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Elementary Theory of L-functions and Eisenstein Series : London Mathematical Society Student Texts - Haruzo Hida

Elementary Theory of L-functions and Eisenstein Series

London Mathematical Society Student Texts


Published: 12th April 1993
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This book is a 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. The approach is basically algebraic, and the treatment is elementary. No deep knowledge from algebraic geometry and representation theory is required. The author's main tool in dealing with these problems is taken from cohomology theory over Riemann surfaces, which is also explained in detail in the book. He also gives a concise but thorough treatment of analytic continuation and functional equation. Graduate students wishing to know more about L-functions will find this a unique introduction to this fascinating branch of mathematics.

"...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

Suggestions to the readerp. xi
Algebraic number theoryp. 1
Linear algebra over ringsp. 1
Algebraic number fieldsp. 5
p-adic numbersp. 17
Classical L-functions and Eisenstein seriesp. 25
Euler's methodp. 25
Analytic continuation and the functional equationp. 33
Hurwitz and Dirichlet L-functionsp. 40
Shintani L-functionsp. 47
L-functions of real quadratic fieldp. 54
L-functions of imaginary quadratic fieldsp. 63
Hecke L-functions of number fieldsp. 66
p-adic Hecke L-functionsp. 73
Interpolation seriesp. 73
Interpolation series in p-adic fieldsp. 75
p-adic measures on Z[subscript p]p. 78
The p-adic measure of the Riemann zeta functionp. 80
p-adic Dirichlet L-functionsp. 82
Group schemes and formal group schemesp. 89
Toroidal formal groups and p-adic measuresp. 96
p-adic Shintani L-functions of totally real fieldsp. 99
p-adic Hecke L-functions of totally real fieldsp. 102
Homological interpretationp. 107
Cohomology groups on G[subscript m](C)p. 107
Cohomological interpretation of Dirichlet L-valuesp. 117
p-adic measures and locally constant functionsp. 118
Another construction of p-adic Dirichlet L-functionsp. 120
Elliptic modular forms and their L-functionsp. 125
Classical Eisenstein series of GL(2)[subscript /Q]p. 125
Rationality of modular formsp. 131
Hecke operatorsp. 139
The Petersson inner product and the Rankin productp. 150
Standard L-functions of holomorphic modular formsp. 157
Modular forms and cohomology groupsp. 160
Cohomology of modular groupsp. 160
Eichler-Shimura isomorphismsp. 167
Hecke operators on cohomology groupsp. 175
Algebraicity theorem for standard L-functions of GL(2)p. 186
Mazur's p-adic Mellin transformsp. 189
Ordinary [Lambda]-adic forms, two variable p-adic Rankin products and Galois representationsp. 194
p-Adic families of Eisenstein seriesp. 195
The projection to the ordinary partp. 200
Ordinary [Lambda]-adic formsp. 208
Two variable p-adic Rankin productp. 221
Ordinary Galois representations into GL[subscript 2](Z[subscript p X])p. 228
Examples of [Lambda]-adic formsp. 234
Functional equations of Hecke L-functionsp. 239
Adelic interpretation of algebraic number theoryp. 239
Hecke characters as continuous idele charactersp. 245
Self-duality of local fieldsp. 249
Haar measures and the Poisson summation formulap. 253
Adelic Haar measuresp. 257
Functional equations of Hecke L-functionsp. 261
Adelic Eisenstein series and Rankin productsp. 272
Modular forms on GL[subscript 2](F[subscript A])p. 272
Fourier expansion of Eisenstein seriesp. 282
Functional equation of Eisenstein seriesp. 292
Analytic continuation of Rankin productsp. 298
Functional equations of Rankin productsp. 306
Three variable p-adic Rankin productsp. 310
Differential operators of Maass and Shimurap. 310
The algebraicity theorem of Rankin productsp. 317
Two variable [Lambda]-adic Eisenstein seriesp. 326
Three variable p-adic Rankin productsp. 331
Relation to two variable p-adic Rankin productsp. 339
Concluding remarksp. 343
Summary of homology and cohomology theoryp. 345
Referencesp. 365
Answers to selected exercisesp. 371
Indexp. 383
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521435697
ISBN-10: 0521435692
Series: London Mathematical Society Student Texts
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 400
Published: 12th April 1993
Country of Publication: GB
Dimensions (cm): 23.5 x 15.88  x 1.91
Weight (kg): 0.57