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 (see [55]). 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 = 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 M = 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 is an element of M-m(Q) : xi >= 0, xi 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 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 >= 2, see [58], [55].