Compactification of Siegel modular varieties with bad reduction

Compactification of Siegel modular varieties with bad reduction We construct arithmetic toroidal compactifications of the moduli stack of principally polarized abelian varieties with parahoric level structure. To this end, we extend the methods of Faltings and Chai [7] to a case of bad reduction. Our compactifications are not smooth near the boundary; the singularities are those of the moduli stacks of abelian varieties with parahoric level structure of lower genus. We modify Faltings and Chai's construction of compactifications without level structure. The key point is that our approximation preserves the p-torsion subgroup of the abelian varieties. As an application, we give a new proof of the existence of the canonical subgroup for some families of abelian varieties.