\text{H}^2(C)/\Lambda$ to the cohomology groups of the auxiliary curves $X$ and $Y$. Since the morphism $\tilde{\mathcal{X}} \to C$ is dominant, the map $\text{H}^2(\tilde{\mathcal{X}}) \to \text{H}^2(C)$ is surjective. Consider the subspace $B \subset \text{H}^2(\tilde{\mathcal{X}})$ spanned by the classes of the components of the exceptional divisor of $\tilde{\mathcal{X}} \to X/\mathbb{F}_{q^a}$ and the strict transform of $\mathcal{E}$ in $\mathcal{X}$. Then

$$ \begin{aligned} \text{H}^2(\tilde{\mathcal{X}}) &\simeq \text{H}^2(\mathcal{X}/\mathbb{F}_{q^a}) \oplus B \\ &\simeq \text{H}^2(\mathcal{X})^{\mathbb{F}_{q^a}} \oplus B \\ &\simeq \text{H}^2(X \times Y)^{\mathbb{F}_{q^a}} \oplus B \\ &\simeq (\text{H}^0(X) \otimes \text{H}^2(Y))^{\mathbb{F}_{q^a}} \oplus (\text{H}^1(X) \otimes \text{H}^1(Y))^{\mathbb{F}_{q^a}} \oplus (\text{H}^2(X) \otimes \text{H}^0(Y))^{\mathbb{F}_{q^a}} \oplus B, \end{aligned} $$

where the last isomorphism follows from the Künneth formula (see [Mil80]). Note that all the isomorphisms above are compatible with the action of $\text{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$. Now $\text{H}^0(X) \otimes \text{H}^2(Y)$ and $\text{H}^2(X) \otimes \text{H}^0(Y)$ are respectively generated by the strict transforms of $X \times \infty_Y$ and $\infty_X \times Y$ in $\tilde{\mathcal{X}}$. Lemma 3.9 implies that $\text{H}^0(X) \otimes \text{H}^2(Y)$, $\text{H}^2(X) \otimes \text{H}^0(Y)$, and $B$ all map to the trivial lattice under $\text{H}^2(\tilde{\mathcal{X}}) \to \text{H}^2(C)$. We deduce from this the existence of a surjective map

$$ \Upsilon : (\text{H}^1(X) \otimes \text{H}^1(Y))^{\mathbb{F}_{q^a}} \to \text{H}^2(C)/\Lambda. $$
Moreover, $\Upsilon$ is Gal($\overline{\mathbb{F}}_q/\mathbb{F}_q$)-equivariant because all the isomorphisms above are compatible with the action of Gal($\overline{\mathbb{F}}_q/\mathbb{F}_q$). We further describe the left-hand side. Recall from Section 3.3 that the cohomology groups $H^1(X)$ and $H^1(Y)$ decompose into isotypical components: (3.4) and (3.6) yield

$$H^1(X) \otimes H^1(Y) = \bigoplus_{(j,\alpha) \in \mathbb{I}_a} \bigoplus_{\beta \in \mathbb{F}_{q^a}^\times} H^1(X)^{(j,\alpha)} \otimes H^1(Y)^{\beta}.$$

The group $\mathbb{F}_{q^a}$ acts on $H^1(X)$ (resp. $H^1(Y)$) by multiplication by the character $\chi$ associated to $(j, \alpha)$ (resp. $\psi_{\mathbb{F}_{q^a}, \beta}$). Thus, among all the planes $H^1(X)^{(j,\alpha)} \otimes H^1(Y)^{\beta}$, the $\mathbb{F}_{q^a}$-invariant ones are those for which $\alpha = \beta$. We get

$$(H^1(X) \otimes H^1(Y))^{\mathbb{F}_{q^a}} = \bigoplus_{\substack{(j,\alpha) \in \mathbb{I}_a \\ \beta \in \mathbb{F}_{q^a}^\times}} (H^1(X)^{(j,\alpha)} \otimes H^1(Y)^{\beta})^{\mathbb{F}_{q^a}} = \bigoplus_{(j,\alpha) \in \mathbb{I}_a} H^1(X)^{(j,\alpha)} \otimes H^1(Y)^{\alpha}.$$

This space has therefore dimension $2|\mathbb{I}_a| = 4(q^a - 1)$. Because of Lemma 4.9, the dimension of $H^2(\mathcal{C})/\Lambda$ equals the degree of $L(S, T)$. Now the Grothendieck-Ogg-Shafarevich formula ([Mil80, Chapter V, Theorem 2.12]) implies that $\deg L(S, T) = \deg N_S - 8 = 4(q^a - 1)$. By equality of dimensions, $\Upsilon$ is actually an isomorphism. We deduce that

$$L(S, T) = \det \left( 1 - \mathrm{Fr}_q T \left| \left( H^1(X) \otimes H^1(Y) \right)^{\mathbb{F}_{q^a}} \right) \right).$$

For any $o \in \mathbf{O}_q(a)$ and any choice of representative $(j_o, \alpha_o) \in o$, the $|o|$-th iterate of $\mathrm{Fr}_q$ stabilizes the plane $H^1(X)^{(j_o, \alpha_o)} \otimes H^1(Y)^{\alpha_o}$ (see Lemmas 3.6 and 3.7). The computation of the eigenvalues of $\mathrm{Fr}_q^{|o|}$ done in Section 3.3 gives

$$\det \left( 1 - \mathrm{Fr}_q^{|o|} T \left| \left( H^1(X)^{(j_o, \alpha_o)} \otimes H^1(Y)^{\alpha_o} \right) \right) \right) = (1 - \gamma(o) \kappa_1(o) T)(1 - \gamma(o) \kappa_2(o) T).$$

Applying [Ulm07, Lemma 2.2] again, we deduce

$$\det \left( 1 - \mathrm{Fr}_q T \left| \left( \bigoplus_{(j,\alpha) \in o} H^1(X)^{(j,\alpha)} \otimes H^1(Y)^{\alpha} \right) \right) \right) = (1 - \gamma(o) \kappa_1(o) T^{|o|}) (1 - \gamma(o) \kappa_2(o) T^{|o|}).$$

Finally, developing the right-hand-side and taking the products over all $o \in \mathbf{O}_q(a)$ we conclude

$$L(S, T) = \prod_{o \in \mathbf{O}_q(a)} \left( 1 - \gamma(o) \mathrm{Kl}(o) T^{|o|} + \gamma(o)^2 q^{|o|} T^{2|o|} \right).$$

□

## 4.4 Special value and rank of S

From now on we fix a complex embedding $\iota: \overline{\mathbb{Q}} \hookrightarrow \mathbb{C}$. Thanks to (Ga 1) and (Kl 3) we know that, for any $o \in \mathbf{O}_q(a)$, $|\iota(\kappa_1(o))| = |\iota(\kappa_2(o))| = |\iota(\gamma(o))| = q^{|o|/2}$ and this implies that $\iota(\kappa_1(o))$ and $\iota(\kappa_2(o))$ are complex conjugate.

**Definition 4.10.** Let $o \in \mathbf{O}_q(a)$, we define the Gauss and Kloosterman angles attached to $o$ as the unique elements $\varepsilon_o \in [0, 2\pi)$ and $\theta_o \in [0, \pi]$ such that

$$\iota(\gamma(o)) = q^{|o|/2} e^{i \varepsilon_o} \quad \text{and} \quad \iota(\mathrm{Kl}(o)) = 2 q^{|o|/2} \cos(\theta_o). \qquad (4.9)$$

Up to relabelling $\kappa_1(o)$ and $\kappa_2(o)$, we may assume that $\theta_o$ is the argument of $\kappa_1(o)$ and $-\theta_o$ is the argument of $\kappa_2(o)$, i.e., $\iota(\kappa_1(o)) = q^{|o|/2} e^{i\theta_o}$ and $\iota(\kappa_2(o)) = q^{|o|/2} e^{-i\theta_o}$.

We describe now the analytic rank of $S_a$ and the special value of $L(S, T)$ in terms of these angles.

**Theorem 4.11.** The order of vanishing of $L(S, T)$ at $T = q^{-1}$ is 0. The special value of $L(S, T)$ equals

$$L^*(S) = \prod_{o \in \mathbf{O}_q(a)} 4 \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|. \qquad (4.10)$$
Proof. Using the notation introduced in Definition 4.10 we rewrite the result of Theorem 4.5:

$$L(S, T) = \prod_{o \in \mathbf{O}_q(a)} \left(1 - e^{i(\varepsilon_o + \theta_o)} (qT)^{|o|}\right) \left(1 - e^{i(\varepsilon_o - \theta_o)} (qT)^{|o|}\right). \quad (4.11)$$

For any $o \in \mathbf{O}_q(a)$, (K14) implies that $\{\mathrm{ord}_p(\kappa_1(o)), \mathrm{ord}_p(\kappa_2(o))\} = \{0, |o|\}$ and Lemma 3.5 gives $\mathrm{ord}_p(\gamma(o)) \in \left\{\frac{|o|}{3}, \frac{|o|}{2}, \frac{2|o|}{3}\right\}$. Thus for $i = 1, 2$ the p-adic valuation of the algebraic integer $\gamma(o)\kappa_i(o)$ is different from $|o|$, hence $\gamma(o)\kappa_i(o)q^{-|o|} \neq 1$. This means that, when evaluating (4.11) at $T = q^{-1}$, no term in the product vanishes. Therefore the order of vanishing of $L(S, T)$ at $T = q^{-1}$ is 0, and the special value is simply $L(S, q^{-1})$. Substituting $T = q^{-1}$ in (4.11) gives

$$L^*(S) = \prod_{o \in \mathbf{O}_q(a)} \left(1 - e^{i(\varepsilon_o + \theta_o)}\right) \left(1 - e^{i(\varepsilon_o - \theta_o)}\right).$$

Elementary trigonometric manipulations then show that

$$L^*(S) = \prod_{o \in \mathbf{O}_q(a)} 4 e^{i\varepsilon_o} \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right). \quad (4.12)$$

The Riemann hypothesis for $L(S, T)$ (see Theorem 4.2) implies that $L^*(S)$ is a positive real number, so $|L^*(S)| = L^*(S)$. Taking absolute values on both sides of (4.12) concludes the proof. $\square$

Since $S$ satisfies the BSD conjecture (see Theorem 4.4), we deduce the rank of its Mordell-Weil group, as well as the value of its Néron-Tate regulator.

**Corollary 4.12.** We have rk$(S(\mathbb{K})) = 0$. In other words, $S(\mathbb{K}) = S(\mathbb{K})_{\mathrm{tor}}$ is finite. Moreover $\mathrm{Reg}(S) = 1$.

# 5 Distribution of angles

Formula (4.10) describing the special value of $L(S, T)$ involves angles of Gauss and Kloosterman sums. In order to study the size of $L^*(S)$ in the following section, we describe here how these angles distribute in the interval $[0, 2\pi)$.

## 5.1 Angles of Gauss sums

As discussed in Section 3.2, when $q \equiv 1 \mod 3$, there is a map $\mathrm{pr}_1 : \mathbf{O}_q(a) \to \{\pm 1\}$; for any $o \in \mathbf{O}_q(a)$, $\mathrm{pr}_1(o)$ is a lift to $\{\pm 1\}$ of the “first coordinate” $j$ of any representative $(j, \alpha) \in \mathbb{I}_a$ of $o$.

**Proposition 5.1.** • If $q \equiv 1 \mod 3$, there exists an irrational number $\phi \in [0, 2\pi)$, depending at most on $q$, such that for any $o \in \mathbf{O}_q(a)$, the angle $\varepsilon_o$ satisfies

$$\varepsilon_o - \mathrm{pr}_1(o)|o|\phi \in \left\{0, \frac{2\pi}{3}, \frac{4\pi}{3}\right\}.$$

Moreover, for any element of the form $\eta + jd\phi$, where $\eta \in \{0, \frac{2\pi}{3}, \frac{4\pi}{3}\}$, $j \in \{\pm 1\}$ and $d|a$ there is some $o \in \mathbf{O}_q(a)$ such that $\varepsilon_o = \eta + jd\phi$.

