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/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(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 →∞ .