The arithmetic of genus two curves with (4,4)-split Jacobians

TL;DR

This paper classifies genus 2 curves with Jacobians that split via (4,4)-isogenies into elliptic curves, providing a comprehensive description of their invariants and Richelot isogenies over various fields.

Contribution

It offers a complete classification of principally polarized abelian surfaces arising from elliptic curve gluings along 4-torsion, extending understanding of Richelot isogenies beyond base field restrictions.

Findings

Full classification of (4,4)-split Jacobians.

Relation between absolute invariants of such Jacobians.

General description of Richelot isogenies over non-algebraically closed fields.

Abstract

In this paper we study genus 2 curves whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.