Generalization of Deuring Reduction Theorem

TL;DR

This paper extends the Deuring reduction theorem to higher-dimensional Abelian varieties with complex multiplication, linking their group scheme decompositions to prime ideal factorizations in CM fields.

Contribution

It generalizes the classical Deuring theorem to Abelian varieties of dimensions 1, 2, and 3, establishing explicit relationships between group scheme decompositions and prime ideal factorizations.

Findings

Established a connection between $A[p]$ decomposition and $p ext{-} ext{O}_K$ prime ideal factorization.

Derived explicit relationships for Abelian varieties of dimensions 1, 2, and 3.

Abstract

In this paper we generalize the Deuring theorem on a reduction of elliptic curve with complex multiplication. More precisely, for an Abelian variety $A$, arising after reduction of an Abelian variety with complex multiplication by a CM field $K$ over a number field at a pace of good reduction. We establish a connection between a decomposition of the first truncated Barsotti-Tate group scheme $A[p]$ and a decomposition of $p\cO_{K}$ into prime ideals. In particular, we produce these explicit relationships for Abelian varieties of dimensions $1, 2$ and 3.