The Maass–Shimura differential operators and congruences between arithmetical Siegel modular forms

We extend further a new method for constructing $p$-adic $L$-functions associated with modular forms. For this purpose, we study congruences between nearly holomorphic Siegel modular forms using an explicit action of the Maass–Shimura arithmetical differential operators. We view nearly holomorphic arithmetical Siegel modular forms as certain formal expansions over $A=\mathbb C_p$. The important property of these arithmetical differential operators is their commutation with the Hecke operators (under an appropriate normalization).
We show in Section 5 that a fine combinatorial structure of the action of these arithmetical differential operators on the $A$-module $\mathcal M=\mathcal M(A)$ of nearly holomorphic Siegel modular forms produces new congruences between nearly holomorphic Siegel modular forms inside a formal $q$-expansion ring of the form ${A[\![q^B]\!][R_{ij}]}$ where $B=B_m=\{\xi={}^t\xi\in M_m(\mathbb Q)\colon\xi\ge 0,\ \xi\text{ half-integral}\}$ is the semi-group, important for the theory of Siegel modular forms), and the nearly holomorphic parameters $(R_{ij})=R$ correspond to the matrix $R=(4\pi\operatorname{Im}(z))^{-1}$ in the Siegel modular case. These congruences produce various $p$-adic $L$-functions attached to modular forms using a general method of canonical projection. We give in Theorem 5.1 a general construction of $h$-admissible measures attached to sequences of special modular distributions. Our construction generalizes at the same time the following two cases:
(1) the standard $L$-function of a Siegel cusp eigenform, [20, Chapter 4];
(2) the Mellin transform of an elliptic cusp eigenform of weight $k\ge 2$, see [58], [55].