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/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 [PU16], J satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon-Ulmer [GU20], 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(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 X(J) is "large" as q → ∞.