Theta operators and a generic entailment for $GSp_4$

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 $W(\overline{\rho})$.