• If $q \equiv 2 \mod 3$ then for any $o \in \mathbf{O}_q(a)$ we have $\varepsilon_o \in \{0, \frac{\pi}{3}, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}, \frac{5\pi}{3}\}$.

**Proof.** We treat the case $q \equiv 1 \mod 3$ first. Let $g_{q,1} := G_{\mathbb{F}_q}(\chi_{\mathbb{F}_q,3}, \psi_{\mathbb{F}_q})$, $g_{q,2} := G_{\mathbb{F}_q}(\chi_{\mathbb{F}_q,3}^2, \psi_{\mathbb{F}_q})$, and $\phi$ to be the unique element of $[0, 2\pi)$ such that $\iota(g_{q,1}) = \sqrt{q}e^{i\phi}$. Since $\chi_{\mathbb{F}_q,3}$ is a character of order 3, $\chi_{\mathbb{F}_q,3}^2 = \overline{\chi_{\mathbb{F}_q,3}}$, and so $\iota(g_{q,1})$ and $\iota(g_{q,2})$ are complex conjugate, implying that $\iota(g_{q,2}) = \sqrt{q}e^{-i\phi}$. Let $o \in \mathbf{O}_q(a)$, and $(j, \alpha) \in \mathbb{I}_a$ be a representative of $o$. By definition we have $\mathrm{pr}_1(o) \equiv j \mod 3$, $\mathrm{pr}_1(o) \in \{\pm 1\}$, and $\gamma(o) = G_{\mathbb{F}_o}(\chi_{\mathbb{F}_o,3}^j, \psi_{\mathbb{F}_o,\alpha})$. Applying (Ga 2), the Hasse-Davenport relation (Ga 4) and the discussion above, this last equality becomes

$$\gamma(o) = (\chi_{\mathbb{F}_o,3}(\alpha))^{-\mathrm{pr}_1(o)} g_{q,1}^{\mathrm{pr}_1(o)|o|}. \quad (5.1)$$
Now $(\chi_{\mathbb{F}_o,3}(\alpha))^{-\mathrm{pr}_1(o)}$ is a third root of unity, so taking arguments of the associated complex numbers in (5.1) shows that $\varepsilon_o - \mathrm{pr}_1(o)|o|\phi \in \{0, \frac{2\pi}{3}, \frac{4\pi}{3}\}$. A straightforward application of Lemma 3.5 when $a=1$ shows that $\mathrm{ord}_{\mathbb{P}}(g_{q,1}) = 2/3$. Therefore $\mathrm{ord}_{\mathbb{P}}(g_{q,1}/\sqrt{q}) = 1/6 \neq 0$, and so $e^{i\phi}$ is not a root of unity, implying that $\phi$ is irrational.

Finally, fix $d|a, j \in \{\pm 1\}$ and $\eta \in \{0, \frac{2\pi}{3}, \frac{4\pi}{3}\}$. If $\eta=0$, then for any $\alpha \in \mathbb{F}_{q^d} \setminus \mathbb{F}_{q^{d-1}}$, equation (5.1) implies that the orbit $o \in \mathbf{O}_q(a)$ of $(j, \alpha^3)$ satisfies $\varepsilon_o = jd\phi$. If $\eta \neq 0$, by an element-counting argument, one can check that there is always some $\alpha \in \mathbb{F}_{q^a}^\times$ such that $[\mathbb{F}_q(\alpha) : \mathbb{F}_q] = d$ and $\alpha$ is not a cube in $\mathbb{F}_{q^d}$. Switching $\alpha$ and $\alpha^2$ if necessary, we may assume that $\iota(\chi_{\mathbb{F}_{q^d,3}}^{-j}(\alpha)) = e^{i\eta}$ and again, using (5.1), the orbit $o \in \mathbf{O}_q(a)$ of $(j, \alpha)$ satisfies the equality $\varepsilon_o = \eta + jd\phi$.

When $q \equiv 2 \bmod 3$, recall from the proof of Lemma 3.5 that for any $o \in \mathbf{O}_q(a)$, any $(j, \alpha) \in o$ we have $\gamma(o) = \pm \chi_{\mathbb{F}_o,3}(\alpha)(-q)^{|o|/2}$. Therefore, $\gamma(o)^6 = q^{3|o|}$ and $(e^{i\varepsilon_o})^6 = 1$. Hence, $\varepsilon_o$ is as stated in the proposition. $\square$

**Remark 5.2.** When $q \equiv 2 \bmod 3$, there are only 6 possible values for the angles $\varepsilon_o$ as $o$ ranges through $\mathbf{O}_q(a)$. When $q \equiv 1 \bmod 3$, the irrationality of $\phi$ implies that $\varepsilon_o$ may take many different values as $o$ runs through elements of $\mathbf{O}_q(a)$. Nevertheless, if one only considers orbits having a fixed size, the angles $\varepsilon_o$ only take at most 6 different possible values. This will be used in Section 6 to estimate the size of $L^*(S)$.

## 5.2 Equidistribution of Kloosterman angles

In this section, we state an equidistribution result concerning the angles of our Kloosterman sums $\mathrm{Kl}(o)$. In Section 6.3 we will manipulate the Kloosterman angles $\theta_o$ associated to orbits in $\mathbf{O}_q(a)$. First, we define and discuss how Kloosterman angles associated to orbits in $\mathbf{P}_q(a)$ distribute. Later, we will explain how these are related to the angles $\theta_o$. Before stating any result, we introduce the necessary theoretical background to discuss equidistribution of Kloosterman angles.

Recall from [Kat88, §4] the construction of the Kloosterman sheaf $\mathcal{Kl} := \mathrm{Kl}_2(1/2)$. This is an $\ell$-adic lisse sheaf on $\mathbb{G}_m/\mathbb{F}_q$ of rank 2 which is pure of weight 0. Consider the morphism $[2] : \mathbb{G}_m \to \mathbb{G}_m, x \mapsto x^2$ and define $\mathcal{F} := [2]^*\mathcal{Kl}$. $\mathcal{F}$ is also a lisse $\overline{\mathbb{Q}}_\ell$-sheaf of rank 2 on $\mathbb{G}_m$ which is pure of weight 0 (see [FL05]). More concretely, $\mathcal{F}$ is characterized by the following condition: at any place $v$ of $\mathbb{G}_m/\mathbb{F}_q$, we have

$$ \mathrm{Tr}(\mathrm{Fr}_v | \mathcal{F}_{\bar{v}}) = |\mathbb{F}_v|^{-1/2} \mathrm{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v}, \beta_v^2), $$

where $\beta_v \in \mathbb{G}_m(\overline{\mathbb{F}}_q)$ is any element in the closed point of $\mathbb{G}_m$ corresponding to $v$, and $\bar{v}$ is a geometric point lying over $v$. Note that the change of variables $x \mapsto \beta_v x$ gives $\mathrm{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v}, \beta_v^2) = \mathrm{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v, \beta_v}, 1)$, which corresponds to the Kloosterman sums introduced in Section 3.2. The lisse sheaf $\mathcal{F}$ gives rise to a $\overline{\mathbb{Q}}_\ell$-representation $\rho_{\mathcal{F}}$ of the arithmetic fundamental group of $\mathbb{G}_m/\mathbb{F}_q$. Remark 1 in [FL05] shows that the Zariski closure $\mathcal{G}$ of $\rho_{\mathcal{F}}(\pi_1(\mathbb{G}_m \otimes_{\mathbb{F}_q} \overline{\mathbb{F}}_q))$ in GL(2) is $\mathcal{G} = \mathrm{SL}(2)$ (same as for the sheaf $\mathcal{Kl}$) and that $\mathcal{F}$ satisfies the assumptions of [Kat88, §3.1 - §3.3], namely that $\mathrm{im}(\rho_{\mathcal{F}}) \subset \mathcal{G}(\overline{\mathbb{Q}}_\ell)$, i.e., $\rho_{\mathcal{F}}$ has trivial determinant.

Via the isomorphism $\overline{\mathbb{Q}}_\ell \simeq \mathbb{C}$ fixed above, we may view $\mathcal{G}$ as an algebraic group over $\mathbb{C}$. The subgroup $K := \mathrm{SU}(2, \mathbb{C})$ is then maximal compact in $\mathcal{G}(\mathbb{C})$. The set of conjugacy classes of $K$, denoted by $K^\sharp$, may be identified with the interval $[0, \pi]$. Indeed any matrix $M \in K$ is conjugate (in $K$) to a diagonal matrix $\mathrm{Diag}(e^{i\theta_M}, e^{-i\theta_M})$ for some $\theta_M \in [0, \pi]$, hence $\mathrm{Tr}(M) = 2 \cos\theta_M$. Denote by $\nu^\sharp$ the measure on $K^\sharp$ defined as the direct image of the Haar measure on $K$, normalised to have total mass 1. The pushforward of $\nu^\sharp$ under the identification $K^\sharp \xrightarrow{\sim} [0, \pi]$ is the Sato-Tate measure $\nu_{\mathrm{ST}} := \frac{2}{\pi} \sin^2\theta \, d\theta$ (see [Kat88, Chap. 13] for details).

We can now define the Kloosterman angle associated to a place. For any place $v \in M_{\mathbb{K}} \setminus \{0, \infty\}$ corresponding to a closed point of $\mathbb{G}_m$, denoted also by $v$, the semi-simplification of $\rho_{\mathcal{F}}(\mathrm{Fr}_v) \in \mathcal{G}(\mathbb{C})$ is conjugate to an element of $K$. Therefore, we can define the conjugacy class (in $K$) of $\rho_{\mathcal{F}}(\mathrm{Fr}_v)^{\mathrm{s.s.}}$, and associate to $v$ (via the identification $K^\sharp \simeq [0, \pi]$) a unique element $\theta_v \in [0, \pi]$. We call $\theta_v$ the Kloosterman angle associated to the place $v$. For any $\beta \in v$, we define $\theta(\beta) := \theta_v$, which is independent of $\beta$ by (K|1) for the angle associated to $v$ as above. Under the identification $K^\sharp \simeq [0, \pi]$ we have

$$ 2 \cos \theta_v = \mathrm{Tr}(\rho_{\mathcal{F}}(\mathrm{Fr}_v)^{\mathrm{s.s.}}) = \mathrm{Tr}(\mathrm{Fr}_v | \mathcal{F}_{\bar{v}}) = |\mathbb{F}_v|^{-1/2} \mathrm{Kl}_{\mathbb{F}_v}(\psi_{\mathbb{F}_v}, \beta^2). \quad (5.2) $$
**Remark 5.3.** We can relate the angles $\theta_v$ to the angles $\theta_o$ defined in Section 4.4. Let $o \in \mathbf{O}_q(a)$ and $v := \Theta(o) \in \mathbf{P}_q(a)$. Equation (5.2) combined with (K12) gives $\theta_o \equiv \begin{cases} \theta_v \mod \pi & \text{if } |o| = |v|, \\ 2\theta_v \mod \pi & \text{if } |o| = 2|v|. \end{cases}$

We associate to the family of angles $(\theta_v)_{v \in \mathbf{P}_q^{\max}(a)}$ a measure on $K^\natural \simeq [0, \pi]$.

**Definition 5.4.** For any $a \ge 1$, denote by $\mathbf{P}_q^{\max}(a) := \{v \in \mathbf{P}_q(a) \mid d_v = a\}$ the subset of orbits of maximal size. We define $\nu_a$ to be the discrete probability measure on $[0, \pi]$ given by

$$ \nu_a := \frac{1}{\Pi_q(a)} \sum_{v \in \mathbf{P}_q^{\max}(a)} \delta\{\theta_v\}, $$

where $\delta\{x\}$ denotes the Dirac delta measure at $x \in [0, \pi]$ and, according to the notation of Section 3.2, we have $\Pi_q(a) = |\mathbf{P}_q^{\max}(a)|$.

