Decompositions of singular abelian surfaces

TL;DR

This paper classifies all decompositions of singular abelian surfaces into elliptic curves, extending previous classifications to Picard number 4 using lattice computations and class group actions, with applications to singular K3 surfaces.

Contribution

It explicitly finds all decompositions of abelian surfaces with Picard number 4, generalizing techniques and providing a new proof of Ma's formula.

Findings

Complete classification of decompositions for Picard number 4

Explicit computation of transcendental lattices for CM elliptic curves

Application to the study of singular K3 surfaces

Abstract

Given an abelian surface, the number of its distinct decompositions into a product of elliptic curves has been described by Ma. Moreover, Ma himself classified the possible decompositions for abelian surfaces of Picard number $1 \leq \rho \leq 3$. We explicitly find all such decompositions in the case of abelian surfaces of Picard number $\rho= 4$. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group on the factors of a given decomposition. We also provide an alternative and simpler proof of Ma's formula, and an application to singular K3 surfaces.