The supersingular locus in Siegel modular varieties with Iwahori level structure

Abstract

Abstract. We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case g is even.

Supersingular Kottwitz–Rapoport strata and Deligne–Lusztig varieties

“…Even the dimension of the supersingular locus is known only for g ≤ 3 (but our results in this paper and in [18] prove that for even g it is g 2 /2). Note that the situation here is definitely more complicated than in the case of good reduction; as an example, in the case g = 2, the supersingular locus coincides with the p-rank 0 locus, but it is not contained in the closure of the p-rank 1 locus.…”

“…8.2, together with the results of Ngô and Genestier [34]. The dimension of the p-rank 0 locus is [g 2 /2] (see [18], Thm. 8.8).…”

“…In Section 4 we specify the subset of superspecial strata inside the set of all KR strata. These strata are supersingular by definition, and in [18] we prove that the set of superspecial KR strata coincides with the set of supersingular KR strata (see Theorem 4.5). In Section 6 we give a very simple geometric description of the superspecial KR strata in terms of DeligneLusztig varieties.…”

“…In [18], we prove that every KR stratum which is entirely contained in the supersingular locus, is superspecial. The proof relies on exploiting the relationship between the KR stratification and the Ekedahl-Oort stratification of A g .…”

“…Furthermore, it can be shown that any irreducible component of maximal dimension of the union of all superspecial KR strata is actually an irreducible component of the p-rank 0 locus, and hence in particular an irreducible component of the supersingular locus. Also see the remarks at the end of Section 4 and in particular [18].…”