$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties

Abstract

For Shimura varieties of Hodge type, we optimally extend algebraic Maass--Shimura differential operators on $p$-integral nearly holomorphic automorphic forms to differential operators on $\mu$-ordinary Mantovan Igusa varieties. We then show that the rank one operators can be integrated to an action of an explicit formal group. Via $p$-adic Fourier theory, this provides a $p$-adic interpolation by extending the action of a symmetric algebra of differential operators to the algebra of functions on the Tate module of the dual $p$-divisible group. Passing to the generic fiber, we obtain an action of an explicit algebra of $p$-adic locally analytic functions, and we show that the action of the subalgebra of locally constant functions is equivalent to a natural Hecke action and thus preserves classical forms. In the ordinary case, we further show that the locally analytic action extends to nearly overconvergent automorphic forms. Our results extend, clarify, and recover prior constructions.