Decomposing abelian varieties into simple factors: algorithms and applications

TL;DR

This paper presents algorithms for decomposing polarized abelian varieties into simple factors, including methods for computing period matrices and applications to Jacobian varieties, without needing full endomorphism algebra data.

Contribution

It introduces effective algorithms for decomposition and period matrix computation, applicable even without full endomorphism algebra knowledge.

Findings

Decomposed a Jacobian of dimension 101, filling a known gap.

Identified a new decomposable Jacobian of CM type, dimension 11.

Provided methods applicable to varieties with and without group actions.

Abstract

We give an effective procedure to explicitly find the decomposition of a polarized abelian variety into its simple factors if a period matrix is known. Since finding this datum is not easy, we also provide two methods to compute the period matrix for a polarized abelian variety, depending on the given geometric information about it. These results work particularly well in combination with our previous work on abelian varieties with group actions, since they allow us to fully decompose such varieties by successively decomposing their factor subvarieties, even when these no longer have a group action. We highlight that we do not require to determine the full endomorphism algebra of any of the (sub)varieties involved. We illustrate the power of our algorithms with two byproducts: we find a completely decomposable Jacobian variety of dimension 101, filling this Ekedahl-Serre gap, and we describe a new completely decomposable Jacobian variety of CM type of dimension 11.