We have the equality<fnref n="1" />

$$ \nu_a = \frac{1}{|\{\beta \in \mathbb{F}_{q^a}^\times \mid [\mathbb{F}_q(\beta) : \mathbb{F}_q] = a\}|} \sum_{[\mathbb{F}_q(\beta) : \mathbb{F}_q] = a} \delta\{\theta(\beta)\}. \quad (5.3) $$

Indeed, $\langle g \rangle$ acts on $\mathbb{F}_{q^a}^\times$ as the Frobenius, so one can partition $\{\beta \in \mathbb{F}_{q^a}^\times \mid [\mathbb{F}_q(\beta) : \mathbb{F}_q] = a\}$ into $\Pi_q(a)$ orbits of size $a$. Regrouping elements by orbits, the right-hand side of (5.3) may be rewritten as

$$ \frac{1}{a \Pi_q(a)} \sum_{v \in \mathbf{P}_q^{\max}(a)} \sum_{\beta \in v} \delta\{\theta(\beta)\} = \frac{1}{a \Pi_q(a)} \sum_{v \in \mathbf{P}_q^{\max}(a)} a \delta\{\theta_v\} $$

and the right-hand side is clearly equal to $\nu_a$.

We cite a theorem of Katz establishing the equidistribution of Kloosterman angles with respect to the Sato-Tate measure (see [Kat88, Theorem 3.6] and [FL05]). Recall that a sequence of Borel measures $(\mu_i)_{i \ge 1}$ on $[0, \pi]$ is said to converge weak-* to $\nu_{\text{ST}}$ if, for every continuous $\mathbb{C}$-valued function $f$ on $[0, \pi]$, the sequence $\int_{[0, \pi]} f d\mu_i$ converges to $\int_{[0, \pi]} f d\nu_{\text{ST}}$ as $i \to \infty$.

**Theorem 5.5.** The sequence of probability measures $(\nu_a)_{a \ge 1}$ converges weak-* to the Sato-Tate measure $\nu_{\text{ST}}$.

*Sketch of the proof.* It suffices to treat the case of characters of irreducible finite dimensional representations of $K$, since these functions are dense in the space of continuous central functions on $K$. The integral of such a character with respect to the Sato-Tate measure equals 0. In [Kat88, FL05], the authors show that for any irreducible finite-dimensional representation $\Psi$ of $K$,

$$ \left| \int_{K^{\natural}} \mathrm{Tr}(\Psi) \, d\nu_a \right| \ll_q \frac{a \dim \Psi}{q^{a/2}}. \quad (5.4) $$

This follows from the Lefschetz trace formula, Deligne's purity theorem and the Grothendieck-Ogg-Shafarevich formula combined to a detailed analysis of the monodromy of Kloosterman sheaves (see [Kat88, FL05] for details). □

To study the size of the special value of $L(S, T)$, we will need an effective version of Theorem 5.5, which we state below:

**Theorem 5.6.** For any continuously differentiable function $g$ on $[0, \pi]$, any $a \ge 1$,

$$ \left| \int_{[0, \pi]} g \, d\nu_a - \int_{[0, \pi]} g \, d\nu_{\text{ST}} \right| \ll_q \frac{a^{1/2}}{q^{a/4}} \int_0^\pi |g'(x)| dx. $$

<footnote n="1">¹In [Kat88, Theorem 3.6], the measure $\nu_a$ is denoted by $Y_a$ and defined by the right-hand side of (5.3).</footnote>
Proof. We follow the proof of Proposition 6.11 in [Gri19]. Recall the definition of the star discrepancy $D_q^*(a)$:

$$D_q^*(a) := \sup_{x \in [0, \pi]} \left| \int_{[0,x)} d\nu_a - \int_{[0,x)} d\nu_{ST} \right|.$$

This quantity measures “how far” the angles $\theta_v$ are from being perfectly equidistributed with respect to the Sato-Tate measure $\nu_{ST}$. Let $g: [0, \pi] \to \mathbb{C}$ be a continuously differentiable function. Corollary 2 in [Nie91] establishes the analogue of Koksma’s inequality in the context of the Sato-Tate measure:

$$\left| \int_{[0, \pi]} g d\nu_a - \int_{[0, \pi]} g d\nu_{ST} \right| \le D_q^*(a) \int_0^\pi |g'(x)| dx. \qquad (5.5)$$

Let $n \ge 1$ be an integer and write $\Psi_n = \text{Symm}^n(\text{std})$ for the $n$-th symmetric power of the standard representation of $K$. We apply (5.4) to $\Psi_n$ and combine it with Lemma 3 in [Nie91] to get the upper bound

$$D_q^*(a) \ll_q \frac{a^{1/2}}{q^{a/4}}. \qquad (5.6)$$

The result follows by putting (5.5) and (5.6) together. $\square$

## 5.3 Kloosterman angles “avoid” Gauss angles

Now we show that the angles of Gauss and Kloosterman sums cannot be equal nor arbitrarily close to each other. This phenomenon was partly observed in [Gri19] or [GdW21], where the authors show that Kloosterman angles avoid certain rational multiples of $\pi$ (which are angles of Gauss sums). Our results follow from Liouville’s inequality and hold in any odd characteristic. That is why, in this section, we work in broader generality and temporarily drop the assumption that $p \ge 7$. **Until the end of the section, we fix a prime number $p \ge 3$ and integers $n, N \ge 1$, with $n$ coprime to $p$.**

Consider a finite extension $\mathbb{F}$ of $\mathbb{F}_p$, a non-trivial multiplicative character $\chi$ on $\mathbb{F}^\times$, a non-trivial additive character $\psi$ on $\mathbb{F}$ and $\alpha \in \mathbb{F}^\times$. Recall from (Ga 1) that $|\iota(\mathbb{G}_F(\chi, \psi))| = \sqrt{|\mathbb{F}|}$. We let $\varepsilon_{\mathbb{F}}(\chi, \psi) \in [0, 2\pi)$ be the argument of $\iota(\mathbb{G}_F(\chi, \psi))$, so that

$$\iota(\mathbb{G}_F(\chi, \psi)) = \sqrt{|\mathbb{F}|} e^{i \varepsilon_{\mathbb{F}}(\chi, \psi)}.$$

Similarly, recall that $\text{Kl}_{\mathbb{F}}(\psi, \alpha)$ decomposes as $\kappa_{1, \mathbb{F}}(\psi, \alpha) + \kappa_{2, \mathbb{F}}(\psi, \alpha)$, where $\kappa_{1, \mathbb{F}}(\psi, \alpha), \kappa_{2, \mathbb{F}}(\psi, \alpha)$ are algebraic integers satisfying $|\kappa_{1, \mathbb{F}}(\psi, \alpha)| = |\kappa_{2, \mathbb{F}}(\psi, \alpha)| = \sqrt{|\mathbb{F}|}$, such that $\iota(\kappa_{1, \mathbb{F}}(\psi, \alpha))$, $\iota(\kappa_{2, \mathbb{F}}(\psi, \alpha))$ are conjugate. Switching $\kappa_{1, \mathbb{F}}(\psi, \alpha), \kappa_{2, \mathbb{F}}(\psi, \alpha)$ if necessary, we may assume that $\iota(\kappa_{1, \mathbb{F}}(\psi, \alpha))$ lies in the upper half plane, and we let $\theta_{\mathbb{F}}(\psi, \alpha) \in [0, \pi]$ denote its argument, so that

$$\iota(\kappa_{1, \mathbb{F}}(\psi, \alpha)) = \sqrt{|\mathbb{F}|} e^{i \theta_{\mathbb{F}}(\psi, \alpha)}, \quad \iota(\kappa_{2, \mathbb{F}}(\psi, \alpha)) = \sqrt{|\mathbb{F}|} e^{-i \theta_{\mathbb{F}}(\psi, \alpha)} \\ \text{and } \iota(\text{Kl}_{\mathbb{F}}(\psi, \alpha)) = 2 \sqrt{|\mathbb{F}|} \cos(\theta_{\mathbb{F}}(\psi, \alpha)).$$

We write $R_N$ for the set of angles of $N$-th roots of unity and $\Gamma_{\mathbb{F},n}$ for the set of angles of Gauss sums associated to characters of order $n$ on $\mathbb{F}^\times$:

$$R_N := \left\{ \frac{2j\pi}{N} ; j = 0, \dots, N-1 \right\}, \\ \Gamma_{\mathbb{F},n} := \left\{ \varepsilon_{\mathbb{F}}(\chi, \psi) ; \chi : \mathbb{F}^\times \to \overline{\mathbb{Q}}^\times \text{ of order } n, \psi : \mathbb{F} \to \overline{\mathbb{Q}}^\times \text{ non-trivial} \right\}.$$

For any finite subset $I \subset [0, 2\pi]$ and any $x \in [0, 2\pi]$, we write $\text{dist}(x, I) = \min_{y \in I} |x - y|$.

**Theorem 5.7.** Let $p \ge 3$ be a prime number and $N, n \ge 1$ be integers with $n$ coprime to $p$. There are explicit constants $\tau_{p,N} > 0$ and $\sigma_{p,n} > 0$ satisfying the following: for any finite extension $\mathbb{F}/\mathbb{F}_p$, any non-trivial additive character $\psi$ on $\mathbb{F}$, any $\alpha \in \mathbb{F}^\times$ we have

$$i) \min\{\text{dist}(\theta_{\mathbb{F}}(\psi, \alpha), R_N), \text{dist}(2\pi - \theta_{\mathbb{F}}(\psi, \alpha), R_N)\} \ge |\mathbb{F}|^{-\tau_{p,N}}.$$
$$ii) \min\{\mathrm{dist}(\theta_{\mathbb{F}}(\psi, \alpha), \Gamma_{\mathbb{F},n}), \mathrm{dist}(2\pi - \theta_{\mathbb{F}}(\psi, \alpha), \Gamma_{\mathbb{F},n})\} \geq |\mathbb{F}|^{-\sigma_{p,n}}.$$

Before proving this theorem, we show that Kloosterman angles are not in $R_N \cup \Gamma_{\mathbb{F},n}$, i.e., the distances appearing in Theorem 5.7 are positive.

**Lemma 5.8.** Under the hypotheses of the previous theorem, we have

$$i) \theta_{\mathbb{F}}(\psi, \alpha) \notin R_N \text{ and } 2\pi - \theta_{\mathbb{F}}(\psi, \alpha) \notin R_N,$$

$$ii) \theta_{\mathbb{F}}(\psi, \alpha) \notin \Gamma_{\mathbb{F},n} \text{ and } 2\pi - \theta_{\mathbb{F}}(\psi, \alpha) \notin \Gamma_{\mathbb{F},n}.$$

*Proof.* Just as in Section 3.1, consider a prime ideal $\mathfrak{p}$ of $\mathbb{Z}$ lying over $p$ and let $\mathrm{ord}_{\mathfrak{p}}$ be the $\mathfrak{p}$-adic valuation normalised so that $\mathrm{ord}_{\mathfrak{p}}(\mathbb{F}) = 1$ (in the notation of Section 3.2 this would correspond to $\mathbb{F} = \mathbb{F}_q$). The angle $\theta_{\mathbb{F}}(\psi, \alpha)$ (resp. $2\pi - \theta_{\mathbb{F}}(\psi, \alpha)$) belongs to $\pi\mathbb{Q}$ if and only if $\kappa_{1,\mathbb{F}}(\psi, \alpha)/\sqrt{|\mathbb{F}|}$ (resp. $\kappa_{2,\mathbb{F}}(\psi, \alpha)/\sqrt{|\mathbb{F}|}$) is a root of unity. But (K|4) states that $\mathrm{ord}_{\mathfrak{p}}(\kappa_{i,\mathbb{F}}(\psi, \alpha)/\sqrt{\mathbb{F}}) = \pm 1/2$ for $i = 1, 2$, so $\kappa_{i,\mathbb{F}}(\psi, \alpha)/\sqrt{|\mathbb{F}|}$ cannot be a root of unity, hence proving the first item. To check item ii), consider a multiplicative character $\chi$ on $\mathbb{F}^\times$ of order $n$, a non-trivial additive character $\psi'$ on $\mathbb{F}$. Stickelberger's theorem implies $\mathrm{ord}_{\mathfrak{p}}(G_{\mathbb{F}}(\chi, \psi')) \in [1/n, 1 - 1/n]$ (see (Ga 5)), and with our normalization of $\mathrm{ord}_{\mathfrak{p}}$, we deduce from (K|4) that $\{\mathrm{ord}_{\mathfrak{p}}(\kappa_{1,\mathbb{F}}(\psi, \alpha)), \mathrm{ord}_{\mathfrak{p}}(\kappa_{2,\mathbb{F}}(\psi, \alpha))\} = \{0, 1\}$. This implies that $G_{\mathbb{F}}(\chi, \psi') \neq \kappa_{i,\mathbb{F}}(\psi, \alpha)$ for $i = 1, 2$, hence $\theta_{\mathbb{F}}(\psi, \alpha), 2\pi - \theta_{\mathbb{F}}(\psi, \alpha) \notin \Gamma_{\mathbb{F},n}$. $\square$

We now turn to the proof of Theorem 5.7, the main tool used being a form of Liouville's inequality.

