On the arithmetic of a family of superelliptic curves

2021/05/06 by Sarah Arpin, Richard Griffon, Arpin, Sarah +5
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.02812

openalex publication_date 2021/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 → ∞.

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