A Deuring criterion for abelian varieties

Chris Blake

Published 2013-11-21Version 1

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.