Irreducibility of the Siegel moduli spaces with parahoric level structure

We prove that the moduli space ${\text{A}}_{g,\Gamma_0{(p)}}\otimes{\overline{\text{F}}}_p$ of principally polarized abelian varieties of dimension g with a Γ0(p)-level structure in characteristic p has 2^g irreducible components. The cases of other parahoric level structures are also considered.