*Proof of Theorem 5.7.* Fix a finite extension $\mathbb{F}/\mathbb{F}_p$, a multiplicative character $\chi$ on $\mathbb{F}^\times$ of order $n$, two non-trivial additive characters $\psi$ and $\psi'$ on $\mathbb{F}$ and $\alpha \in \mathbb{F}^\times$. Define three algebraic numbers $g := G_{\mathbb{F}}(\chi, \psi)/\sqrt{|\mathbb{F}|}$ and $k_i := \kappa_{i,\mathbb{F}}(\psi, \alpha)/\sqrt{|\mathbb{F}|}$ for $i = 1, 2$. Let $P_N(T) = T^N - 1 \in \mathbb{Q}[T]$, and $\pi_g \in \mathbb{Q}[T]$ be the minimal polynomial of $g$ over $\mathbb{Q}$.

*Proof of i)* We know from Lemma 5.8 that $P_N(k_i) \neq 0$ for $i = 1, 2$. Using the convexity of $z \mapsto e^{iz}$ we get, for any $j \in \{0, \dots, N-1\}$,

$$|\iota(P_N(k_1))| = \left| e^{iN\theta_{\mathbb{F}}(\psi, \alpha)} - e^{iN\frac{2j\pi}{N}} \right| \le N \left| \theta_{\mathbb{F}}(\psi, \alpha) - \frac{2j\pi}{N} \right|, \quad (5.7)$$

and similarly

$$|\iota(P_N(k_2))| \le N \left| 2\pi - \theta_{\mathbb{F}}(\psi, \alpha) - \frac{2j\pi}{N} \right|. \quad (5.8)$$

On the other hand, Liouville's inequality (see [Wal00, Proposition 3.14]) gives, for $i = 1, 2$,

$$\log|\iota(P_N(k_i))| \ge -[\mathbb{Q}(k_i) : \mathbb{Q}](\log\|P_N\|_1 + N \mathrm{ht}(k_i)). \quad (5.9)$$

Here, we have denoted by $ht: \overline{\mathbb{Q}} \to \mathbb{R}$ the absolute logarithmic Weil height and by $\|P_N\|_1$ the sum of the sizes of the coefficients of $P_N$. Lemma 5.3 in [Gri19] gives the upper bounds $[\mathbb{Q}(k_i) : \mathbb{Q}] \le 2(p-1)$ and $ht(k_i) \le \log\sqrt{|\mathbb{F}|}$, and we clearly have $\|P_N\|_1 = 2$. Combining these estimates we obtain

$$\log|\iota(P(k_i))| \ge -2(p-1)\left(\log 2 + N \log \sqrt{|\mathbb{F}|}\right) \ge -2(p-1)\left(\frac{\log 2}{\log 3} + \frac{N}{2}\right) \log|\mathbb{F}|, \quad (5.10)$$

where the last inequality uses the fact that $|\mathbb{F}| \ge 3$. Combining (5.7) with (5.10) we get

$$\left| \theta_{\mathbb{F}}(\psi, \alpha) - \frac{j\pi}{N} \right| \ge \frac{|\mathbb{F}|^{-2(p-1)(\log 2 / \log 3 + N/2)}}{N} \ge |\mathbb{F}|^{-2(p-1)(\log 2 / \log 3 + N/2) - \log(N) / \log(3)},$$

(the second inequality uses again $|\mathbb{F}| \ge 3$). Setting $\tau_{p,N} := 2(p-1)(\log 2 / \log 3 + N/2) + \log(N) / \log(3)$ gives the desired inequality. Finally, combining (5.8) and (5.10) and following the same line of reasoning, we conclude that $\mathrm{dist}(2\pi - \theta_{\mathbb{F}}(\psi, \alpha), R_N) \ge |\mathbb{F}|^{-\tau_{p,N}}$ too.

*Proof of ii)* We proceed as above, by first noticing that $\pi_g(k_i) \neq 0$ for $i = 1, 2$. Indeed, for any $\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ we have $\sigma(G_{\mathbb{F}}(\chi, \psi)) = G_{\mathbb{F}}(\sigma \circ \chi, \sigma \circ \psi)$. The character $\sigma \circ \chi$ on $\mathbb{F}^\times$ has order $n$, so that the argument of $\sigma(G_{\mathbb{F}}(\chi, \psi))$ belongs to $\Gamma_{\mathbb{F},n}$ and Lemma 5.8 shows that $\sigma(g) \neq k_i$.
Let $c \in \mathbb{Z}$ be the lcm of the denominators of coefficients of $\pi_g$ and define $P := c\pi_g \in \mathbb{Z}[T]$. Write $d = \deg(\pi_g)$ for the degree of $\pi_g$ and $g_1 = g, g_2, \dots, g_d$ for the Galois conjugates of $g$. We have

$$|\iota(P(k_1))| = |c| |\iota(k_1 - g_1)| \prod_{j=2}^{d} |\iota(k_1 - g_j)|.$$

Since $|\iota(k_1)| = |\iota(g_j)| = 1$ for all $j$ (cf. (Ga 1) and (Kl 3)), the convexity of $z \mapsto e^{iz}$ implies

$$|\iota(P(k_1))| \le 2^{d-1}|c| \left| e^{i \theta_F(\psi, \alpha)} - e^{i \varepsilon_F(\chi, \psi')} \right| \le 2^{d-1}|c| |\theta_F(\psi, \alpha) - \varepsilon_F(\chi, \psi')|, \quad (5.11)$$

and, similarly,

$$|\iota(P(k_2))| \le 2^{d-1}|c| |2\pi - \theta_F(\psi, \alpha) - \varepsilon_F(\chi, \psi')|. \quad (5.12)$$

In this setting, Liouville's inequality yields:

$$\log|\iota(P(k_i))| \ge -[\mathbb{Q}(k_i) : \mathbb{Q}](\log\|P\|_1 + d \, \mathrm{ht}(k_i)). \quad (5.13)$$

Recall that $G_F(\chi, \psi) \in \mathbb{Q}(\zeta_{np})$ so $g \in \mathbb{Q}(\zeta_{np}, \sqrt{p})$ and therefore $d \le 2np$. Using elementary symmetric polynomials, we have $\|P\|_1 \le |c| \sum_{r=0}^d \binom{d}{r} = |c| 2^d$, since all of the $\iota(g_j)$ have absolute value 1. In order to bound $|c|$, notice that $|\mathbb{F}|g = G_F(\chi, \psi)\sqrt{|\mathbb{F}|}$ is an algebraic integer so its minimal polynomial $\pi_{|\mathbb{F}|g}$ has integral coefficients. On the other hand $|\mathbb{F}|^{-d}\pi_{|\mathbb{F}|g}(|\mathbb{F}|T) \in \mathbb{Q}[T]$ vanishes at all $g_j$, is unitary, and has the same degree as $\pi_g$. We deduce that

$$|\mathbb{F}|^d \pi_g(T) = \pi_{|\mathbb{F}|g}(|\mathbb{F}|T) \in \mathbb{Z}[T],$$

implying that $|c| \le |\mathbb{F}|^d$. Plugging the bounds above into (5.13) gives

$$\begin{aligned} \log|\iota(P(k_i))| &\ge -2(p-1)\left(\log((2|\mathbb{F}|)^d) + 2np \log \sqrt{|\mathbb{F}|}\right) \\ &\ge -2(p-1)(2np \log(2|\mathbb{F}|) + np \log|\mathbb{F}|) \\ &\ge -2np(p-1)(3 \log|\mathbb{F}| + 2 \log 2) \\ &\ge -2np(p-1)\left(3 + \frac{2 \log 2}{\log 3}\right) \log|\mathbb{F}| \quad \text{as } |\mathbb{F}| \ge 3. \end{aligned}$$

Combining this with (5.11) gives

$$|\theta_F(\psi, \alpha) - \varepsilon_F(\chi, \psi')| \ge \frac{|\mathbb{F}|^{-2np(p-1)(3+2\log 2/\log 3)}}{2^{2np}|\mathbb{F}|^{2np}} \ge |\mathbb{F}|^{-\sigma_{p,n}},$$

where $\sigma_{p,n} := 2np(\frac{\log 2}{\log 3} + (p-1)(3 + \frac{2\log 2}{\log 3}) + 1)$ (again we used the fact that $|\mathbb{F}| \ge 3$). In a similar way, we also have $|2\pi - \theta_F(\psi, \alpha) - \varepsilon_F(\chi, \psi)| \ge |\mathbb{F}|^{-\sigma_{p,n}}$. $\square$

Here is the special case of Theorem 5.7 that we need in Section 6.

**Corollary 5.9.** Recall the notation from Section 4.4 and define $\sigma_p := \sigma_{p,3}$. For any $o, o' \in \mathbf{O}_q(a)$ such that $|o| = |o'|$ we have

$$\min(|\theta_o - \varepsilon_{o'}|, |\theta_o + \varepsilon_{o'}|, |2\pi - \theta_o - \varepsilon_{o'}|, |2\pi - \theta_o + \varepsilon_{o'}|) \ge q^{-\sigma_p|o|}.$$

*Proof.* Let $\mathbb{F}_o = \mathbb{F}_{o'}$ be the unique extension of $\mathbb{F}_q$ of degree $|o|$, so that $|\mathbb{F}_o| = q^{|o|}$. Since $\varepsilon_{o'} \in \Gamma_{\mathbb{F}_o,3}$, we deduce from Theorem 5.7 that $|\theta_o - \varepsilon_{o'}| \ge q^{-\sigma_p|o|}$ and $|2\pi - \theta_o - \varepsilon_{o'}| \ge q^{-\sigma_p|o|}$. In order to show the two other inequalities, we simply need to check that $-\varepsilon_{o'} \in \Gamma_{\mathbb{F}_o,3}$. Fix a representative $(j', \alpha') \in o'$. By definition, $\varepsilon_{o'} = \varepsilon_{\mathbb{F}_o}(\chi_{\mathbb{F}_o,3}^{j'}, \psi_{\mathbb{F}_o,\alpha'})$ and since $\chi_{\mathbb{F}_o,3}$ is of order 3, we have $-\varepsilon_{o'} = \varepsilon_{\mathbb{F}_o}(\chi_{\mathbb{F}_o,3}^{-j'}, \psi_{\mathbb{F}_o,\alpha'}) \in \Gamma_{\mathbb{F}_o,3}$, hence the result. $\square$
# 6 Analytic estimates on the special value

In this section we estimate the asymptotic behaviour of the special value of $L(S, T)$ as the parameter $a$ tends to $\infty$. To make the dependence on $a$ more explicit, we write $S_a$ with the subscript $a$ instead of just $S$. The goal of this section is to prove the following theorem:

**Theorem 6.1.** Let $\mathbb{F}_q$ be a finite field of characteristic $p \ge 7$, $\mathbb{K} := \mathbb{F}_q(t)$. For any $a \ge 1$, let $C_a$ be the hyperelliptic curve defined by (2.3), and $S_a := \text{Jac}(C_a)$ be its Jacobian. There are positive constants $c_1, c_2$, depending at most on $q$, such that for any $a \ge 1$, we have

$$-\frac{c_1}{a} \le \frac{\log L^*(S_a)}{\log H(S_a)} \le \frac{c_2}{a}. \qquad (6.1)$$

**Remark 6.2.** By definition we have $\log H(S_a) = h(S_a) \log q$. Recall from Proposition 2.15 that $\log H(S_a)$ and $q^a$ have the same order of magnitude. Thus, Theorem 6.1 is equivalent to the following statement: there are positive constants $c'_1, c'_2$ depending at most on $q$ such that $-\frac{c'_1}{a} \le \frac{\log L^*(S_a)}{q^a} \le \frac{c'_2}{a}$. We will study the ratio $\log L^*(S_a)/q^a$ rather than $\log L^*(S_a)/\log H(S_a)$.

