Stefan Reppen: On the Hasse locus of Shimura varieties mod p

Abstract

This thesis studies the order of vanishing of the Hasse invariant on the geometric special fibre of certain Shimura varieties. More precisely, let \(A\to S\) denote the universal abelian variety over the geometric special fibre of either; a Hilbert modular variety associated to a totally real field, \(F\), at a prime unramified in \(F\); a unitary Shimura variety associated to an imaginary quadratic extension \(F^{+}/\mathbb{Q}\) of signature \((n-1,1)\), at a prime split in \(F^{+}\); or a three dimensional Siegel Shimura variety. Let h denote the Hasse invariant on S and let \(\text{Fil}^{\bullet}H_{\text{dR}}^{\dim (A/S)}(A/S)\) and \({}_{\text{conj}}\text{Fil}_{\bullet}H_{\text{dR}}^{\dim (A/S)}(A/S)\) denote the Hodge respectively conjugate filtration on \(H_{\text{dR}}^{\dim (A/S)}(A/S)\). In this thesis we prove that for any closed point \(s\in S\), the order of vanishing of h at s is equal to the largest integer, m, such that \({}_{\text{conj}}\text{Fil}_{0}H_{\text{dR}}^{\dim (A_s)}(A_s)\hookrightarrow \text{Fil}^{m}H_{\text{dR}}^{\dim (A_s)}(A_s)\). This result is the direct analogue of Ogus' on certain families of Calabi-Yau varieties in positive characteristic.