A theta operator for the group $\mathrm{GSp}_4$

We construct a differential operator on sheaves of $p$-adic modular forms defined over the locus of $p$-rank $\ge 1$ of the Siegel threefold, by applying a revisited version of the approach that Sean Howe recently introduced in his paper "A unipotent circle action on $p$-adic modular forms" (2020, Trans. Am. Math. Soc.) to construct the theta operator in the elliptic case.