Adeles and Algebraic Groups

Adeles and Algebraic Groups

Author: A. Weil

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 137

ISBN-13: 1468491563

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This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel’s work on quadratic forms. Serving as an introduction to the subject, these notes may also provide stimulation for further research.


Algebraic Groups and Number Theory

Algebraic Groups and Number Theory

Author: Vladimir Platonov

Publisher: Academic Press

Published: 1993-12-07

Total Pages: 629

ISBN-13: 0080874592

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This milestone work on the arithmetic theory of linear algebraic groups is now available in English for the first time. Algebraic Groups and Number Theory provides the first systematic exposition in mathematical literature of the junction of group theory, algebraic geometry, and number theory. The exposition of the topic is built on a synthesis of methods from algebraic geometry, number theory, analysis, and topology, and the result is a systematic overview ofalmost all of the major results of the arithmetic theory of algebraic groups obtained to date.


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

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

Author: Stephen S. Gelbart

Publisher: Princeton University Press

Published: 2016-03-02

Total Pages: 280

ISBN-13: 1400881617

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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?


Rational Points on Varieties

Rational Points on Varieties

Author: Bjorn Poonen

Publisher: American Mathematical Soc.

Published: 2017-12-13

Total Pages: 358

ISBN-13: 1470437732

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This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere.


Basic Number Theory.

Basic Number Theory.

Author: Andre Weil

Publisher: Springer Science & Business Media

Published: 2013-12-14

Total Pages: 332

ISBN-13: 3662059789

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Itpzf}JlOV, li~oxov uoq>ZUJlCJ. 7:WV Al(JX., llpoj1. AE(Jj1. The first part of this volume is based on a course taught at Princeton University in 1961-62; at that time, an excellent set ofnotes was prepared by David Cantor, and it was originally my intention to make these notes available to the mathematical public with only quite minor changes. Then, among some old papers of mine, I accidentally came across a long-forgotten manuscript by ChevaIley, of pre-war vintage (forgotten, that is to say, both by me and by its author) which, to my taste at least, seemed to have aged very welt It contained abrief but essentially com plete account of the main features of c1assfield theory, both local and global; and it soon became obvious that the usefulness of the intended volume would be greatly enhanced if I inc1uded such a treatment of this topic. It had to be expanded, in accordance with my own plans, but its outline could be preserved without much change. In fact, I have adhered to it rather c10sely at some critical points.


Algebraic Geometry IV

Algebraic Geometry IV

Author: A.N. Parshin

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 291

ISBN-13: 366203073X

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Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.


A Brief Guide to Algebraic Number Theory

A Brief Guide to Algebraic Number Theory

Author: H. P. F. Swinnerton-Dyer

Publisher: Cambridge University Press

Published: 2001-02-22

Total Pages: 164

ISBN-13: 9780521004237

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Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.


Algebraic Groups and Number Theory

Algebraic Groups and Number Theory

Author: Vladimir Platonov

Publisher: Cambridge University Press

Published: 2023-08-31

Total Pages: 379

ISBN-13: 052111361X

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The first volume of a two-volume book offering a comprehensive account of the arithmetic theory of algebraic groups.


Algebraic Groups and Number Theory: Volume 1

Algebraic Groups and Number Theory: Volume 1

Author: Vladimir Platonov

Publisher: Cambridge University Press

Published: 2023-08-31

Total Pages: 380

ISBN-13: 1009380656

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The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements, as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas.