Abstract
AbstractWe 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.
Highlights
The study of weight-shifting operations on modular forms has a rich and fruitful history. Besides those naively obtained from the graded algebra structure on the space of classical modular forms of all weights, there is a deeper construction due to Ramanujan [30] which shifts the weight by two using differentiation, leading to a more general theory of Maass–Shimura operators
In order to define the partial Θ-operators, we first define a canonical factorisation of the partial Hasse invariants over a finite flat (Igusa) cover of the Hilbert modular variety over F
We explain how the construction of Θ-operators in [12] directly generalises to the case where p is ramified in F, yielding an operator that shifts the weight k by (1,1) in the final two components corresponding to embeddings with the same reduction; that is, θp,i,ep−1, θp,i,ep
Summary
The study of weight-shifting operations on modular forms has a rich and fruitful history. Analogous weight-shifting operations in characteristic p, first studied by Swinnerton-Dyer and Serre [34], take on special significance in the context of congruences between modular forms and the implications for associated Galois representations. Multiplication by a Hasse invariant H, to forms of weight k + p − 1; a differential operator Θ, to forms of weight k + p + 1; a linearised p-power map V, to forms of weight pk. Following the work of Swinnerton-Dyer and Serre, there were further significant developments to the theory due to Katz [23, 24] (interpreting the constructions more geometrically), Jochnowitz [21, 22] (on the weight filtration and Tate’s Θ-cycles) and Downloaded from https://www.cambridge.org/core.
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