Let us estimate the logarithm of the special value. Recall from Theorem 4.11 that we have, for any $a \ge 1$,

$$L^*(S_a) = \prod_{o \in \mathcal{O}_q(a)} 4 \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|. \qquad (6.2)$$

The special value is a positive real number, so applying the logarithm to (6.2) and dividing by $q^a$ gives

$$\frac{\log L^*(S)}{q^a} = \log 4 \frac{|\mathbf{O}_q(a)|}{q^a} + \frac{1}{q^a} \sum_{o \in \mathcal{O}_q(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|. \qquad (6.3)$$

We can already prove the upper bound in Theorem 6.1.

*Proof of the upper bound in Theorem 6.1.* For any orbit $o \in \mathcal{O}_q(a)$, we have the trivial upper bound $\log|\sin(\frac{\varepsilon_o+\theta_o}{2})\sin(\frac{\varepsilon_o-\theta_o}{2})| \le 0$, which implies $\frac{\log L^*(S)}{q^a} \le \frac{|\mathcal{O}_q(a)|}{q^a} \log 4$. We conclude by using the estimate $|\mathcal{O}_q(a)| \ll_q q^a/a$ given in Lemma 3.3 iii). $\square$

We devote the remainder of the section to proving the lower bound in Theorem 6.1. Since $\log 4|\mathcal{O}_q(a)|/q^a = O(1/a)$, we focus on the sum appearing in the right-hand side of (6.3).

## 6.1 Reduction to maximal orbits

In this section, we introduce the notion of maximal orbits and we show that the contribution of non-maximal orbits to the logarithm of the size of the special value is negligible.

**Definition 6.3.** For any $a \ge 1$, define the set of maximal orbits

$$\mathbf{O}_q^{\max}(a) := \{o \in \mathcal{O}_q(a) \mid |o| \text{ is maximal among orbits in } \mathcal{O}_q(a)\}.$$

Any orbit $o \in \mathcal{O}_q^{\max}(a)$ has length

$$|o| = \begin{cases} a & \text{if } q \equiv 1 \bmod 3 \text{ or } a \text{ is even,} \\ 2a & \text{if } q \equiv 2 \bmod 3 \text{ and } a \text{ is odd.} \end{cases}$$

**Remark 6.4.** We defined in Section 5.2 the notion of a maximal orbit in $\mathbf{P}_q(a)$. Using $\Theta$, we can relate maximal orbits in $\mathbf{O}_q(a)$ to maximal orbits in $\mathbf{P}_q(a)$ as follows. Recall from Section 3.2 that there is a map $\Theta: \mathbf{O}_q(a) \to \mathbf{P}_q(a)$ induced by the natural map $(\mathbb{Z}/3\mathbb{Z})^\times \times \mathbb{F}_{q^a}^\times \to \mathbb{F}_{q^a}^\times$. Fix $o \in \mathbf{O}_q(a)$ and let $v := \Theta(o) \in \mathbf{P}_q(a)$.

i) When $q \equiv 1 \bmod 3$, we have $o \in \mathcal{O}_q^{\max}(a) \Leftrightarrow v \in \mathbf{P}_q^{\max}(a)$, in which case $|o| = |v| = a$ and $|\Theta^{-1}(v)| = 2$.

ii) When $q \equiv 2 \bmod 3$ and $a$ is odd, we have $o \in \mathcal{O}_q^{\max}(a) \Leftrightarrow v \in \mathbf{P}_q^{\max}(a)$, in which case $|o| = 2a$, $|v| = a$, and $|\Theta^{-1}(v)| = 1$.
iii) When $q \equiv 2 \mod 3$ and $4|a$, we have $o \in \mathbf{O}_q^{\max}(a) \Leftrightarrow v \in \mathbf{P}_q^{\max}(a)$, in which case $|o| = |v| = a$ and $|\Theta^{-1}(v)| = 2$.

iv) When $q \equiv 2 \mod 3$ and $2$ divides $a$ exactly once, we have $o \in \mathbf{O}_q^{\max}(a) \Leftrightarrow |v| \in \{a, \frac{a}{2}\}$. If $|v| = a$ then $|\Theta^{-1}(v)| = 2$ and if $|v| = \frac{a}{2}$ then $|\Theta^{-1}(v)| = 1$.

All these results are a direct application of (3.1) and (3.2).

Using the definition of the orbits given in Section 3.2, we have $\sum_{o \in \mathbf{O}_q(a)} |o| = |\mathbb{I}_a| = 2(q^a - 1)$. As a consequence of the Prime Number Theorem for $\mathbb{F}_q[t]$, Lemma 3.3 iii) states that $|\mathbf{O}_q(a)|$ has order of magnitude $q^a/a$. Thus, the average length of an orbit is $a$, so that, on average, orbits in $\mathbf{O}_q(a)$ are as large as they can be. On the other hand, using the above remark, we estimate the typical size of a non-maximal orbit:

**Lemma 6.5.** For any $a \ge 1$,

$$ \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} |o| \ll_q q^{a/2}. $$

*Proof.* We distinguish cases according to $q \mod 3$ and the parity of $a$.

When $q \equiv 1 \mod 3$, we combine the estimate $\Pi_q(k) \le q^k/k$ from Lemma 3.3 i) with the first case in Remark 6.4 to get

$$ \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} |o| = \sum_{1 \le k < a \atop k \mid a} \sum_{o \in \mathbf{O}_q(a) \atop |\Theta(o)|=k} |o| = \sum_{1 \le k < a \atop k \mid a} 2 \sum_{v \in \mathbf{P}_q(a) \atop |v|=k} |v| \\ = 2 \sum_{1 \le k \le [a/2] \atop k \mid a} k \Pi_q(k) \le 2 \sum_{1 \le k \le a/2} q^k \ll_q q^{a/2}. $$

The third equality uses the fact that a divisor of $a$ strictly smaller than $a$ is smaller than $\lfloor \frac{a}{2} \rfloor$.

When $q \equiv 2 \mod 3$ and $a$ is odd then $v \in \mathbf{P}_q(a)$ is not maximal if and only if $|v| \le \lfloor a/3 \rfloor$ (the smallest integer $> 1$ that could divide $a$ is 3). Combining Lemma 3.3 i) with Remark 6.4 ii) we get

$$ \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} |o| = \sum_{1 \le k \le \lfloor a/3 \rfloor \atop k \mid a} \sum_{o \in \mathbf{O}_q(a) \atop |\Theta(o)|=k} |o| = \sum_{1 \le k \le \lfloor a/3 \rfloor \atop k \mid a} 2|v| \le \sum_{1 \le k \le a/3 \atop k \mid a} 2k \Pi_q(k) \ll_q q^{a/3}. $$

When $q \equiv 2 \mod 3$ and $a$ is even, again, a divisor of $a$ is strictly smaller than $a$ if and only if it is smaller than $a/2$. Following the same line of reasoning as above we have

$$ \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} |o| = \sum_{1 \le k \le a/2 \atop k \mid a} \sum_{o \in \mathbf{O}_q(a) \atop |\Theta(o)|=k} |o| + \sum_{1 \le k \le a/2 \atop k \mid a \text{ odd}} \sum_{o \in \mathbf{O}_q(a) \atop |\Theta(o)|=k} |o| \\ = \sum_{1 \le k \le a/2 \atop k \mid a \text{ even}} 2 \sum_{v \in \mathbf{P}_q(a) \atop |v|=k} |v| + \sum_{1 \le k \le a/2 \atop k \mid a \text{ odd}} \sum_{v \in \mathbf{P}_q(a) \atop |v|=k} 2|v| \ll_q q^{a/2}. $$

In all three cases, independently of $q \mod 3$ and the parity of $a$, the sum $\sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} |o|$ is indeed bounded from above by $O(q^{a/2})$. $\square$

The interest of introducing $\mathbf{O}_q^{\max}(a)$ is that restricting to the “subfamily” of angles $(\varepsilon_o)_{o \in \mathbf{O}_q^{\max}(a)}$ reduces the amount of possible values these angles can take (cf. Remark 5.2). As the result below proves, the contribution of the non-maximal orbits to the sum appearing in (6.3) is negligible:

**Proposition 6.6.** As $a \to \infty$, we have

$$ \left| \frac{1}{q^a} \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} \log \left| \sin \left( \frac{\varepsilon_o + \theta_o}{2} \right) \sin \left( \frac{\varepsilon_o - \theta_o}{2} \right) \right| \right| \ll_q \frac{1}{q^{a/2}}. $$
Proof. For any $o \in \mathbf{O}_q(a)$, we have by Lemma 5.8 ii), $\frac{\varepsilon_o+\theta_o}{2}$, $\frac{\varepsilon_o-\theta_o}{2} \notin \{0, \pi\}$, implying the inequalities
$0 < |\sin(\frac{\varepsilon_o+\theta_o}{2})\sin(\frac{\varepsilon_o-\theta_o}{2})| \le 1$ for any $o \in \mathbf{O}_q(a)$. All the terms in the sum are non-positive, so

$$
\left| \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| \right| \\
= \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} \log \left( \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|^{-1} \right).
$$

Let $o \in \mathbf{O}_q(a)$, recall that $\varepsilon_o$ belongs to the interval $[0, 2\pi)$ and $\theta_o$ belongs to $[0, \pi]$ so $\frac{\varepsilon_o+\theta_o}{2}, \frac{\varepsilon_o-\theta_o}{2}$ both belong to $[-\frac{\pi}{2}, \frac{3\pi}{2}) \setminus \{0, \pi\}$. For any $x \in [-\frac{\pi}{2}, \frac{3\pi}{2}) \setminus \{0, \pi\}$, we have $\frac{1}{|\sin(x)|} \le \frac{\pi^2}{|x(\pi-x)|}$. Applied to $x = \frac{\varepsilon_o+\theta_o}{2}$ and $x = \frac{\theta_o-\varepsilon_o}{2}$, this upper bound gives

$$
\left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|^{-1} \le \frac{(2\pi)^4}{\left| (\varepsilon_o + \theta_o)(2\pi - \theta_o - \varepsilon_o)(\varepsilon_o - \theta_o)(2\pi - \theta_o + \varepsilon_o) \right|}.
$$

Using Corollary 5.9 we get

$$
\log \left( \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|^{-1} \right) \le \log\left(\frac{(2\pi)^4}{q^{-4\sigma_p}|o|}\right) \ll_q |o|.
$$

Summing over non-maximal orbits $o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)$, we obtain

$$
\left| \frac{1}{q^a} \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| \right| \le \frac{1}{q^a} \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\max}(a)} |o|,
$$

and Lemma 6.5 allows to conclude.

<figure><img src="image_8.png" /></figure>

6.2 Lower bounds on the sum over maximal orbits

In this section we give lower bounds on $\frac{1}{q^a} \sum_{o \in \mathbf{O}_q^{\max}(a)} \log |\sin(\frac{\varepsilon_o+\theta_o}{2}) \sin(\frac{\varepsilon_o-\theta_o}{2})|$, which is the term we still need to control in order to establish Theorem 6.1.

