Geometry of Hilbert Modular Varieties over Totally Ramified Primes

Let L be a totally real field with ring of integers OL. Let N ≥ 4 be an integer and let M(μN) be the fine moduli scheme over Z of polarized abelian varieties with real multiplication (RM) and μN-level structure, satisfying the Deligne-Pappas condition. For every scheme S, we let M(S, μN) = M(μN) ×Z S be the moduli scheme over S; see Definition 2.1. Many aspects of the geometry of the modular varieties M(Fp, μN) are obtained via local deformation theory that factorizes according to the decomposition of p in OL. The unramified case was considered in [9] (see also [8]). Given that, one may restrict one’s attention to the case p = p in OL. We discuss here only the case e = g, that is, p is totally ramified in L. The ramified case was first treated by Deligne and Pappas in [6] (the case g = 2 was considered in [2]). We recall some of their results under the assumption that p is totally ramified. Let A/k be a polarized abelian variety with RM, defined over a field k of characteristic p. Fix an isomorphism OL⊗Zk ∼ = k[T ]/(T). One knows that HdR(A) is a free k[T ]/(T)-module of rank 2. The elementary divisors theorem furnishes us with k[T ]/(T)generators α and β for HdR(A) such that