Families of elliptic curves with genus 2 covers of degree 2

TL;DR

This paper investigates genus 2 covers of elliptic curves of degree 2 over arbitrary bases, establishing uniqueness, generalizing classical theorems, and proving a Torelli theorem in this context.

Contribution

It provides a unique determination of genus 2 covers from basic data, generalizes classical degree 2 cover theorems to relative cases, and proves a Torelli theorem over arbitrary bases.

Findings

The data in the basic construction uniquely determine the cover.

A classical theorem is generalized to the relative setting.

A Torelli theorem is established for genus 2 curves over arbitrary bases.

Abstract

We study genus 2 covers of relative elliptic curves over an arbitrary base in which 2 is invertible. Particular emphasis lies on the case that the covering degree is 2. We show that the data in the "basic construction" of genus 2 covers of relative elliptic curves determine the cover in a unique way (up to isomorphism). A classical theorem says that a genus 2 cover of an elliptic curve of degree 2 over a field of characteristic different from 2 is birational to a product of two elliptic curves over the projective line. We formulate and prove a generalization of this theorem for the relative situation. We also prove a Torelli theorem for genus 2 curves over an arbitrary base.