Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms

Title......Page 3
Preface......Page 7
Contents......Page 9
Kummer Congruences and Bernoulli Numbers......Page 11
$p$–adic $L$–Functions and Mellin Transforms......Page 12
$p$–adic $L$–Functions of Siegel Modular Forms......Page 14
Distributions with Values in Siegel Modular Forms......Page 16
Families of Siegel Modular Forms......Page 18
Remarks on modular forms of positive slope......Page 19
$U(p)$–Operator and Method of Canonical projection......Page 20
Automorphic $L$–Functions......Page 21
1.1.1 $p$-adic numbers......Page 23
1.1.3 The structure of the multiplicative group ${\\mathbb Q}^\\times_p$ and $K^\\times$......Page 25
1.1.5 The Tate field......Page 26
1.2.1 Continuous functions......Page 27
1.2.2 Analytic functions and power series......Page 28
1.2.3 Newton polygons......Page 30
1.3.1 Distributions......Page 31
1.3.2 Measures......Page 34
1.3.3 The $S$-adic Mazur measure......Page 36
1.4.1 The domain of definition of the non-Archimedean zeta functions......Page 38
1.4.2 The analytic structure of $X_S$......Page 39
1.4.3 The non-Archimedean Mellin transform......Page 41
1.4.4 The Iwasawa algebra......Page 42
1.4.5 Formulas for coefficients of power series......Page 43
1.4.6 Example. The $S$-adic Mazur measure and the non-Archimedean Kubota-Leopoldt zeta function......Page 44
1.5.1 Non-Archimedean integration......Page 45
1.5.2 $h$-admissible measure......Page 47
1.6.1 Dirichlet series......Page 48
1.6.2 Concluding remarks......Page 54
2 Siegel modular forms and the holomorphic projection operator......Page 55
2.1.1 Symplectic group and Siegel upper half plane......Page 56
2.1.2 Siegel modular forms......Page 57
2.1.4 Hecke operators......Page 61
2.1.5 Hecke polynomials......Page 62
2.1.6 The spinor zeta function and the standard zeta function......Page 63
2.1.7 Non-commutative extension of the Hecke algebra and the Satake isomorphism......Page 65
2.1.8 Action of the Hecke operators on Fourier expansions......Page 68
2.2.1 Theta series......Page 72
2.2.2 Siegel-Eisenstein series......Page 75
2.2.3 The Rankin zeta function......Page 76
2.2.4 The standard zeta function $D(s, f, \\chi )$ as the Rankin convolution......Page 78
2.3.1 Rationality properties of Fourier coefficients......Page 80
2.3.2 Preparation: the confluent hypergeometric function......Page 82
2.3.3 Critical values of the confluent hypergeometric function......Page 85
2.3.4 Normalized Siegel-Eisenstein series......Page 89
2.4.1 Holomorphic projection operator......Page 93
2.4.2 Poincaré series of two variables (of exponential type) of higher level......Page 95
2.4.3 Reduction of theorem 2.16 to properties of Poincaré series......Page 97
2.4.4 Fourier expansion of the holomorphic projection of special modular forms......Page 99
2.5.1 The polynomial $R(z; r, \\beta )$......Page 101
3.1 Description of the Shimura differential operators......Page 104
3.2.1 Algebraic nearly holomorphic Siegel modular forms......Page 108
3.2.3 Action of the U-operator......Page 111
3.3.1 Differential operators on ${\\mathbb H}_m$......Page 115
3.3.2 The polynomial $Rm(z; r, \\beta )$......Page 116
3.3.3 Proof of Lemma 3.7......Page 118
3.3.4 Action of the Shimura differential operator on formal Fourier expansions......Page 119
3.3.5 Commutation of the Shimura operator with Hecke operators......Page 124
3.4.1 Arithmetical nearly holomorphic Siegel modular forms......Page 126
3.4.2 Action on Fourier expansion......Page 129
3.4.3 Differentiation on monomials......Page 130
4.1.1 Integration in nearly holomorphic Siegel modular forms......Page 135
4.1.3 The group......Page 136
4.1.4 Canonical projection......Page 137
4.1.5 The standard zeta function of a Siegel cusp eigenform......Page 138
4.2.1 Operatots of Maass and Shimura......Page 141
4.2.3 Siegel-Eisenstein series......Page 142
4.2.4 Normaized Siegel-Eisenstein series......Page 144
4.2.5 Distributions with values in nearly holomorphic Siegel modular forms......Page 148
4.2.6 Convolutions of distributions with values in nearly holomorphic Siegel modular forms......Page 151
4.3.2 Measures and sequences of distributions......Page 152
4.4.2 Theta series......Page 156
4.4.3 The Rankin zeta function......Page 157
4.4.4 The standard zeta function ${\\cal D}(s, f, \\chi )$ as the Rankin convolution......Page 158
4.4.5 Algebraic properties of the special values of normalized distributions......Page 160
4.4.6 Integral representation for the functions ${\\cal D}^\\pm (s, f, \\chi )$......Page 162
4.4.7 Action of the group Aut({\\mathbb C}) on scalar products of modular forms......Page 164
4.4.8 Algebraicity properties and Fourier coeffcients......Page 171
4.5.1 Convolutions of theta distributions and Eisenstein distributions with values in nearly holomorphic Siegel modular forms......Page 173
4.5.2 Evaluation of algebraic linear forms......Page 176
4.6.1 Regularized distributions in Siegel modular forms......Page 182
4.6.3 Fourier expansions of distributions with values in nearly holomorphic Siegel modular forms......Page 184
4.6.5 Main congruences for the Fourier expansions of regularized distributions......Page 186
4.6.6 Kummer congruences and Mazur’s measure......Page 187
4.6.7 Reduction of the Main congruence to congruences for partial sums......Page 189
4.6.8 Proof of the Main congruence......Page 191
4.6.9 Proof of Theorem 4.23......Page 193
References......Page 195
Index......Page 202