Elementary Theory of L-functions and Eisenstein Series

Elementary Theory of L-functions and Eisenstein Series

by Haruzo Hida
     
 

ISBN-10: 0521435692

ISBN-13: 9780521435697

Pub. Date: 02/28/1993

Publisher: Cambridge University Press

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.  See more details below

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.

Product Details

ISBN-13:
9780521435697
Publisher:
Cambridge University Press
Publication date:
02/28/1993
Series:
London Mathematical Society Student Texts Series, #26
Edition description:
New Edition
Pages:
400
Product dimensions:
5.98(w) x 8.98(h) x 0.91(d)

Table of Contents

Suggestions to the reader
Ch. 1Algebraic number theory1
1.1Linear algebra over rings1
1.2Algebraic number fields5
1.3p-adic numbers17
Ch. 2Classical L-functions and Eisenstein series25
2.1Euler's method25
2.2Analytic continuation and the functional equation33
2.3Hurwitz and Dirichlet L-functions40
2.4Shintani L-functions47
2.5L-functions of real quadratic field54
2.6L-functions of imaginary quadratic fields63
2.7Hecke L-functions of number fields66
Ch. 3p-adic Hecke L-functions73
3.1Interpolation series73
3.2Interpolation series in p-adic fields75
3.3p-adic measures on Z[subscript p]78
3.4The p-adic measure of the Riemann zeta function80
3.5p-adic Dirichlet L-functions82
3.6Group schemes and formal group schemes89
3.7Toroidal formal groups and p-adic measures96
3.8p-adic Shintani L-functions of totally real fields99
3.9p-adic Hecke L-functions of totally real fields102
Ch. 4Homological interpretation107
4.1Cohomology groups on G[subscript m](C)107
4.2Cohomological interpretation of Dirichlet L-values117
4.3p-adic measures and locally constant functions118
4.4Another construction of p-adic Dirichlet L-functions120
Ch. 5Elliptic modular forms and their L-functions125
5.1Classical Eisenstein series of GL(2)[subscript /][subscript Q]125
5.2Rationality of modular forms131
5.3Hecke operators139
5.4The Petersson inner product and the Rankin product150
5.5Standard L-functions of holomorphic modular forms157
Ch. 6Modular forms and cohomology groups160
6.1Cohomology of modular groups160
6.2Eichler-Shimura isomorphisms167
6.3Hecke operators on cohomology groups175
6.4Algebraicity theorem for standard L-functions of GL(2)186
6.5Mazur's p-adic Mellin transforms189
Ch. 7Ordinary [Lambda]-adic forms, two variable p-adic Rankin products and Galois representations194
7.1p-Adic families of Eisenstein series195
7.2The projection to the ordinary part200
7.3Ordinary [Lambda]-adic forms208
7.4Two variable p-adic Rankin product221
7.5Ordinary Galois representations into GL[subscript 2](Z[subscript p][[X]])228
7.6Examples of [Lambda]-adic forms234
Ch. 8Functional equations of Hecke L-functions239
8.1Adelic interpretation of algebraic number theory239
8.2Hecke characters as continuous idele characters245
8.3Self-duality of local fields249
8.4Haar measures and the Poisson summation formula253
8.5Adelic Haar measures257
8.6Functional equations of Hecke L-functions261
Ch. 9Adelic Eisenstein series and Rankin products272
9.1Modular forms on GL[subscript 2](F[subscript A])272
9.2Fourier expansion of Eisenstein series282
9.3Functional equation of Eisenstein series292
9.4Analytic continuation of Rankin products298
9.5Functional equations of Rankin products306
Ch. 10Three variable p-adic Rankin products310
10.1Differential operators of Maass and Shimura310
10.2The algebraicity theorem of Rankin products317
10.3Two variable [Lambda]-adic Eisenstein series326
10.4Three variable p-adic Rankin products331
10.5Relation to two variable p-adic Rankin products339
10.6Concluding remarks343
Appendix: Summary of homology and cohomology theory345
References365
Answers to selected exercises371
Index383

Read More

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >