On arithmetic of the superspecial locus

Abstract

We develop a method for describing the Galois action on the superspecial locus of the Siegel moduli space in characteristic $p$. Using this description, we give a modern treatment for the results of Ibukiyama and Katsura [Compos. Math., 1994] concerning the ${\mathbb{F}}_p$-rational points and the trace of a Hecke operator of Atkin-Lehner type. This also leads to analogues with level-$N$ structure. The trace of the Hecke operator can be reduced into one term (instead of finitely many terms a priori) by the simple trace formula when $N$ is large.

35 pages, title changed, same results, Sections 2, 3, 8 shortened substantially

References in corpus (2)

• Ekedahl-Oort and Newton strata for Shimura varieties of PEL type
• Hecke Eigensystems for (mod p) Modular Forms of PEL-type and Algebraic Modular Forms

Cited by in corpus (1)

• On fields of definition of components of the Siegel supersingular locus