Stephen S. Gelbart's Books

Automorphic Forms on Adele Groups. (AM-83), Volume 83

By: Stephen S. Gelbart

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Automorphic Forms on Adele Groups. (AM-83), Volume 83

By: Stephen S. Gelbart

This volume investigates the interplay between the classical theory of automorphic forms and the modern theory of representations of adele groups. Interpreting important recent contributions of Jacquet and Langlands, the author presents new and previously inaccessible results, and systematically develops explicit consequences and connections with the classical theory. The underlying theme is the decomposition of the regular representation of the adele group of GL(2). A detailed proof of the celebrated trace formula of Selberg is included, with a discussion of the possible range of applicability of this formula. Throughout the work the author emphasizes new examples and problems that remain open within the general theory. TABLE OF CONTENTS: 1. The Classical Theory 2. Automorphic Forms and the Decomposition of L2(PSL(2,R) 3. Automorphic Forms as Functions on the Adele Group of GL(2) 4. The Representations of GL(2) over Local and Global Fields 5. Cusp Forms and Representations of the Adele Group of GL(2) 6. Hecke Theory for GL(2) 7. The Construction of a Special Class of Automorphic Forms 8. Eisenstein Series and the Continuous Spectrum 9. The Trace Formula for GL(2) 10. Automorphic Forms on a Quaternion Algebr?

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Page Count

280

Publisher

Princeton University Press

Published Date

ISBN 10

1400881617

ISBN 13

9781400881611

Lectures on the Arthur-Selberg Trace Formula

By: Stephen S. Gelbart

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Lectures on the Arthur-Selberg Trace Formula

By: Stephen S. Gelbart

The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy classes of a group and the spectral terms given by the induced representations. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. The formulas are difficult in general and even the case of $GL$(2) is nontrivial. The book gives proof of Arthur's trace formula of the 1970s and 1980s, with special attention given to $GL$(2). The problem is that when the truncated terms converge, they are also shown to be polynomial in the truncation variable and expressed as ``weighted'' orbital and ``weighted'' characters. In some important cases the trace formula takes on a simple form over $G$. The author gives some examples of this, and also some examples of Jacquet's relative trace formula. This work offers for the first time a simultaneous treatment of a general group with the case of $GL$(2). It also treats the trace formula with the example of Jacquet's relative formula. Features: Discusses why the terms of the geometric and spectral type must be truncated, and why the resulting truncations are polynomials in the truncation of value $T$. Brings into play the significant tool of ($G, M$) families and how the theory of Paley-Weiner is applied. Explains why the truncation formula reduces to a simple formula involving only the elliptic terms on the geometric sides with the representations appearing cuspidally on the spectral side (applies to Tamagawa numbers). Outlines Jacquet's trace formula and shows how it works for $GL$(2).

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Page Count

112

Publisher

American Mathematical Soc.

Published Date

ISBN 10

0821805711

ISBN 13

9780821805718

Analytic Properties of Automorphic L-Functions

By: Stephen Gelbart,Freydoon Shahidi

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Analytic Properties of Automorphic L-Functions

By: Stephen Gelbart,Freydoon Shahidi

Analytic Properties of Automorphic L-Functions is a three-chapter text that covers considerable research works on the automorphic L-functions attached by Langlands to reductive algebraic groups. Chapter I focuses on the analysis of Jacquet-Langlands methods and the Einstein series and Langlands’ so-called “Euler products . This chapter explains how local and global zeta-integrals are used to prove the analytic continuation and functional equations of the automorphic L-functions attached to GL(2). Chapter II deals with the developments and refinements of the zeta-inetgrals for GL(n). Chapter III describes the results for the L-functions L (s, ?, r), which are considered in the constant terms of Einstein series for some quasisplit reductive group. This book will be of value to undergraduate and graduate mathematics students.

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Page Count

142

Publisher

Academic Press

Published Date

ISBN 10

1483261034

ISBN 13

9781483261034

The Descent Map From Automorphic Representations Of Gl(n) To Classical Groups

By: David Soudry,David Ginzburg,Stephen Rallis

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The Descent Map From Automorphic Representations Of Gl(n) To Classical Groups

By: David Soudry,David Ginzburg,Stephen Rallis

This book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of Gelfand-Graev type, or of Fourier-Jacobi type when applied to certain residual Eisenstein series. This book contains a complete account of this automorphic descent, with complete, detailed proofs. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars.

