Partial Hasse invariants on splitting models of Hilbert modular varieties

TL;DR

This paper constructs generalized partial Hasse invariants on splitting models of Hilbert modular varieties, enabling the study of Galois pseudo-representations in ramified cases, extending prior unramified results.

Contribution

It introduces new partial Hasse invariants on splitting models for ramified primes, broadening the scope of Galois representation constructions in Hilbert modular varieties.

Findings

Constructed $g$ partial Hasse invariants on splitting models.

Proved existence of Galois pseudo-representations in ramified cases.

Extended previous unramified results to ramified primes.

Abstract

Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-representations attached to Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties $\mathrm{mod}$ $p^m$, extending a previous result of M. Emerton and the authors which required $p$ to be unramified in $F$.