A Deuring criterion for abelian varieties

Abstract

We give several generalisations of the Deuring reduction criterion for elliptic curves to abelian varieties of higher dimension. In particular the Newton polygon of the reduction of an abelian variety A with complex multiplication by F at a prime above p can be explicitly computed in terms of the CM type of A and the decomposition group of a suitable prime above p in a Galois closure of F. Conversely, it is shown that the formal isogeny type of the reduction of A at a prime above p places certain restrictions on the decomposition of p in F.