On a generalization of Deuring's results

TL;DR

This paper generalizes Deuring's classical theorem on CM-elliptic curves by studying reductions of abelian varieties with complex multiplication using Dieudonne theory, and discusses conditions for maximality of CM curves.

Contribution

It extends Deuring's results to higher-dimensional abelian varieties with complex multiplication using Dieudonne theory.

Findings

Generalization of Deuring's theorem to abelian varieties

Conditions for a CM curve to be maximal

Application of Dieudonne theory in reduction analysis

Abstract

Using the Dieudonne theory we will study a reduction of an abelian variety of complex multiplication. Our results may be regarded as a generalization of a classical theorem due to Deuring for a CM-elliptic curve. We will also discuss a sufficient condition that is a proper smooth curve with CM to be maximal.