**Colloquium Publications**

Volume: 58;
2010;
210 pp;
Hardcover

MSC: Primary 11;
Secondary 22

Print ISBN: 978-0-8218-4989-7

Product Code: COLL/58

List Price: $58.00

Individual Member Price: $46.40

**Electronic ISBN: 978-1-4704-1767-3
Product Code: COLL/58.E**

List Price: $58.00

Individual Member Price: $46.40

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#### Supplemental Materials

# Eisenstein Series and Automorphic $L$-Functions

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*Freydoon Shahidi*

This book presents a treatment of the theory of
\(L\)-functions developed by means of the theory of Eisenstein
series and their Fourier coefficients, a theory which is usually
referred to as the Langlands–Shahidi method. The information
gathered from this method, when combined with the converse theorems of
Cogdell and Piatetski-Shapiro, has been quite sufficient in
establishing a number of new cases of Langlands functoriality
conjecture; at present, some of these cases cannot be obtained by any
other method. These results have led to far-reaching new estimates
for Hecke eigenvalues of Maass forms, as well as definitive solutions
to certain problems in analytic and algebraic number theory.

This book gives a detailed treatment of important parts of this theory,
including a rather complete proof of Casselman–Shalika's formula for
unramified Whittaker functions as well as a general treatment of the
theory of intertwining operators. It also covers in some detail the global
aspects of the method as well as some of its applications to group
representations and harmonic analysis.

This book is addressed to graduate students and researchers who are
interested in the Langlands program in automorphic forms and its
connections with number theory.

#### Table of Contents

# Table of Contents

## Eisenstein Series and Automorphic $L$-Functions

- Cover Cover11
- Title page iii4
- Contents v6
- Introduction 18
- Reductive groups 714
- Satake isomorphisms 2330
- Generic representations 4552
- Intertwining operators 8188
- Local coefficients 99106
- Eisenstein series 107114
- Fourier coefficients of Eisenstein series 121128
- Functional equations 129136
- Further properties of 𝐿–functions 153160
- Applications to functoriality 167174
- Appendices: Tables of Dynkin diagrams 181188
- Bibliography 199206
- Index 207214
- Back Cover Back Cover1218

#### Readership

Graduate students and research mathematicians interested in automorphic forms and \(L\)-functions, number theory, and representation theory.

#### Reviews

Written by one of the main players in the game, Eisenstein Series and Automorphic L-functions is nigh on indispensable for anyone wishing to get into this beautiful and exciting part of number theory. A glance at the table of contents of Freydoon Shahidi's book shows that it is something of a pedagogical dream: it seems to cover just about everything one might want to learn about the interface between automorphic representations and L-functions - a theme of huge importance in number theory in the last 50 years. Of course, there is more to the subject than can be covered in the book's 200 pages, but Shahidi touches on a great number of critically important topics, arranges them extremely efficiently, and then expounds them brilliantly. . . . I cannot imagine a better trajectory toward learning this material than to go through the pages of Eisenstein Series and Automorphic L-functions with a lot of note paper handy and a comfortable writing pen, to follow up on Shahidi's numerous cross-references when so indicated (and to the right degree, so to speak), and of course to fill out the sketches and do the exercises that the author presents. This very beautiful mathematics is well worth the effort.

-- Michael Berg, MAA Reviews