We begin by bounding the term inside the logarithm. Let us briefly recall the description of Gauss angles given in Proposition (5.1). When $q \equiv 1 \pmod 3$, there is a map $\text{pr}_1 : \mathbf{O}_q(a) \to \{\pm 1\}$ lifting the first coordinate of any orbit in $\mathbf{O}_q(a)$. Moreover, in this case, there is an irrational number $\phi \in [0, 2\pi)$ such that for any $o \in \mathbf{O}_q^{\max}(a)$, the angle $\varepsilon_o$ is of the form

$$
\varepsilon_o = \begin{cases} \text{pr}_1(o)a\phi + \frac{2j\pi}{3} & \text{with } j \in \{0, 1, 2\} \text{ if } q \equiv 1 \pmod 3, \\ \frac{j\pi}{3} & \text{with } j \in \{0, \dots, 5\} \text{ if } q \equiv 2 \pmod 3. \end{cases} \tag{6.4}
$$

As a simple exercise in real analysis reveals, we have, for any $x \in [0, 2\pi)$,

$$
\left( \frac{\sin(6x)}{4} \right)^2 \le \min_{j=0, \dots, 5} \left| \sin \left( x + \frac{j\pi}{6} \right) \right|. \quad (6.5)
$$

Define $\lambda_a := a\phi$ if $q \equiv 1 \pmod 3$ and $\lambda_a := 0$ if $q \equiv 2 \pmod 3$. We can write, independently of $q \pmod 3$
and the parity of $a$, $\varepsilon_o = \text{pr}_1(o)\lambda_a + j\pi/3$, for some $j \in \{0, \dots, 5\}$. With this notation

$$
\frac{\varepsilon_o + \theta_o}{2} = \frac{\theta_o \pm \lambda_a}{2} + \frac{j\pi}{6} \quad \text{and} \quad \frac{\varepsilon_o - \theta_o}{2} = \frac{-\theta_o \pm \lambda_a}{2} + \frac{j\pi}{6}
$$

where $j \in \{0, \dots, 5\}$ and the sign before $\lambda_a$ is $\text{pr}_1(o)$ for both terms. Since $y \mapsto (\sin y)^2$ is even and $\pi$-periodic, applying inequality (6.5) to $x = (\theta_o \pm \lambda_a)/2$ we get

$$
\left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| \ge \left(\frac{\sin(3(\theta_o + \lambda_a))}{4}\right)^2 \left(\frac{\sin(3(\theta_o - \lambda_a))}{4}\right)^2. \quad (6.6)
$$

The right-hand side is positive, because Proposition 5.1 ensures that any $\pm\lambda_a + j\pi/3$ for $j \in \{0, \dots, 5\}$ is the angle $\varepsilon_o'$ of a Gauss sum $\gamma(o')$ for some $o' \in \mathbf{O}_q^{\max}(a)$. Thus Lemma 5.8 guarantees that $3(\theta_o \pm \lambda_a) \not\equiv 0 \pmod \pi$.
**Definition 6.7.** For any $r \in \mathbb{Z} \setminus \{0\}$, any $\lambda \in [0, \pi]$, define a function $W^{r,\lambda} : [0, \pi] \to (-\infty, 0]$ by

$$W^{r,\lambda}(x) = \begin{cases} \log|\sin(r(x + \lambda))| & \text{if } r(x + \lambda) \not\equiv 0 \mod \pi, \\ 0 & \text{if } r(x + \lambda) \equiv 0 \mod \pi. \end{cases}$$

Since both terms in (6.6) are positive, we use the definition above to write

$$\sum_{o \in \mathbf{O}_q^{\max}(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| \\ \geq 2 \sum_{o \in \mathbf{O}_q^{\max}(a)} (W^{3,\lambda_a}(\theta_o) + W^{3,-\lambda_a}(\theta_o)) - 4 \log 4 |\mathbf{O}_q^{\max}(a)|. \quad (6.7)$$

The sum over $o \in \mathbf{O}_q^{\max}(a)$ in the right-hand side of (6.7) may be interpreted as an integral over $[0, \pi]$. Recall indeed from Definition 5.4 that for any $a \ge 1$, the measure $\nu_a$ is given by $\nu_a := \frac{1}{\Pi_q(a)} \sum_{v \in \mathbf{P}_q^{\max}(a)} \delta\{\theta_v\}$.

**Proposition 6.8.** For any $a \ge 1$, let $\lambda_a$ be as above and define $r_a := 3$ if $q \equiv 1 \mod 3$ or $a$ is even, and $r_a = 6$ if $q \equiv 2 \mod 3$ and $a$ is odd. Define also $\varpi(a) := 1$ if $q \equiv 2 \mod 3$ and $a$ is divisible exactly once by 2, and $\varpi(a) := 0$ if not. Then we have

$$\frac{1}{q^a} \sum_{o \in \mathbf{O}_q^{\max}(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| \\ \geq \frac{4\Pi_q(a)}{q^a} \int_{[0,\pi]} W^{r_a, \lambda_a} + W^{r_a, -\lambda_a} d\nu_a + \varpi(a) \frac{4\Pi_q([a/2])}{q^a} \int_{[0,\pi]} W^{r_{[a/2]}, \lambda_{[a/2]}} d\nu_{[a/2]} + O\left(\frac{1}{a}\right).$$

*Proof.* We start by rewriting the sum in the right-hand side of (6.7) over $o \in \mathbf{O}_q^{\max}(a)$ as a sum (or sums) over $v \in \mathbf{P}_q^{\max}(a)$. To do so, we use the case-by-case analysis done in Remark 6.4. Remark 5.3 states that for any $o \in \mathbf{O}_q^{\max}(a)$ letting $v := \Theta(v)$ we have $\theta_o \equiv \frac{|o|}{|v|} \theta_v \mod \pi$. Treating separately each of the different cases in Remark 6.4 gives:

i) if $q \equiv 1 \mod 3$, then $\sum_{o \in \mathbf{O}_q^{\max}(a)} W^{3,\lambda_a}(\theta_o) = 2 \sum_{v \in \mathbf{P}_q^{\max}(a)} W^{3,\lambda_a}(\theta_v)$,

ii) if $q \equiv 2 \mod 3$ and $a$ is odd, then $\sum_{o \in \mathbf{O}_q^{\max}(a)} W^{3,0}(\theta_o) = \sum_{v \in \mathbf{P}_q^{\max}(a)} W^{6,0}(\theta_v)$,

iii) if $q \equiv 2 \mod 3$ and $4|a$, then $\sum_{o \in \mathbf{O}_q^{\max}(a)} W^{3,0}(\theta_o) = 2 \sum_{v \in \mathbf{P}_q^{\max}(a)} W^{3,0}(\theta_v)$,

iv) if $q \equiv 2 \mod 3$ and $2$ divides $a$ exactly once, then

$$\sum_{o \in \mathbf{O}_q^{\max}(a)} W^{3,0}(\theta_o) = 2 \sum_{v \in \mathbf{P}_q^{\max}(a)} W^{3,0}(\theta_v) + \sum_{\substack{v \in \mathbf{P}_q(a) \\ |v|=a/2}} W^{6,0}(\theta_v).$$

In the first case, the same formula holds if one replaces $W^{3,\lambda_a}$ by $W^{3,-\lambda_a}$. In the last three cases, $\lambda_a = 0$, so we simply have $W^{3,\lambda_a} + W^{3,-\lambda_a} = 2W^{3,0}$. Using the definition of $r_a$ given in the proposition, we obtain in cases i), ii) and iii)

$$\sum_{o \in \mathbf{O}_q^{\max}(a)} (W^{3,\lambda_a}(\theta_o) + W^{3,-\lambda_a}(\theta_o)) \geq 2 \sum_{v \in \mathbf{P}_q^{\max}(a)} (W^{r_a, \lambda_a}(\theta_v) + W^{r_a, -\lambda_a}(\theta_v)). \quad (6.8)$$

In the fourth case we have the extra term $\sum_{|v|=a/2} W^{6,0}(\theta_v) = \sum_{|v|=a/2} W^{r_a/2, \lambda_a/2}(\theta_v)$. Using the term $\varpi(a)$ defined in the proposition and dividing by $q^a$, we get for any value of $a$

$$\frac{1}{q^a} \sum_{o \in \mathbf{O}_q^{\max}(a)} (W^{3,\lambda_a}(\theta_o) + W^{3,-\lambda_a}(\theta_o)) \\ \geq \frac{2}{q^a} \sum_{v \in \mathbf{P}_q^{\max}(a)} (W^{r_a, \lambda_a}(\theta_v) + W^{r_a, -\lambda_a}(\theta_v)) + \varpi(a) \frac{2}{q^a} \sum_{|v|=|a/2|} W^{6,0}(\theta_v) \\ = \frac{2\Pi_q(a)}{q^a} \int_{[0,\pi]} W^{r_a, \lambda_a} + W^{r_a, \lambda_a} d\nu_a + \varpi(a) \frac{2\Pi_q([a/2])}{q^a} \int_{[0,\pi]} W^{6,0} d\nu_{[a/2]},$$
where the equality uses the definition of $\nu_a$. Combining this with (6.7) and using the estimate $4\log 4 |O_q^{\max}(a)|/q^a = O(1/a)$ gives the desired result. $\square$

## 6.3 Bounding a sequence of integrals

In this section, we bound the integrals appearing in Proposition 6.8.

**Proposition 6.9.** Keep notation as in the previous section and choose $\epsilon \in \{\pm 1\}$. Then the sequence of integrals $\left(\int_{[0,\pi]} W^{r_a, \epsilon\lambda_a} d\nu_a\right)_{a \ge 1}$ is bounded.

If we were able to apply the equidistribution results from Section 5.2 to $W^{r_a, \lambda_a}$, then Proposition 6.9 would easily follow. Indeed, Theorem 5.5 would imply that the sequence of integrals appearing in Proposition 6.9 converges and is therefore bounded. However, the function $W^{r_a, \lambda_a}$ is not continuous on $[0, \pi]$, so we cannot directly apply Theorem 5.5. Instead, we approximate $W^{r_a, \lambda_a}$ by continuously differentiable functions on $[0, \pi]$ whose integral with respect to $\nu_a$ we control.

For $r \in \mathbb{Z} \setminus \{0\}$ and $\lambda \in [0, \pi]$, let $\mathcal{D}_{r,\lambda} := \{\frac{j\pi}{r} - \lambda; j \in \mathbb{Z}\} \cap [0, \pi]$. The function $W^{r,\lambda}$ is $\frac{\pi}{r}$-periodic and has $r$ points of discontinuity, namely the elements of $\mathcal{D}_{r,\lambda}$. Moreover, $W^{r,\lambda}$ is symmetric around each of the elements of $\mathcal{D}_{r,\lambda}$, meaning that if $x_0 \in \mathcal{D}_{r,\lambda}$, then

$$W^{r,\lambda}(x_0 - x) = W^{r,\lambda}(x_0 + x) \quad \text{for any } x \in [0, \pi]. \qquad (6.9)$$

We first give an explicit bound on the integral of $W^{r,\lambda}$ with respect to the Sato-Tate measure:

**Lemma 6.10.** For any $r \in \mathbb{Z} \setminus \{0\}$ and any $\lambda \in [0, \pi]$, we have

$$\left| \int_{[0, \pi]} W^{r, \lambda} d\nu_{\text{ST}} \right| \le \log 4.$$

*Proof.* Using the trivial upper bound $\sin^2 \le 1$, the fact that $W^{r,\lambda}$ is non-positive on $[0, \pi]$ and doing the change of variables $u = r(x + \lambda)$ gives

$$\begin{aligned} \left| \int_{[0, \pi]} W^{r, \lambda} d\nu_{\text{ST}} \right| &= \left| \frac{2}{\pi} \int_{0}^{\pi} \log |\sin(r(x + \lambda)) \sin^2(x)| dx \right| \\ &\le -\frac{2}{r\pi} \int_{r\lambda}^{r\pi+r\lambda} \log|\sin(u)| du = -\frac{2}{\pi} \int_{0}^{\pi} \log|\sin(u)| du, \end{aligned}$$

where the last equality follows by $\pi$-periodicity of $u \mapsto \log|\sin(u)|$. Now it is well-known that $\int_0^\pi \log(\sin(u))du = -\pi \log 2$ (see [GR15, 4.224.3]), hence the result. $\square$

We now construct continuously differentiable functions on $[0, \pi]$ approximating $W^{r,\lambda}$ and vanishing around the points of discontinuity of $W^{r,\lambda}$. Let $\beta_0 : [0, 1] \to [0, 1]$ be an infinitely differentiable function such that $\beta_0(x) = 0$ for $x \in [0, \frac{1}{3}]$ and $\beta_0(x) = 1$ for $x \in [\frac{2}{3}, 1]$. We explicitly choose $\beta_0$ as follows: for any $x \in (0, 1)$, let $h(x) := \exp(-1/x)$ and $k(x) := h(x)/(h(1-x)+h(x))$. For any $x \in (1/3, 2/3)$ we let $\beta_0(x) := k(3x-1)$. For any $\eta \in (0, \frac{\pi}{2})$, consider the function $\beta_\eta : [0, \pi] \to [0, 1]$ defined as follows. For any $x \in [0, \pi]$, let

$$\beta_{\eta}(x) := \begin{cases} \beta_0\left(\frac{x}{\eta}\right) & \text{if } x \in [0, \eta], \\ 1 & \text{if } x \in [\eta, \pi - \eta], \\ \beta_0\left(\frac{\pi - x}{\eta}\right) & \text{if } x \in [\pi - \eta, \pi]. \end{cases}$$

Consider now $r \in \mathbb{Z} \setminus \{0\}$, $\lambda \in [0, \pi]$, and $\eta \in (0, \frac{\pi}{2r})$. Let $\beta_\eta^{r,\lambda} : [0, \pi] \to [0, 1]$ by $\beta_\eta^{r,\lambda}(x) := \beta_{r\eta}(r(x+\lambda))$ and $\Delta_\eta^{r,\lambda} := \{x \in [0, \pi] \mid \text{dist}(x, \mathcal{D}_{r,\lambda}) \ge \eta\}$, we have $\beta_\eta^{r,\lambda}(x) = 1$ for any $x \in \Delta_\eta^{r,\lambda}$.

