Irreducibility of the Siegel moduli spaces with parahoric level structure

Abstract

We prove that the moduli space ${\mathcal{A}}_{g,{\Gamma }_0(p)}\otimes {\bar{\mathbb{F}}}_p$ of principally polarized abelian varieties of dimension $g$ with a ${\Gamma }_0(p)$-level structure in characteristic $p$ has $2^g$ irreducible components.The cases of other parahoric level structures are also considered.

4 pages