N-Covers of hyperelliptic curves

Abstract

For a hyperelliptic curve ${\hbox{\ax C}}$ of genus $g$ with a divisor class of order $n = g + 1$, we shall consider an associated covering collection of curves ${\hbox{\ax D}}_\delta$, each of genus $g^2$. We describe, up to isogeny, the Jacobian of each ${\hbox{\ax D}}_\delta$ via a map from ${\hbox{\ax D}}_\delta$ to ${\hbox{\ax C}}$, and two independent maps from ${\hbox{\ax D}}_\delta$ 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 ${\Bbb Q}$-rational points on a curve of genus 2 for which 2-covering techniques would be impractical.