Abelian Varieties with $p$-rank Zero

TL;DR

This paper generalizes classical theorems on elliptic curves to higher-dimensional abelian varieties with complex multiplication, classifying their reduction types and providing algorithms for constructing special supersingular varieties.

Contribution

It extends Deuring and Goren's theorems to higher dimensions, classifies $p$-torsion structures, and develops algorithms for constructing supersingular abelian varieties.

Findings

Classified $p$-torsion group schemes for 3-dimensional CM abelian varieties.

Provided algorithms to construct supersingular non-superspecial and superspecial varieties.

Showed all such varieties have non-integer endomorphisms of small degree.

Abstract

There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction. We generalise this and a similar theorem by Goren in dimension 2, and classify the $p$-torsion group scheme of the reduction of 3-dimensional abelian varieties with CM by the ring of integers of a cyclic sextic CM field. We also prove a theorem in arbitrary dimension $g$ that distinguishes ordinary and superspecial reduction for abelian varieties with CM by a cyclic CM field of degree $2g$. As an application, we give algorithms to construct supersingular non-superspecial, and superspecial abelian varieties of dimension 2 (surfaces) and dimension 3, and show that all such varieties have non-integer endomorphisms of small degree.