On compatibility between isogenies and polarizations of abelian varieties

We discuss the notion of polarized isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarizations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarized isogenies can be reduced to questions about unpolarized 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 polarized isogeny A→B whose degree is polynomially bounded in n, if there exist both an unpolarized isogeny A→B of degree n and a polarized isogeny A→B of unknown degree. As a further result, we prove that given any two principally polarized abelian varieties related by an unpolarized isogeny, there exists a polarized 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.