Families of elliptic curves with genus 2 covers of degree 2
Claus Diem

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Cryptography and Residue Arithmetic · Historical and Political Studies
