Richard Griffon

We exhibit new sets of conditions which ensure that a family of number fields unconditionally satisfies the Brauer-Siegel theorem, as generalised by Tsfasman and Vladuts. We also give a few explicit examples of such families. In particular, we prove that the Brauer-Siegel theorem holds for any infinite global field contained in a $p$-class field tower.

Let $\mathbb{F}_r$ be a finite field of characteristic $p$. For any pair of coprime integers $a,b$, both prime to $p$, and any power $q$ of $p$, consider the superelliptic curve $C$ defined by $x^a+y^b=t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of the Jacobian $J$ of $C$.

Let $K$ be the function field of a smooth projective algebraic curve. We first describe the relation between the heights of two isogenous elliptic curves defined over $K$ (for various notions of height: modular height, differential height, stable height). In contrast to the case of elliptic curves over number fields the height is preserved under isogeny, unless the characteristic of $K$ divides the degree of the isogeny.

Secondly, we prove an "isogeny estimate" for elliptic curves over $K$ (in the vein of results of Masser-Wüstholz and Gaudron-Rémond concerning elliptic curves over a number field). In particular, if $K$ has characteristic $0$, this theorem says that, between two isogenous elliptic curves over $K$, there is an isogeny of degree $\leq c(K)$, where $c(K)$ is a constant depending at most on $K$.

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large.

More precisely, we consider certain infinite sequences of explicit elliptic curves $E$, for which we prove that their Tate-Shafarevich group $Ш(E)$ is finite and satisfies $|Ш(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes the exponential differential height of $E$. The elliptic curves in these sequences are pairwise neither isogenous nor geometrically isomorphic. We further show that the $p$-primary part of their Tate-Shafarevich group is trivial.

The proof involves explicitly computing the $L$-functions of these elliptic curves, proving the BSD conjecture for them, and obtaining estimates on the size of the central value of their $L$-function.

Let $\mathbb{F}_r$ be a finite field of characteristic $p>3$. For any power $q$ of $p$, consider the elliptic curve $E=E_{q,r}$ defined by $y^2=x^3 + t^q -t$ over $K=\mathbb{F}_r(t)$. We describe several arithmetic invariants of $E$ such as the rank of its Mordell-Weil group $E(K)$, the size of its Néron-Tate regulator $\mathrm{Reg}(E)$, and the order of its Tate-Shafarevich group $Ш(E)$ (which we prove is finite).

These invariants have radically different behaviours depending on the congruence class of $p$ modulo $6$. For instance $Ш(E)$ either has trivial $p$-part or is a $p$-group. On the other hand, we show that the product $|Ш(E)|\mathrm{Reg}(E)$ has size comparable to $r^{q/6}$ as $q\to\infty$, regardless of $p\,\mathrm{mod}\,6$.

Our approach relies on the BSD conjecture, an explicit expression for the $L$-function of $E$, and a geometric analysis of the Néron model of $E$.

Let $\mathbb{F}_q$ be a finite field of odd characteristic and $K= \mathbb{F}_q(t)$. For any integer $d\geq 1$, consider the elliptic curve $E_d$ over $K$ defined by $y^2=x\cdot\big(x^2+t^{2d}\cdot x-4t^{2d}\big)$.

The curves $E_d$ satisfy the BSD conjecture, so that their rank equals the order of vanishing of their $L$-function at the central point. We provide an explicit expression for the $L$-function of $E_d$, and use it to study this order of vanishing in terms of $d$. We deduce that the rank of the Mordell-Weil group $E_d(K)$ is unbounded as $d$ varies (this answers a question of L. Berger). We also prove that the average Mordell-Weil rank is unbounded.

We study the Artin-Schreier family of elliptic curves over $\mathbb{F}_q(t)$ defined by $y^2=x\big(x+1\big)\big(x+(t^{q^a}-t)^2\big)$ for all $a\geq 1$. We prove an asymptotic estimate on the size of the special value of their $L$-function in terms of the degree of their conductor; loosely speaking, we show that the special values are "asymptotically as large as possible".

We also provide an explicit expression for the $L$-function of the elliptic curves in the family. The proof of the main result uses this expression and a detailed study of the distribution of some character sums related to Kloosterman sums. Via the BSD conjecture (proved by Pries and Ulmer in this case), the main result translates into an analogue of the Brauer-Siegel theorem for these elliptic curves.

For a finite field $\mathbb{F}_q$ of characteristic $p\geq 5$ and $K=\mathbb{F}_q(t)$, we consider the family of elliptic curves $E_d$ over $K$ given by $y^2+xy - t^dy=x^3$ for all integers $d$ coprime to $q$. We provide an explicit expression for the $L$-functions of these curves in terms of Jacobi sums.

Moreover, we deduce from this calculation that the curves $E_d$ satisfy an analogue of the Brauer-Siegel theorem. More precisely, we estimate the asymptotic growth of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) by its Néron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.

We prove an analogue of the Brauer-Siegel theorem for the Legendre elliptic curves over $\mathbb{F}_q(t)$. More precisely, if $d \ge 1$ is an integer, we denote by $E_d$ the elliptic curve with model $y^2=x(x+1)(x+t^d)$ over $K=\mathbb{F}_q(t)$. We give an asymptotic estimate of the product of the order of the Tate-Shafarevich group of $E_d$ (which is known to be finite) with its Néron-Tate regulator, in terms of the exponential differential height of $E_d$, as $d\to\infty$.

We study Fermat surfaces over a finite field. Let $\mathbb{F}_q$ be a finite field and, for all $d$ coprime to $d$, let $F_d\subset \mathbb{P}^3$ be the Fermat surface of degree $d$ over $\mathbb{F}_q$, defined by $x_0^d+x_1^d+x_2^d+x_3^d=0$. We show that $$ \log\big( |\mathrm{Br}(F_d)|\cdot \mathrm{Reg}(F_d)\big) \sim \log q^{p_g(F_d)}\qquad \text{as }d\to\infty,$$ where $p_g(F_d)$ is the geometric genus of $F_d$, $\mathrm{Br}(F_d)$ denotes the Brauer group of $F_d/\mathbb{F}_q$ (which is known to be finite) and $\mathrm{Reg}(F_d)$ is the regulator of $F_d/\mathbb{F}_q$ (i.e. the absolute value of a Gram determinant of the Néron-Severi group of $F_d$ with respect to the intersection form).