Abelian surfaces admitting an (l,l)-endomorphism

TL;DR

This paper classifies all principally polarized abelian surfaces with self-(l,l)-isogenies, providing explicit methods for the case l=2 and for surfaces with multiplication by specific imaginary quadratic orders.

Contribution

It offers a complete classification of such abelian surfaces and explicit computational techniques, especially for the case l=2 and for surfaces with certain imaginary quadratic multiplications.

Findings

Explicit classification of abelian surfaces with (l,l)-isogenies

Method to compute all such surfaces for l=2

Procedure to find surfaces with given imaginary quadratic multiplication

Abstract

We give a classification of all principally polarized abelian surfaces that admit an $(l,l)$-isogeny to themselves, and show how to compute all the abelian surfaces that occur. We make the classification explicit in the simplest case $l=2$. As part of our classification, we also show how to find all principally polarized abelian surfaces with multiplication by a given imaginary quadratic order.