On the Prym variety of genus 3 covers of genus 1 curves

TL;DR

This paper constructs an explicit genus 2 curve related to a genus 3 cover of an elliptic curve, revealing the structure of the Jacobian as an isogenous product, thus advancing understanding of Prym varieties in algebraic geometry.

Contribution

It provides an explicit construction of a genus 2 curve associated with a genus 3 cover of an elliptic curve, extending previous theoretical results.

Findings

Jac(C) is isogenous to Jac(D) x Jac(X)

Explicit genus 2 curve X constructed from the cover

Generalizes a degenerate case of Bruin's result

Abstract

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin.