Mass formula and Oort's conjecture for supersingular abelian threefolds

Abstract

Katsura and Oort obtained an explicit description of the supersingular locus S3,1 of the Siegel modular variety of degree 3 in terms of class numbers. In this paper we study an alternative stratification of S3,1, the so-called mass stratification. We show that when p≠2, there are eleven strata (one of a-number 3, two of a-number 2 and eight of a-number 1). We give an explicit mass formula for each stratum and classify possible automorphism groups on each stratum of a-number one. On the largest open stratum we show that every automorphism group is {±1} if and only if p≠2; that is, we prove that Oort's conjecture on the automorphism groups of generic supersingular abelian threefolds holds precisely when p>2.