Un modele semi-stable de la variete de Siegel de genre 3 avec structures de niveau de type Γ_0(p)

Abstract

Let S(g,N,p) be the Siegel modular variety of principally polarized abelian varieties of dimension g with a Γ_0(p)-level structure and a full N-level structure (where p is a prime not dividing N \geq 3 and Γ_0(p) is the inverse image of a Borel subgroup of Sp(2g,F_p) in Sp(2g,Z)). This variety has a natural integral model over Z[1/N] which is not semi-stable over the prime p if g>1. Using the theory of local models of Rapoport-Zink, we construct a semi-stable model of S(g,N,p) over Z[1/N] for g=2 and g=3. For g=2, our construction differs from de Jong's one though the resulting model is the same.