On the arithmetic of a family of superelliptic curves

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 \(y^b+x^a=t^q-t\), and let J be the Jacobian of C. By work of Pries and Ulmer (Trans Am Math Soc 368(12):8553–8595, 2016), J satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon and Ulmer (Pacific J Math 305(2):597–640, 2020) , 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 grows exponentially fast in q as \(q \rightarrow \infty \).