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arXiv · 2410.09602

Theta operators and a generic entailment for $GSp_4$

Martin Ortiz
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Abstract

We construct a new family of mod $p$ weight shifting differential operators on the Siegel threefold. In particular, we construct one operator which generalizes the classical theta cycle, whose weight shift allows for maps between $p$-restricted weights, and which is generically injective on global sections. As an application we produce a generic entailment of Serre weights, i.e. any Hecke eigenform which is modular for a generic Serre weight in the lowest alcove is also modular for a Serre weight in one of the upper alcoves. The entailed Serre weight corresponds to a shadow weight of the lowest alcove Serre weight, in Herzig's conjectural description of $W(\overline{\rho})$.

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Martin Ortiz. 2024-10-12. Theta operators and a generic entailment for $GSp_4$. https://arxiv.org/abs/2410.09602

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