Abelian surfaces over $\mathbb{F}_{q}(t)$ with large Tate-Shafarevich groups

Abstract

We produce an explicit sequence $\left(S_a \right)_{a \geq 1}$ 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 $\left \arrowvert\mathrm{III}(S_a) \right \arrowvert = H(S_a)^{1 + o(1)}$ as $a \rightarrow \infty$, 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 N\'eron 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 $\mathrm{III}(S_a)$.