On the arithmetic of a family of superelliptic curves

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 $C\mathrm{/}{\mathbb{F}}_r(t)$ 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 $q\to \mathrm{\infty }\mathrm{.}$