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The Maass-Shimura differential operators and congruences between arithmetical Siegel modular forms. (English) Zbl 1129.11021

The author studies congruences between nearly holomorphic Siegel modular forms using an explicit action of the Maass-Shimura arithmetical differential operators. \(p\)-adic congruences between these forms produce various \(p\)-adic \(L\)-functions using a general method of canonical projection. A general construction of \(h\)-admissible measures attached to sequences of special modular distributions is stated in section 5 (Theorem 5.1). This construction includes (as special cases) the standard \(L\)-functions of Siegel cusp forms (of small slope) and \(L\)-functions of elliptic cusp forms.

MSC:

11F60 Hecke-Petersson operators, differential operators (several variables)
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
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