Families of (3,3)-split Jacobians

TL;DR

This paper classifies special cases of (3,3)-split Jacobians of genus two curves and parametrizes their invariants when related to elliptic curves from the Hesse pencil, enhancing understanding of their structure.

Contribution

It provides a complete classification of special (3,3)-split Jacobians and explicit parametrizations of their invariants in relation to elliptic curves from the Hesse pencil.

Findings

All special (3,3)-split Jacobians are computed.

Explicit parametrizations of Igusa-Clebsch invariants are provided.

Connections to elliptic curves in the Hesse pencil are established.

Abstract

We compute all the "special" cases of (3,3)-split Jacobians and we parametrize the Igusa-Clebsch invariants of curves of genus two whose Jacobian is (3,3)-isogenous to a product of two elliptic curves from the Hesse pencil.