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Elementary Theory of L-functions and Eisenstein Series

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Overview

The approach is basically algebraic and the treatment elementary in this comprehensive and systematic account of the theory of p-adic and classical modular forms and the theory of the special values of arithmetic L-functions and p-adic L-functions.
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Editorial Reviews

From the Publisher
"...its style is unusually lively; even in the exposition of classical results, one feels that the proof has been reinvented and is often illuminating...a large part of the text explains theories and results due to the author; behind a classical title are hidden many theorems never published in book form until now...one must be thankful to the author to have written down the first accessible presentation of the various aspects of his theory...highly reommended to graduate students and more advanced researchers wishing to learn this powerful theory." Jacques Tilouine, Mathematical Reviews

"...this is a comprehensive and important book-one that deserves to be studied carefully by any serious student of L-functions and modular forms." Glen Stevens,Bulletin of the American Mathematical Society

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Product Details

Table of Contents

Suggestions to the reader
Ch. 1 Algebraic number theory 1
1.1 Linear algebra over rings 1
1.2 Algebraic number fields 5
1.3 p-adic numbers 17
Ch. 2 Classical L-functions and Eisenstein series 25
2.1 Euler's method 25
2.2 Analytic continuation and the functional equation 33
2.3 Hurwitz and Dirichlet L-functions 40
2.4 Shintani L-functions 47
2.5 L-functions of real quadratic field 54
2.6 L-functions of imaginary quadratic fields 63
2.7 Hecke L-functions of number fields 66
Ch. 3 p-adic Hecke L-functions 73
3.1 Interpolation series 73
3.2 Interpolation series in p-adic fields 75
3.3 p-adic measures on Z[subscript p] 78
3.4 The p-adic measure of the Riemann zeta function 80
3.5 p-adic Dirichlet L-functions 82
3.6 Group schemes and formal group schemes 89
3.7 Toroidal formal groups and p-adic measures 96
3.8 p-adic Shintani L-functions of totally real fields 99
3.9 p-adic Hecke L-functions of totally real fields 102
Ch. 4 Homological interpretation 107
4.1 Cohomology groups on G[subscript m](C) 107
4.2 Cohomological interpretation of Dirichlet L-values 117
4.3 p-adic measures and locally constant functions 118
4.4 Another construction of p-adic Dirichlet L-functions 120
Ch. 5 Elliptic modular forms and their L-functions 125
5.1 Classical Eisenstein series of GL(2)[subscript /][subscript Q] 125
5.2 Rationality of modular forms 131
5.3 Hecke operators 139
5.4 The Petersson inner product and the Rankin product 150
5.5 Standard L-functions of holomorphic modular forms 157
Ch. 6 Modular forms and cohomology groups 160
6.1 Cohomology of modular groups 160
6.2 Eichler-Shimura isomorphisms 167
6.3 Hecke operators on cohomology groups 175
6.4 Algebraicity theorem for standard L-functions of GL(2) 186
6.5 Mazur's p-adic Mellin transforms 189
Ch. 7 Ordinary [Lambda]-adic forms, two variable p-adic Rankin products and Galois representations 194
7.1 p-Adic families of Eisenstein series 195
7.2 The projection to the ordinary part 200
7.3 Ordinary [Lambda]-adic forms 208
7.4 Two variable p-adic Rankin product 221
7.5 Ordinary Galois representations into GL[subscript 2](Z[subscript p][[X]]) 228
7.6 Examples of [Lambda]-adic forms 234
Ch. 8 Functional equations of Hecke L-functions 239
8.1 Adelic interpretation of algebraic number theory 239
8.2 Hecke characters as continuous idele characters 245
8.3 Self-duality of local fields 249
8.4 Haar measures and the Poisson summation formula 253
8.5 Adelic Haar measures 257
8.6 Functional equations of Hecke L-functions 261
Ch. 9 Adelic Eisenstein series and Rankin products 272
9.1 Modular forms on GL[subscript 2](F[subscript A]) 272
9.2 Fourier expansion of Eisenstein series 282
9.3 Functional equation of Eisenstein series 292
9.4 Analytic continuation of Rankin products 298
9.5 Functional equations of Rankin products 306
Ch. 10 Three variable p-adic Rankin products 310
10.1 Differential operators of Maass and Shimura 310
10.2 The algebraicity theorem of Rankin products 317
10.3 Two variable [Lambda]-adic Eisenstein series 326
10.4 Three variable p-adic Rankin products 331
10.5 Relation to two variable p-adic Rankin products 339
10.6 Concluding remarks 343
Appendix: Summary of homology and cohomology theory 345
References 365
Answers to selected exercises 371
Index 383
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