On torsion of superelliptic Jacobians

TL;DR

This paper provides a detailed structure theorem for the torsion points on superelliptic Jacobians, explores torsion existence on specific curves over finite fields, and applies these results to bound Mordell-Weil ranks over rationals.

Contribution

It introduces a new structure theorem for the m-torsion of superelliptic Jacobians and applies it to bound ranks of Jacobians over  and .

Findings

Structured description of m-torsion in superelliptic Jacobians

Existence results for torsion points on specific curves over finite fields

Lower bounds for Mordell-Weil ranks of certain Jacobians over 

Abstract

We give a structure theorem for the $m$-torsion of the Jacobian of a general superelliptic curve $y^m=F(x)$. We study existence of torsion on curves of the form $y^q=x^p-x+a$ over finite fields of characteristic $p$. We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over $\mathbb Q$.