Jacobians among abelian threefolds: a geometric approach

TL;DR

This paper provides a geometric construction of a canonical extension field over which a given abelian threefold becomes isomorphic to a Jacobian, clarifying the relationship between abelian threefolds and Jacobians.

Contribution

It introduces a geometric method to explicitly construct the minimal extension over which an abelian threefold is a Jacobian.

Findings

Existence of a canonical extension of degree 1 or 2

Construction of the extension via geometric methods

Clarification of when an abelian threefold is a Jacobian

Abstract

Let A be a principally polarized abelian threefold over a perfect field k, not isomorphic to a product over the algebraic closure of k. There exists a canonical extension k' of k, of degree 1 or 2, such that A becomes isomorphic to a Jacobian over k'. The aim of this note is to give a geometric construction of this extension.