4 Admissible measures for standard L–functions and nearly holomorphic Siegel modular forms

Abstract

4.1 Congruences between modular forms and p-adic integration 4.1.1 Integration in nearly holomorphic Siegel modular forms 4.1.2 Arithmetical nearly holomorphic Siegel modular forms 4.1.3 The group 4.1.4 Canonical projection 4.1.5 The standard zeta function of a Siegel cusp eigenform 4.2 Algebraic differential operators and Siegel-Eisenstein distributions 4.2.1 Operatots of Maass and Shimura 4.2.2 Formulas for Fourier expansions 4.2.3 Siegel-Eisenstein series. 4.2.4 Normaized Siegel-Eisenstein series 4.2.5 Distributions with values in nearly holomorphic Siegel modular forms. 4.2.6 Convolutions of distributions with values in nearly holomorphic Siegel modular forms. 4.3 A general result on admissible measures 4.3.1 Profinite group 4.3.2 Measures and sequences of distributions 4.4 The standard L-function 4.4.1 The standard L function 4.4.2 Theta series 4.4.3 The Rankin zeta function 4.4.4 The standard zeta function D(s,f,x) as the Rankin convolution 4.4.5 Algebraic properties of the special values of normalized distributions. 4.4.6 Integral representation for the functions D±(s,f,x) 4.4.7 Action of the group Autℂ on scalar products of modular forms. 4.4.8 Algebraicity properties and Fourier coefficients 4.5 Algebraic linear forms on modular forms 4.5.1 Convolutions of theta distributions and Eisenstein distributions with values in nearly holomorphic Siegel modular forms. 4.5.2 Evaluation of algebraic linear forms 4.6 Congruences and proof of the Main theorem 4.6.1 Regularized distributions in Siegel modular forms. 4.6.2 Sufficient conditions for admissibility of measures with values in nearly holomorphic Siegel modular forms. 4.6.3 Fourier expansions of distributions with values in nearly holomorphic Siegel modular forms. 4.6.4 Fourier expansions of regularized distributions. 4.6.5 Main congruences for the Fourier expansions of regularized distributions. 4.6.6 Kummer congruences and Mazur’s measure. 4.6.7 Reduction of the Main congruence to congruences for partial sums. 4.6.8 Proof of the Main congruence. 4.6.9 Proof of Theorem 4.23