Define $W_\eta^{r,\lambda} := W^{r,\lambda} \beta_\eta^{r,\lambda}$, this is a continuously differentiable function on $[0, \pi]$ which coincides with $W^{r,\lambda}$ on $\Delta_\eta^{r,\lambda}$ and which vanishes in the neighbourhood of elements of $\mathcal{D}_{r,\lambda}$. More precisely, for any $x_0 \in \mathcal{D}_{r,\lambda}$ and any $x \in [x_0 - \eta/3, x_0 + \eta/3]$, we have $W_\eta^{r,\lambda}(x) = 0$. In addition to that, $W_\eta^{r,\lambda}$ is also $\frac{\pi}{r}$-periodic and symmetric around each of the points of discontinuity of $W^{r,\lambda}$, meaning that equation (6.9) holds if one replaces $W^{r,\lambda}$ by $W_\eta^{r,\lambda}$.
**Lemma 6.11.** Let $r \in \mathbb{Z} \setminus \{0\}$, and $\lambda \in [0, \pi]$. There is a positive constant $c(r)$ depending only on $r$ such that, for any $\eta \in (0, \frac{\pi}{2r})$, we have

$$ \int_{0}^{\pi} |(W_{\eta}^{r, \lambda})'(x)| dx \leq c(r)|\log \eta|. $$

*Proof.* First of all, notice that $(W_{\eta}^{r,\lambda})'$ is $\pi$-periodic (it is even $\frac{\pi}{r}$-periodic), so that it suffices to treat the special case $\lambda = 0$. When $\lambda = 0$, the points of discontinuity of $W_{r,0}$ are those in $D_{r,0} = \{\frac{j\pi}{r}; j \in \{0, \dots, r-1\}\}$. To lighten notation, we write $W$ for $W^{r,0}$, $\beta_{\eta}$ for $\beta_{\eta}^{r,0}$ and $W_{\eta}$ for $W_{\eta}^{r,0}$. By $\frac{\pi}{r}$-periodicity and symmetry of $W'$ around each $\frac{j\pi}{r}$, it is enough to prove that

$$ \int_{0}^{\pi/2r} |W'_{\eta}(x)| dx \ll_{r} |\log \eta|. \qquad (6.10) $$

For any $x \in [0, \frac{\pi}{2r}]$, we have $\beta_{\eta}(x) \le 1$ so $|W'_{\eta}(x)| \le |W'(x)| + |\beta'_{\eta}(x)| |W(x)|$. Write $\|\beta'_{0}\|_{\infty}$ for the sup norm of $\beta'_{0}$ on $[0,1]$, then we have $\beta'_{\eta}(x) = 0$ for any $x \in [\eta, \frac{\pi}{2r}]$, and $|\beta'_{\eta}| \le \frac{\|\beta'_{0}\|_{\infty}}{\eta}$ for $x \in [0, \eta]$. With the specific choice for $\beta_{0}$ made above, one can check that $\|\beta'_{0}\|_{\infty} = 6$. Secondly, we bound $W$. It is well-known that $\sin(u) \ge \frac{2u}{\pi}$ for any $u \in [0, \pi/2]$. This implies that, for any $x \in (0, \frac{\pi}{2r}]$, we have $|W(x)| = \log(\frac{1}{\sin(rx)}) \le \log(\frac{\pi}{2rx})$. Finally, to control $W'$, we use the same lower bound on $\sin$ to deduce, for any $x \in (0, \frac{\pi}{2r}]$,

$$ |W'(x)| = \left| \frac{r \cos(rx)}{\sin(rx)} \right| \le \frac{r}{\sin(rx)} \le \frac{\pi}{2x}. $$

With these inequalities at hand, we can estimate the integral displayed in (6.10), which we split into integrals over $(0, \eta]$ and $[\eta, \frac{\pi}{2r}]$. $W_{\eta}$ is constant equal to 0 over $[0, \frac{\eta}{3}]$, hence $W'_{\eta} = 0$ on this interval, and

$$ \begin{aligned} \int_{0}^{\eta} |W'_{\eta}(x)| dx &= \int_{\eta/3}^{\eta} |W'_{\eta}(x)| dx + \int_{\eta/3}^{\eta} |\beta'_{\eta}(x)| |W(x)| dx \\ &\le \int_{\eta/3}^{\eta} |W'(x)| dx + \int_{\eta/3}^{\eta} |\beta'_{\eta}(x)| |W(x)| dx \\ &\le \int_{\eta/3}^{\eta} \frac{\pi}{2x} dx + \frac{6}{\eta} \int_{\eta/3}^{\eta} \log\left(\frac{\pi}{2rx}\right) dx \ll_{r} |\log \eta|. \end{aligned} $$

To estimate the integral of $|W'_{\eta}|$ on $[\eta, \frac{\pi}{2r}]$ we combine the bounds established above with the fact that $\beta'_{\eta}(x) = 0$ for any $x \in [\eta, \frac{\pi}{2r}]$:

$$ \int_{\eta}^{\pi/2r} |W'_{\eta}(x)| dx \le \int_{\eta}^{\pi/2r} |W'(x)| dx \le \int_{\eta}^{\pi/2r} \frac{\pi}{2x} dx \ll_{r} |\log \eta|. $$

Both implicit constants depend only on $r$, so adding the two bounds gives the desired result. $\square$

We have now all the necessary tools to prove Proposition 6.9.

**Proof of Proposition 6.9.** We show that the sequence of integrals of $W^{r_a, \lambda_a}$ is bounded, the proof would be the same if one replaced $W^{r_a, \lambda_a}$ by $W^{r_a, -\lambda_a}$ (this corresponds to $\epsilon = -1$ in the notation of the proposition). We use the triangle inequality to obtain

$$ \left| \int_{[0, \pi]} W^{r_a, \lambda_a} d\nu_a \right| \le \left| \int_{[0, \pi]} W^{r_a, \lambda_a} - W_{\eta}^{r_a, \lambda_a} d\nu_a \right| + \left| \int_{[0, \pi]} W_{\eta}^{r_a, \lambda_a} d\nu_a - \int_{[0, \pi]} W_{\eta}^{r_a, \lambda_a} d\nu_{ST} \right| + \left| \int_{[0, \pi]} W_{\eta}^{r_a, \lambda_a} d\nu_{ST} \right|. $$

We claim that the first integral of the right-hand side vanishes if $\eta$ is small enough. Recall from Section 5.3 the definition of the constants $\sigma_p$ and $\tau_{p,N}$. Fix $\eta := (\min((q^a)^{-\sigma_p}, (q^a)^{-\tau_{p,3}}, (q^a)^{-\tau_{p,6}}))^{1/2}$ (using the assumption $p \ge 7$, one can check that $\eta < \pi/12$). The points of discontinuity of $W^{r_a, \lambda_a}$
are either angles of Gauss sums or of the form $\frac{j\pi}{6}$ for some $j \in \{0, \dots, 5\}$. Theorem 5.7 implies that $\mathrm{dist}(\theta_v, \mathcal{D}_{r_a, \lambda_a}) \ge \eta$ for any $v \in \mathbf{P}_q^{\mathrm{max}}(a)$. This means that the support of the measure $\nu_a$ is contained in $\Delta_{\eta_a}^{r_a, \lambda_a}$. By construction of $W_{\eta}^{r_a, \lambda_a}$ we have $W^{r_a, \lambda_a}(x) = W_{\eta}^{r_a, \lambda_a}(x)$ for any $x \in \Delta_{\eta_a}^{r_a, \lambda_a}$, so we deduce that $\int_{[0,\pi]} |W^{r_a, \lambda_a} - W_{\eta}^{r_a, \lambda_a}| d\nu_a = 0$.

We show that the second term is small when $a \to \infty$. Indeed, the function $W_{\eta}^{r_a, \lambda_a}$ is continuously differentiable on $[0, \pi]$, and we may apply Theorem 5.6. Combining it with Lemma 6.11 gives

$$ \left| \int_{[0,\pi]} W_{\eta}^{r_a, \lambda_a} d\nu_a - \int_{[0,\pi]} W_{\eta}^{r_a, \lambda_a} d\nu_{\mathrm{ST}} \right| \ll_q \frac{a^{1/2}}{q^{a/4}} \int_0^{\pi} |(W_{\eta}^{r_a, \lambda_a})'(x)| dx \ll_q c(r_a) \frac{a^{1/2}}{q^{a/4}} |\log \eta|. $$

Having chosen $\eta = (\min((q^a)^{-\sigma_p}, (q^a)^{-\tau_{p,3}}, (q^a)^{-\tau_{p,6}}))^{1/2}$, we have $|\log \eta| \ll_q a$, so

$$ \left| \int_{[0,\pi]} W_{\eta}^{r_a, \lambda_a} d\nu_a - \int_{[0,\pi]} W_{\eta}^{r_a, \lambda_a} d\nu_{\mathrm{ST}} \right| \ll_q c(r_a) \frac{a^{3/2}}{q^{a/4}} \ll_q \max(c(3), c(6)) \frac{a^{3/2}}{q^{a/4}}. \quad (6.11) $$

Finally, the third term is bounded independently of $a$. Indeed, $|W_{\eta}^{r_a, \lambda_a}(x)| \le |W^{r_a, \lambda_a}(x)|$ for any $x \in [0, \pi]$, so Lemma 6.10 implies $\left|\int_{[0,\pi]} W_{\eta}^{r_a, \lambda_a} d\nu_{\mathrm{ST}}\right| \le \log 4$ for any $a \ge 1$. Adding up the three contributions:

$$ \left| \int_{[0,\pi]} W^{r_a, \lambda_a} d\nu_a \right| \le 0 + O\left(\frac{a^{3/2}}{q^{a/4}}\right) + \log 4 \ll_q 1. $$

□

## 6.4 Conclusion of the proof of Theorem 6.1

In this section we conclude the proof of Theorem 6.1 by establishing the lower bound appearing in it.

