Alexei A. Panchishkin

Non-Archimedean L-functions and arithmetical Siegel modular forms

Introduction.- Non-Archimedean analytic functions, measures and distributions.- Siegel modular forms and the holomorphic projection operator.- Arithmetical differential operators on nearly holomorphic Siegel modular forms.- Admissible measures for standard L-functions and nearly holomorphic Siegel modular forms.

On the Siegel-Eisenstein measure and its applications

An Eisenstein measure on the symplectic group over rational number field is constructed which interpolatesp-adically the Fourier expansion of Siegel-Eisenstein series. The proof is based on explicit computation of the Fourier expansions by Siegel, Shimura and Feit. As an application of this result ap-adic family of Siegel modular forms is given which interpolates Klingen-Eisenstein series of degree two using Boecherer’s integral representation for the Klingen-Eisenstein series in terms of the Siegel-Eisenstein series.

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSpr × GSp2m to GSpr+2m for some positive integers r and m.

Rankin’s Lemma of Higher Genus and Explicit Formulas for Hecke Operators

We develop explicit formulas for Hecke operators of higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebras and in a tensor product of such algebras. In particular, we formulate and prove Rankin’s lemma in genus two. An application to a holomorphic lifting from GSp2 ×GSp2 to GSp4 is given using Ikeda–Miyawaki constructions.

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

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

3 Arithmetical differential operators on nearly holomorphic Siegel modular forms

3.1 Description of the Shimura differential operators 3.2 Nearly holomorphic Siegel modular forms 3.2.1 Algebraic nearly holomorphic Siegel modular forms 3.2.2 Formal expansions of nearly holomorphic forms. 3.2.3 Action of the U-operator. 3.3 Algebraic differential operators of Maass and Shimura 3.3.1 Differential operators on ℍ m . 3.3.2 The polynomial R m (z; r,β ). 3.3.3 Proof of Lemma 3.7 3.3.4 Action of the Shimura differential operator on formal Fourier expansions 3.3.5 Commutation of the Shimura operator with Hecke operators 3.4 Arithmetical variables of nearly holomorphic forms 3.4.1 Arithmetical nearly holomorphic Siegel modular forms 3.4.2 Action on Fourier expansion 3.4.3 Differentiation on monomials

1 Non-Archimedean analytic functions, measures and distributions

1.1 p-adic numbers and the Tate field 1.1.1 p-adic numbers 1.1.2 Topology of p-adic fields 1.1.3 The structure of the multiplicative group ℚ p × and K× 1.1.4 The S-adic numbers 1.1.5 The Tate field 1.2 Continuous and analytic functions 1.2.1 Continuous functions 1.2.2 Analytic functions and power series 1.2.3 Newton polygons 1.3 Distributions, measures, and the abstract Kummer congruences 1.3.1 Distributions 1.3.2 Measures 1.3.3 The S-adic Mazur measure 1.4 Iwasawa algebra and the non-Archimedean Mellin transform 1.4.1 The domain of definition of the non-Archimedean zeta functions 1.4.2 The analytic structure of X S 1.4.3 The non-Archimedean Mellin transform 1.4.4 The Iwasawa algebra 1.4.5 Formulas for coefficients of power series 1.4.6 Example. The S-adic Mazur measure and the non-Archimedean Kubota-Leopoldt zeta function 1.4.7 Measures associated with Dirichlet characters 1.5 Admissible measures and their Mellin transform 1.5.1 Non-Archimedean integration 1.5.2 h-admissible measure 1.6 Complex valued distributions, associated with Euler products 1.6.1 Dirichlet series 1.6.2 Concluding remarks