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Page Count

350

Publisher

World Scientific

Published Date

ISBN 10

9814464945

ISBN 13

9789814464949

Explicit Constructions of Automorphic L-Functions

By: Stephen Gelbart,Ilya Piatetski-Shapiro,Stephen Rallis

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Explicit Constructions of Automorphic L-Functions

By: Stephen Gelbart,Ilya Piatetski-Shapiro,Stephen Rallis

The goal of this research monograph is to derive the analytic continuation and functional equation of the L-functions attached by R.P. Langlands to automorphic representations of reductive algebraic groups. The first part of the book (by Piatetski-Shapiro and Rallis) deals with L-functions for the simple classical groups; the second part (by Gelbart and Piatetski-Shapiro) deals with non-simple groups of the form G GL(n), with G a quasi-split reductive group of split rank n. The method of proof is to construct certain explicit zeta-integrals of Rankin-Selberg type which interpolate the relevant Langlands L-functions and can be analyzed via the theory of Eisenstein series and intertwining operators. This is the first time such an approach has been applied to such general classes of groups. The flavor of the local theory is decidedly representation theoretic, and the work should be of interest to researchers in group representation theory as well as number theory.

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Page Count

158

Publisher

Springer

Published Date

ISBN 10

3540478809

ISBN 13

9783540478805

Modular Forms and Special Cycles on Shimura Curves. (AM-161)

By: Stephen S. Kudla,Michael Rapoport,Tonghai Yang

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Modular Forms and Special Cycles on Shimura Curves. (AM-161)

By: Stephen S. Kudla,Michael Rapoport,Tonghai Yang

Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

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Page Count

384

Publisher

Princeton University Press

Published Date

ISBN 10

1400837162

ISBN 13

9781400837168

Automorphic Representations, L-functions and Applications

Progress and Prospects : Proceedings of a Conference Honoring Steve Rallis on the Occasion of His 60th Birthday, the Ohio State University, March 27-30, 2003

By: Stephen Rallis

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Automorphic Representations, L-functions and Applications

Progress and Prospects : Proceedings of a Conference Honoring Steve Rallis on the Occasion of His 60th Birthday, the Ohio State University, March 27-30, 2003

By: Stephen Rallis

This volume is the proceedings of the conference on Automorphic Representations, L-functions and Applications: Progress and Prospects, held at the Department of Mathematics of The Ohio State University, March 27-30, 2003, in honor of the 60th birthday of Steve Rallis. The theory of automorphic representations, automorphic L-functions and their applications to arithmetic continues to be an area of vigorous and fruitful research. The contributed papers in this volume represent many of the most recent developments and directions, including Rankin-Selberg L-functions (Bump, Ginzburg-Jiang-Rallis, Lapid-Rallis) the relative trace formula (Jacquet, Mao-Rallis) automorphic representations (Gan-Gurevich, Ginzburg-Rallis-Soudry) representation theory of p-adic groups (Baruch, Kudla-Rallis, Moeglin, Cogdell-Piatetski-Shapiro-Shahidi) p-adic methods (Harris-Li-Skinner, Vigneras), and arithmetic applications (Chinta-Friedberg-Hoffstein). The survey articles by Bump, on the Rankin-Selberg method, and by Jacquet, on the relative trace formula, should be particularly useful as an introduction to the key ideas about these important topics. This volume should be of interest both to researchers and students in the area of automorphic representations, as well as to mathematicians in other areas interested in having an overview of current developments in this important field.

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Page Count

442

Publisher

Walter de Gruyter

Published Date

ISBN 10

3110179393

ISBN 13

9783110179392

Automorphic Forms on Adele Groups. (AM-83), Volume 83

By: Stephen S. Gelbart

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Automorphic Forms on Adele Groups. (AM-83), Volume 83

By: Stephen S. Gelbart

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Solve jigsaw puzzle

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280

Publisher

Princeton University Press

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ISBN 10

1400881617

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9781400881611

Book's by Stephen S. Gelbart

Automorphic Forms on Adele Groups. (AM-83), Volume 83

Automorphic Forms on Adele Groups. (AM-83), Volume 83

Lectures on the Arthur-Selberg Trace Formula

Lectures on the Arthur-Selberg Trace Formula

Analytic Properties of Automorphic L-Functions

Analytic Properties of Automorphic L-Functions

The Descent Map From Automorphic Representations Of Gl(n) To Classical Groups

The Descent Map From Automorphic Representations Of Gl(n) To Classical Groups

Explicit Constructions of Automorphic L-Functions

Explicit Constructions of Automorphic L-Functions

Modular Forms and Special Cycles on Shimura Curves. (AM-161)

Modular Forms and Special Cycles on Shimura Curves. (AM-161)

Automorphic Representations, L-functions and Applications

Automorphic Representations, L-functions and Applications

Automorphic Forms on Adele Groups. (AM-83), Volume 83

Automorphic Forms on Adele Groups. (AM-83), Volume 83

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