GEOMETRIC WEIGHT-SHIFTING OPERATORS ON HILBERT MODULAR FORMS IN CHARACTERISTIC p

Abstract

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. 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.

Kodaira-Spencer isomorphisms and degeneracy maps on Iwahori-level Hilbert modular varieties: the saving trace

“…We adopt much of the notation and conventions of [Dia21] for notions and constructions associated to embeddings of F . In particular, for each p ∈ S p , we let Σ p denote the set of embeddings F p → Q p , and we identify Σ := p∈Sp Σ p with the set of embeddings F ֒→ Q p via the canonical bijection.…”

“…for each τ ∈ Σ 0 , where H τ,i is the τ -component of the locally free O F ⊗ O S -module H 1 dR (A i /S) for i = 1, 2, the superscript ∨ denotes its O S [u]/(E τ )-dual, and the µ τ,i are the τ -components of the isomorphisms obtained from the polarizations and Poincaré duality (see [Dia21,(4)…”

“…The scheme Y is equipped with line bundles 3 ω and δ arising from its interpretation as a coarse moduli space parametrizing abelian schemes with additional data including an O F -action. Since Y is smooth over O, its relative dualizing sheaf K Y /O is identified with ∧ [F :Q] Ω 1 Y /O , and in [RX17] (see also [Dia21,§3.3]), Reduzzi and Xiao establish an integral version of the Kodaira-Spencer isomorphism, taking the form…”