Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization (1,2)

TL;DR

This paper provides an explicit algebraic description of genus 2 curves whose Jacobians are degree 2 isogenous to Prym varieties from certain genus 3 coverings, linking algebraic geometry with integrable systems.

Contribution

It introduces a novel algebraic framework for describing Jacobians related to Prym varieties of genus 3 curves with polarization (1,2).

Findings

Explicit algebraic descriptions for genus 2 curves related to Prym varieties.

Identification of cases where Prym varieties contain pairs of elliptic curves.

Examples demonstrating the algebraic descriptions and isogenies.

Abstract

For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated to such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties. We also consider some special cases of the covering C -> E, in particular, when the corresponding Prym varieties contain pairs of elliptic curves and the Jacobian of C is isogeneous (but not isomorphic) to the product of 3 different elliptic curves. Our description is accompanied with explicit numerical examples.