Decompositions of singular abelian surfaces

Given a singular abelian surface $A$, we find all possible decompositions of $A$ into a product of two mutually isogenous elliptic curves with complex multiplication. This is done by computing the transcendental lattice of arbitrary such products, and by studying the action of a certain class group on the factors of a given decomposition. We also give an alternative proof of the formula for the number of decompositions of $A$, which is originally due to Ma.

Keywords

abelian surfaces, complex multiplication, elliptic curves, class field theory, quadratic forms

2010 Mathematics Subject Classification

14K12, 14K22

Full Text (PDF format)

Received 5 October 2016

Accepted 8 August 2017

Published 3 May 2019