On the arithmetic of a family of superelliptic curves

TL;DR

This paper studies the arithmetic properties of a family of superelliptic curves over finite fields, computing their L-functions, ranks, and other invariants, and providing new examples of abelian varieties with unbounded rank.

Contribution

It generalizes previous work to explicitly compute L-functions and invariants of Jacobians of superelliptic curves, revealing new families with unbounded rank and growth of Tate-Shafarevich groups.

Findings

L-function expressed via Gauss sums

Existence of families with unbounded rank

Growth of Tate-Shafarevich group as q increases

Abstract

Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/\mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of $J$ appearing in BSD, including the rank of the Mordell--Weil group $J(\mathbb F_{r}(t))$, the Faltings height of $J$, and the Tamagawa numbers of $J$ in terms of the parameters $a,b,q$. For any $p$ and $r$, we show that for certain $a$ and $b$ depending only on $p$ and $r$, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as $q$ varies through powers of $p$. Under a different set of criteria on $a$ and $b$, we prove that the order of the Tate--Shafarevich group of $J$ grows quasilinearly in $q$ as $q \to \infty.$