N-covers of hyperelliptic curves

Abstract

For a hyperelliptic curve C of genus g with a divisor class of order n=g+1, we shall consider an associated covering collection of curves D, each of genus g. We describe, up to isogeny, the Jacobian of each D via a map from D to C, and two independent maps from D to a curve of genus g(g-1)/2. For some curves, this allows covering techniques that depend on arithmetic data of number fields of smaller degree than standard 2-coverings; we illustrate this by using 3-coverings to find all Q-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.