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Automorphic Forms and Galois Representations: Volume 2

Author: Fred Diamond

Publisher: Cambridge University Press

Published: 2014-10-16

Total Pages: 387

ISBN-13: 1316062341

Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representations' in July 2011, the aim of which was to explore recent developments in this area. The expository articles and research papers across the two volumes reflect recent interest in p-adic methods in number theory and representation theory, as well as recent progress on topics from anabelian geometry to p-adic Hodge theory and the Langlands program. The topics covered in volume two include curves and vector bundles in p-adic Hodge theory, associators, Shimura varieties, the birational section conjecture, and other topics of contemporary interest.

Towards a Modulo $p$ Langlands Correspondence for GL$_2$

Author: Christophe Breuil

Publisher: American Mathematical Soc.

Published: 2012-02-22

Total Pages: 127

ISBN-13: 0821852272

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The authors construct new families of smooth admissible $\overline{\mathbb{F}}_p$-representations of $\mathrm{GL}_2(F)$, where $F$ is a finite extension of $\mathbb{Q}_p$. When $F$ is unramified, these representations have the $\mathrm{GL}_2({\mathcal O}_F)$-socle predicted by the recent generalizations of Serre's modularity conjecture. The authors' motivation is a hypothetical mod $p$ Langlands correspondence.

Number Theory, Analysis and Geometry

Author: Dorian Goldfeld

Publisher: Springer Science & Business Media

Published: 2011-12-21

Total Pages: 715

ISBN-13: 1461412609

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Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas of the field, namely Number Theory, Analysis, and Geometry, representing Lang’s own breadth of interest and impact. A special introduction by John Tate includes a brief and fascinating account of the Serge Lang’s life. This volume's group of 6 editors are also highly prominent mathematicians and were close to Serge Lang, both academically and personally. The volume is suitable to research mathematicians in the areas of Number Theory, Analysis, and Geometry.

Lectures on Modular Forms

Author: Joseph J. Lehner

Publisher: Courier Dover Publications

Published: 2017-05-17

Total Pages: 96

ISBN-13: 0486821404

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Concise book offers expository account of theory of modular forms and its application to number theory and analysis. Substantial notes at the end of each chapter amplify the more difficult subjects. 1969 edition.

Coefficient Systems and Supersingular Representations of GL2(F)

Author: Vytautas Paskunas

Publisher:

Published: 2004

Total Pages: 102

ISBN-13:

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Let $F$ be a non-Archimedean local field with the residual characteristic $p$. The author constructs a good number of smooth irreducible $\overline {\mathbf {F}}_p$-representations of $\mathrm {GL}_2(F)$, which are supersingular in the sense of Barthel and Livne. If $F=\mathbf {Q}_p$ then results of Breuil imply that our construction gives all the supersingular representations up to the twist by an unramified quasi-character. The author conjectures that this is true for an arbitrary $F$. The book is suitable for graduate students and research mathematicians interested in algebra and algebraic geometry.

Elliptic Curves and Arithmetic Invariants

Author: Haruzo Hida

Publisher: Springer Science & Business Media

Published: 2013-06-13

Total Pages: 464

ISBN-13: 1461466571

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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-invariant, L-invariant, and similar topics. This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties. Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader. Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory. Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

Modular Forms and Fermat’s Last Theorem

Author: Gary Cornell

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 592

ISBN-13: 1461219744

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This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

Finite Fields and Applications

Author: Gary L. Mullen

Publisher: American Mathematical Soc.

Published: 2007

Total Pages: 190

ISBN-13: 0821844180

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Finite fields Combinatorics Algebraic coding theory Cryptography Background in number theory and abstract algebra Hints for selected exercises References Index.

Arithmetic Algebraic Geometry

Author: Brian David Conrad

Publisher: American Mathematical Soc.

Published:

Total Pages: 588

ISBN-13: 9780821886915

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The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.

Categories and Sheaves

Author: Masaki Kashiwara

Publisher: Springer Science & Business Media

Published: 2005-12-19

Total Pages: 496

ISBN-13: 3540279504

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Categories and sheaves appear almost frequently in contemporary advanced mathematics. This book covers categories, homological algebra and sheaves in a systematic manner starting from scratch and continuing with full proofs to the most recent results in the literature, and sometimes beyond. The authors present the general theory of categories and functors, emphasizing inductive and projective limits, tensor categories, representable functors, ind-objects and localization.