Geometric weight-shifting operators on Hilbert modular forms in characteristic p

TL;DR

This paper studies weight-shifting operators on Hilbert modular forms in characteristic p, extending previous work to ramified primes, and uses geometric tools to optimize their effect on weights and analyze kernels.

Contribution

It generalizes the construction of partial Theta-operators for ramified primes and describes their kernels using geometric Frobenius operators, improving understanding of weight shifts.

Findings

Constructed optimal partial Theta-operators in ramified cases.

Described kernels of Theta-operators via geometric Frobenius images.

Proved a partial positivity result for minimal weights of mod p Hilbert modular forms.

Abstract

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic $p$, generalizing the author's prior work with Sasaki to the case where $p$ is ramified in the totally real field $F$. In particular we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial $\Theta$-operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore we describe the kernels of partial $\Theta$-operators in terms of images of geometrically constructed partial Frobenius operators. Finally we apply our results to prove a partial positivity result for minimal weights of mod $p$ Hilbert modular forms.