On arithmetic of the superspecial locus

TL;DR

This paper develops a method to describe the Galois action on the superspecial locus of the Siegel moduli space in characteristic p, providing new insights into rational points and Hecke operator traces with applications to level structures.

Contribution

It introduces a modern approach to analyze the Galois action on the superspecial locus and simplifies trace calculations of Hecke operators for large level N.

Findings

Galois action description on superspecial locus

Simplified trace formula for Hecke operators at large N

Extensions to level-N structures

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.