Hyperelliptic curves covering an elliptic curve twice

TL;DR

This paper proves the existence of hyperelliptic curves of genus 5 with two independent maps to a given elliptic curve over fields of characteristic not 2 or 3, and also identifies genus 3 cases for geometrically hyperelliptic curves.

Contribution

It establishes the existence of hyperelliptic curves with multiple maps to elliptic curves, expanding understanding of coverings in algebraic geometry.

Findings

Existence of genus 5 hyperelliptic curves with two independent maps to elliptic curves.

Existence of genus 3 hyperelliptic curves with degree two maps to a conic.

Results hold over fields with characteristic not 2 or 3.

Abstract

We show that for any elliptic curve (with j invariant not 0 or 1728) over any field of characteristic different from 2 and 3, there exists an hyperelliptic curve H of genus 5 with two independent maps to the given elliptic curve. We also show that, if we want such a curve to be just geometrically hyperelliptic, so having a degree two map to a conic, then there are some with genus 3.