On isogenies among certain abelian surfaces

TL;DR

This paper constructs a three-parameter family of genus-three curves with Prym varieties two-isogenous to genus-two Jacobians, using elliptic fibrations and Kummer surfaces to interpret the two-isogeny geometrically.

Contribution

It introduces a new geometric construction linking genus-three curves and genus-two Jacobians via explicit two-isogenies using elliptic fibrations.

Findings

Explicit family of genus-three curves with special Prym varieties

Geometric interpretation of two-isogenies via elliptic fibrations

Connection between Prym varieties and Jacobians of genus-two curves

Abstract

We construct a three-parameter family of non-hyperelliptic and bielliptic plane genus-three curves whose associated Prym variety is two-isogenous to the Jacobian variety of a general hyperelliptic genus-two curve. Our construction is based on the existence of special elliptic fibrations with the section on the associated Kummer surfaces that provide a simple geometric interpretation for the rational double cover induced by the two-isogeny between the abelian surfaces.