Theta operators and a generic entailment for $GS{p}_4$

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 $W(\overline{\ddot{I}})$.

This version contains only the required elements to prove the generic entailment. The rest of the previous version's content is superseded by our paper "Theta operators on Hodge type Shimura varieties"