Eisenstein Series and Automorphic $L$-Functions
Eisenstein Series and Automorphic $L$-Functions

Author: Freydoon Shahidi

Publisher: American Mathematical Soc.

Published: 2010

Total Pages: 218

ISBN-13: 0821849891

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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.

Eisenstein Series and Applications
Eisenstein Series and Applications

Author: Wee Teck Gan

Publisher: Springer Science & Business Media

Published: 2007-12-22

Total Pages: 317

ISBN-13: 0817646396

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Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. Bringing together contributions from areas which do not usually interact with each other, this volume introduces diverse users of Eisenstein series to a variety of important applications. With this juxtaposition of perspectives, the reader obtains deeper insights into the arithmetic of Eisenstein series. The central theme of the exposition focuses on the common structural properties of Eisenstein series occurring in many related applications.

Eisenstein Series and Automorphic Representations
Eisenstein Series and Automorphic Representations

Author: Philipp Fleig

Publisher: Cambridge Studies in Advanced

Published: 2018-07-05

Total Pages: 587

ISBN-13: 1107189926

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Detailed exposition of automorphic representations and their relation to string theory, for mathematicians and theoretical physicists.

Eisenstein Series and Automorphic Representations
Eisenstein Series and Automorphic Representations

Author: Philipp Fleig

Publisher: Cambridge University Press

Published: 2018-07-05

Total Pages: 588

ISBN-13: 1108118992

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This introduction to automorphic forms on adelic groups G(A) emphasises the role of representation theory. The exposition is driven by examples, and collects and extends many results scattered throughout the literature, in particular the Langlands constant term formula for Eisenstein series on G(A) as well as the Casselman–Shalika formula for the p-adic spherical Whittaker function. This book also covers more advanced topics such as spherical Hecke algebras and automorphic L-functions. Many of these mathematical results have natural interpretations in string theory, and so some basic concepts of string theory are introduced with an emphasis on connections with automorphic forms. Throughout the book special attention is paid to small automorphic representations, which are of particular importance in string theory but are also of independent mathematical interest. Numerous open questions and conjectures, partially motivated by physics, are included to prompt the reader's own research.

Some Applications of Modular Forms
Some Applications of Modular Forms

Author: Peter Sarnak

Publisher: Cambridge University Press

Published: 1990-11-15

Total Pages: 124

ISBN-13: 1316582442

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The theory of modular forms and especially the so-called 'Ramanujan Conjectures' have been applied to resolve problems in combinatorics, computer science, analysis and number theory. This tract, based on the Wittemore Lectures given at Yale University, is concerned with describing some of these applications. In order to keep the presentation reasonably self-contained, Professor Sarnak begins by developing the necessary background material in modular forms. He then considers the solution of three problems: the Ruziewicz problem concerning finitely additive rotationally invariant measures on the sphere; the explicit construction of highly connected but sparse graphs: 'expander graphs' and 'Ramanujan graphs'; and the Linnik problem concerning the distribution of integers that represent a given large integer as a sum of three squares. These applications are carried out in detail. The book therefore should be accessible to a wide audience of graduate students and researchers in mathematics and computer science.

Spectral Decomposition and Eisenstein Series
Spectral Decomposition and Eisenstein Series

Author: Colette Moeglin

Publisher: Cambridge University Press

Published: 1995-11-02

Total Pages: 382

ISBN-13: 9780521418935

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A self-contained introduction to automorphic forms, and Eisenstein series and pseudo-series, proving some of Langlands' work at the intersection of number theory and group theory.

Elementary Theory of L-functions and Eisenstein Series
Elementary Theory of L-functions and Eisenstein Series

Author: Haruzo Hida

Publisher: Cambridge University Press

Published: 1993-02-11

Total Pages: 404

ISBN-13: 9780521435697

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The theory of p-adic and classic modular forms, and the study of arithmetic and p-adic L-functions has proved to be a fruitful area of mathematics over the last decade. Professor Hida has given courses on these topics in the USA, Japan, and in France, and in this book provides the reader with an elementary but detailed insight into the theory of L-functions. The presentation is self contained and concise, and the subject is approached using only basic tools from complex analysis and cohomology theory. Graduate students wishing to know more about L-functions will find that this book offers a unique introduction to this fascinating branch of mathematics.