Laurent Clozel's Books

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

By: James Arthur,Laurent Clozel

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Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

By: James Arthur,Laurent Clozel

A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the "automorphic representations" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences. This book studies one of the simplest general problems in the theory, that of relating automorphic forms on arithmetic subgroups of GL(n,E) and GL(n,F) when E/F is a cyclic extension of number fields. (This is known as the base change problem for GL(n).) The problem is attacked and solved by means of the trace formula. The book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms.

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248

Publisher

Princeton University Press

Published Date

ISBN 10

1400882400

ISBN 13

9781400882403

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

By: Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight

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Elliptic Curves, Hilbert Modular Forms and Galois Deformations

By: Laurent Berger,Gebhard Böckle,Lassina Dembélé,Mladen Dimitrov,Tim Dokchitser,John Voight

The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

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257

Publisher

Springer Science & Business Media

Published Date

ISBN 10

3034806183

ISBN 13

9783034806183

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula

By: James Arthur,Laurent Clozel

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Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula

By: James Arthur,Laurent Clozel

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247

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Princeton University Press

Published Date

ISBN 10

0691085188

ISBN 13

9780691085180

Asian-French Summer School on Algebraic Geometry and Number Theory

By: Minhyong Kim,R. Sujatha,Laurent Lafforgue,Takeshi Saitō,Alain Genestier,Jörg Wildeshaus,Báo Châu Ngô,Laurent Clozel,Marc Levine,Bruno Kahn

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Asian-French Summer School on Algebraic Geometry and Number Theory

By: Minhyong Kim,R. Sujatha,Laurent Lafforgue,Takeshi Saitō,Alain Genestier,Jörg Wildeshaus,Báo Châu Ngô,Laurent Clozel,Marc Levine,Bruno Kahn

This volume contains the third part of the lecture notes of the Asian-French Summer School on Algebraic Geometry and Number Theory, which was held at the Institut des Hautes Etudes Scientifiques (Bures-sur-Yvette) and the Universite Paris-Sud XI (Orsay) in July 2006. This summer school was devoted to the theory of motives and its recent developments and to related topics, notably Shimura varieties and automorphic representations.

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ISBN 10

285629846X

ISBN 13

9782856298466

Book's by Laurent Clozel

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula

Asian-French Summer School on Algebraic Geometry and Number Theory

Asian-French Summer School on Algebraic Geometry and Number Theory

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