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arXiv · 2011.14128

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

Fred Diamond
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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.

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Fred Diamond. 2020-11-28. Geometric weight-shifting operators on Hilbert modular forms in characteristic p. https://doi.org/10.1017/s1474748021000530

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