*Proof of the lower bound in 6.1.* As discussed in Remark 6.2, it is sufficient to prove that we have $\log L^*(S_a)/q^a \gg_q 1/a$. Recall from (6.3) that

$$ \frac{\log L^*(S)}{q^a} = \log 4 \left|\frac{\mathbf{O}_q(a)}{q^a}\right| + \frac{1}{q^a} \sum_{o \in \mathbf{O}_q(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right|. \quad (6.12) $$

Lemma 3.3 iii) implies that $\log 4|\mathbf{O}_q(a)|/q^a = O(1/a)$, and Proposition 6.6 gives

$$ \frac{1}{q^a} \sum_{o \in \mathbf{O}_q(a) \setminus \mathbf{O}_q^{\mathrm{max}}(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| = O\left(\frac{1}{q^{a/2}}\right). $$

Plugging these estimates into (6.12) we get

$$ \frac{\log L^*(S_a)}{q^a} = \frac{1}{q^a} \sum_{o \in \mathbf{O}_q^{\mathrm{max}}(a)} \log \left| \sin\left(\frac{\varepsilon_o + \theta_o}{2}\right) \sin\left(\frac{\varepsilon_o - \theta_o}{2}\right) \right| + O\left(\frac{1}{a}\right). $$

Recall from Lemma 3.3 i) that $\Pi_q(a)/q^a \ge 1/a$. Combining this with Proposition 6.8 implies that

$$ \frac{\log L^*(S_a)}{q^a} \ge \frac{4}{a} \int_{[0,\pi]} (W^{r_a, \lambda_a} + W^{r_a, -\lambda_a}) d\nu_a + \frac{8\pi(a)}{a} \int_{[0,\pi]} W^{r_{[a/2]}, \lambda_{[a/2]}} d\nu_{[a/2]} + O\left(\frac{1}{a}\right). $$

Proposition 6.9 states that for any $\epsilon \in \{\pm 1\}$, the integrals $\int_{[0,\pi]} W^{r_a, \epsilon\lambda_a} d\nu_a$ form a bounded sequence, so the right-hand side in the inequality above is a $O(1/a)$, finally giving the desired result. □

## 7 Large Tate–Shafarevich groups and the Brauer–Siegel ratio

In this section we conclude the proof of Theorem 1.2, and deduce various other properties on the Tate–Shafarevich groups of the surfaces $S_a$. We finally reinterpret Theorem 1.2 as an analogue of the Brauer–Siegel theorem.
## 7.1 Large Tate–Shafarevich groups

From Section 4 we know that the surface $S_a$ satisfies the BSD conjecture for all $a \ge 1$ (see Theorem 4.4): this implies in particular that its Tate–Shafarevich group $\text{III}(S_a)$ is finite. We now deduce from Theorem 6.1 the main result of our article, which describes the rate of growth of the Tate–Shafarevich groups of $S_a$.

**Theorem 7.1.** As $a \to \infty$, we have the estimate

$$|\text{III}(S_a)| = H(S_a)^{1+O(1/a)}, \qquad (7.1)$$

The implicit constant in (7.1) is effective and depends only on $q$.

*Proof.* By Corollary 4.12, we have $\text{Reg}(S_a) = 1$. Taking the logarithm of both sides of the BSD formula (4.1) and rearranging terms gives

$$\frac{\log|\text{III}(S_a)|}{\log H(S_a)} = 1 + \frac{\log L^*(S)}{\log H(S_a)} - \frac{\log(q^2 \prod_{v \in M_K} c_v(S_a)) - 2 \log|S_a(\mathbb{K})_{\text{tor}}|}{\log H(S_a)}. \qquad (7.2)$$

From Section 2.4 we know that $\prod_{v \in M_K} c_v(S_a) = c_\infty(S_a)|9$. On the other hand, Theorem 3.8 in [HP16] states that $|S_a(\mathbb{K})_{\text{tor}}|^2 \ll_q h(S_a)^4$. Proposition 2.15 yields $h(S_a) = q^a + 1$, so we deduce that $\frac{2\log|S_a(\mathbb{K})_{\text{tor}}|}{\log H(S_a)} \ll_q \frac{a}{q^a}$. On the other hand, Theorem 6.1 states $\log L^*(S_a)/\log H(S_a) = O(1/a)$, so substituting all these estimates in (7.2) gives

$$\frac{\log|\text{III}(S_a)|}{\log H(S_a)} = 1 + O\left(\frac{1}{a}\right). \qquad (7.3)$$

The statement follows by exponentiating (7.3). $\square$

## 7.2 An analogue of the Brauer–Siegel theorem

In this section we view Theorem 7.1 as an analogue of the Brauer–Siegel theorem. In [HP16], Hindry and Pacheco defined the Brauer–Siegel ratio of an abelian variety $A/\mathbb{K}$ whose Tate–Shafarevich group is finite. It is given by

$$\mathfrak{B}_5(A) := \frac{\log(|\text{III}(A)| \text{Reg}(A))}{\log H(A)}.$$

In our setting, for any $a \ge 1$, the abelian surface $S_a$ has Mordell–Weil rank 0, so its Néron–Tate regulator is trivial and the Brauer–Siegel ratio is $\mathfrak{B}_5(S_a) = \log|\text{III}(S_a)|/\log H(S_a)$. With this notation, Theorem 7.1 may be rewritten in the following way:

**Corollary 7.2.** As $a \to \infty$, we have

$$\mathfrak{B}_5(S_a) = 1 + O(1/a).$$

We refer the reader to [Hin07, HP16] for a detailed discussion on the definition and properties of the Brauer–Siegel ratio and more on the analogy with the original Brauer–Siegel theorem.

In [Ulm19], Ulmer associates a new invariant, called the dimension of III, to an abelian variety $A/\mathbb{K}$ with finite Tate–Shafarevich group. For any $n \ge 1$, let $\mathbb{K}_n := \mathbb{F}_{q^n}(t)$ and define

$$\dim \text{III}(S_a) := \lim_{n \to \infty} \frac{\log|\text{III}(S_a \times_{\mathbb{K}} \mathbb{K}_n)[p^\infty]|}{\log(q^n)}.$$

Ulmer has proved (see [Ulm19, Proposition 4.1]) that this limit indeed exists and is a non-negative integer. One of the interests of $\dim \text{III}(S_a)$ is that it provides a lower bound on $\mathfrak{B}_5(S_a)$ (see Proposition 4.6 in [Ulm19]):

$$\frac{\dim \text{III}(S_a)}{h(S_a)} \le \mathfrak{B}_5(S_a) + o(1) \quad \text{as } a \to \infty. \qquad (7.4)$$

**Proposition 7.3.** For any $a \ge 1$, we have

$$\dim \text{III}(S_a) = 0.$$
Proof. We use Proposition 4.2 from [Ulm19], which gives a closed formula for dim III(S$_{a}$) in terms of h(S$_{a}$) and the p-adic slopes of L(S$_{a}$, T). For any o ∈ O$_{q}$ (a) and any i ∈ {1, 2}, the Newton polygon of 1 − γ (o) κ$_{i}$ (o) T | o | ∈ Z [ T ] is a segment of length | o | and slope ord$_{p}$ ( γ (o) κ$_{i}$ (o)). With our notation, the proposition mentioned above implies

$$
\dim \mathrm{III}(S_a) = h(S_a) - 2 - \sum_{o \in \mathbf{O}_q(a)} \sum_{i=1}^{2} \max\{0, |o| - \mathrm{ord}_p(\gamma(o)\kappa_i(o))\}.
$$

Using (K|4), and switching $\kappa_1(o)$ and $\kappa_2(o)$ if necessary, we may assume that $\mathrm{ord}_p(\kappa_1(o)) = 0$ and
$\mathrm{ord}_p(\kappa_2(o)) = |o|$.

When $q \equiv 1 \mod 3$, Lemma 3.5 implies that for any $o \in \mathbf{O}_q(a)$, we have

$$
\mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_1(o)) = \begin{cases} \frac{2|o|}{3} & \text{if } \mathrm{pr}_1(o) = 1, \\ \frac{|o|}{3} & \text{if } \mathrm{pr}_1(o) = -1, \end{cases} \quad \text{and} \quad \mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_2(o)) = \begin{cases} \frac{5|o|}{3} & \text{if } \mathrm{pr}_1(o) = 1, \\ \frac{4|o|}{3} & \text{if } \mathrm{pr}_1(o) = -1. \end{cases}
$$

Therefore, we get

$$
\sum_{o \in \mathbf{O}_q(a)} \sum_{i=1}^{2} \max\{0, |o| - \mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_i(o))\} = \frac{1}{3} \sum_{\substack{o \in \mathbf{O}_q(a) \\ \mathrm{pr}_1(o)=1}} |o| + \frac{2}{3} \sum_{\substack{o \in \mathbf{O}_q(a) \\ \mathrm{pr}_1(o)=-1}} |o|.
$$

Since $q \equiv 1 \mod 3$, the map $\Theta$ establishes a bijection between $\{o \in \mathbf{O}_q(a) \mid \mathrm{pr}_1(o) = 1\}$ and $\mathbf{P}_q(a)$ (in the same way $\{o \in \mathbf{O}_q(a) \mid \mathrm{pr}_1(o) = -1\}$ and $\mathbf{P}_q(a)$ are also in bijection). Moreover, we have $|o| = |\Theta(o)|$ for any $o \in \mathbf{O}_q(a)$, so we get

$$
\sum_{o \in \mathbf{O}_q(a)} \sum_{i=1}^{2} \max\{0, |o| - \mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_i(o))\} = \left(\frac{1}{3} + \frac{2}{3}\right) \sum_{v \in \mathbf{P}_q(a)} |v| = |\mathbb{F}_{q^a}^\times| = q^a - 1.
$$

When $q \equiv 2 \mod 3$, we deduce from Lemma 3.5 that

$$
\mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_1(o)) = |o|/2 \quad \text{and} \quad \mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_2(o)) = 3|o|/2.
$$

Following the same line of reasoning as above:

$$
\begin{align*}
\sum_{o \in \mathbf{O}_q(a)} \sum_{i=1}^{2} \max\{0, |o| - \mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_i(o))\} &= \sum_{o \in \mathbf{O}_q(a)} (|o| - \mathrm{ord}_{\mathbf{p}}(\gamma(o)\kappa_1(o))) \\
&= \sum_{o \in \mathbf{O}_q(a)} \frac{|o|}{2} = \frac{1}{2} \left| \left(\mathbb{Z}/3\mathbb{Z}\right)^\times \times \mathbb{F}_{q^a}^\times \right| = q^a - 1.
\end{align*}
$$

In both cases we obtain

$$
\dim \mathrm{III}(S_a) = q^a + 1 - 2 - (q^a - 1) = 0.
$$

<figure><img src="image_9.png" /></figure>

In [Ulm19], Ulmer shows that different families of abelian varieties have Brauer–Siegel ratios that tend to 1 by proving that their dimension of III are large enough. Notice that, in the case of the sequence $(S_a)_{a \ge 1}$, Proposition 7.3 implies that $\dim \mathrm{III}(S_a)/h(S_a) = 0$ for any $a \ge 1$, meaning that the algebraic approach presented in [Ulm19] is not sufficient to prove Corollary 7.2.

The Brauer–Siegel ratio has been studied for different sequences of elliptic curves over function
fields (see [HP16, Gri19, GU20, GdW21]) for which the Brauer–Siegel ratio is almost always near 1.
In [AGTT23, Ulm19], the authors study some families of abelian varieties of dimension greater than
1 whose Brauer–Siegel ratios tend to 1. The family $(S_a)_{a>1}$ studied in this article constitutes a new
example of higher-dimensional abelian varieties over $\mathbb{K}$ that are non-isotrivial and whose Brauer–
Siegel ratios tend to 1. This provides some more evidence that the Brauer–Siegel ratio of an abelian
variety over $\mathbb{K}$ should “often” be close to 1 when the exponential height is large, a phenomenon
already observed in the articles mentioned above (see [Hin07, HP16] for a discussion on why this is
the case).
**Acknowledgements** It is a pleasure to thank Richard Griffon for his encouragements and useful comments on earlier versions of this work. Special thanks go to Qing Liu and Céline Maistret for taking the time to explain details about the geometry of curves. The author would also like to thank Marc Hindry and Douglas Ulmer for fruitful discussions about various parts of the paper. This work was funded by the École Doctorale des Sciences Fondamentales, Université Clermont Auvergne.

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MARTIN AZON (martin.azon@uca.fr) – LABORATOIRE DE MATHÉMATIQUES B. PASCAL, UNIVERSITÉ CLERMONT AUVERGNE, Campus des Cézeaux. 3 place Vasarely, TSA 60026 CS 60026, 63178 Aubière Cedex (France).