On the Prym variety of genus 3 covers of genus 1 curves
Christophe Ritzenthaler, Matthieu Romagny

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology
