On compatibility between isogenies and polarisations of abelian varieties

TL;DR

This paper explores the relationship between polarised and unpolarised isogenies of abelian varieties, showing how questions about one can be reduced to the other, with results on degree bounds and existence of polarised isogenies.

Contribution

It establishes new links between polarised and unpolarised isogenies, including degree bounds and existence results, using endomorphism algebra calculations.

Findings

Existence of a polarised isogeny with degree polynomially bounded by an unpolarised one.

Existence of a polarised isogeny between fourth powers of abelian varieties related by an unpolarised isogeny.

Reduction of questions about polarised isogenies to unpolarised isogenies and vice versa.

Abstract

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As a further result, we prove that given any two principally polarised abelian varieties related by an unpolarised isogeny, there exists a polarised isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras.