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

Time: Wed 2022-06-15 09.00

Location: Albano, House 4, Room 6 and Zoom

Doctoral student: Stefan Reppen

Opponent: Marc-Hubert Nicole (Université de Caen)

Supervisor: Wushi Goldring

### 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.