Etale Cohomology and the Weil Conjecture

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Publisher : Springer Science & Business Media
ISBN 13 : 3662025418
Total Pages : 336 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Etale Cohomology and the Weil Conjecture by : Eberhard Freitag

Download or read book Etale Cohomology and the Weil Conjecture written by Eberhard Freitag and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Some years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec tures. For the convenience of the speakers the present authors - who were also the organisers of that meeting - prepared short notes containing the central definitions and ideas of the proofs. The unexpected interest for these notes and the various suggestions to publish them encouraged us to work somewhat more on them and fill out the gaps. Our aim was to develop the theory in as self contained and as short a manner as possible. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils. Of course, all the central ideas are due to the people who created the theory, especially Grothendieck and Deligne. The main references are the SGA-notes [64-69]. With the kind permission of Professor J. A. Dieudonne we have included in the book that finally resulted his excellent notes on the history of the Weil conjectures, as a second introduction. Our original notes were written in German. However, we finally followed the recommendation made variously to publish the book in English. We had the good fortune that Professor W. Waterhouse and his wife Betty agreed to translate our manuscript. We want to thank them very warmly for their willing involvement in such a tedious task. We are very grateful to the staff of Springer-Verlag for their careful work.

Etale Cohomology and the Weil Conjecture

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Publisher : Springer
ISBN 13 : 9783662025437
Total Pages : 320 pages
Book Rating : 4.0/5 (254 download)

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Book Synopsis Etale Cohomology and the Weil Conjecture by : Eberhard Freitag

Download or read book Etale Cohomology and the Weil Conjecture written by Eberhard Freitag and published by Springer. This book was released on 2012-12-03 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Some years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjec tures. For the convenience of the speakers the present authors - who were also the organisers of that meeting - prepared short notes containing the central definitions and ideas of the proofs. The unexpected interest for these notes and the various suggestions to publish them encouraged us to work somewhat more on them and fill out the gaps. Our aim was to develop the theory in as self contained and as short a manner as possible. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils. Of course, all the central ideas are due to the people who created the theory, especially Grothendieck and Deligne. The main references are the SGA-notes [64-69]. With the kind permission of Professor J. A. Dieudonne we have included in the book that finally resulted his excellent notes on the history of the Weil conjectures, as a second introduction. Our original notes were written in German. However, we finally followed the recommendation made variously to publish the book in English. We had the good fortune that Professor W. Waterhouse and his wife Betty agreed to translate our manuscript. We want to thank them very warmly for their willing involvement in such a tedious task. We are very grateful to the staff of Springer-Verlag for their careful work.

Etale Cohomology and the Weil Conjecture

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Publisher :
ISBN 13 : 9783662025420
Total Pages : 344 pages
Book Rating : 4.0/5 (254 download)

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Book Synopsis Etale Cohomology and the Weil Conjecture by : Eberhard Freitag

Download or read book Etale Cohomology and the Weil Conjecture written by Eberhard Freitag and published by . This book was released on 1987-12-29 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is concerned with one of the most important developments in algebraic geometry during the last decades. In 1949 AndrA(c) Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. He himself proved his conjectures by means of an algebraic theory of Abelian varieties in the one-variable case. In 1960 appeared the first chapter of the "ElA(c)ments de GA(c)ometrie AlgA(c)braique" par A. Grothendieck (en collaboration avec J. DieudonnA(c)). In these "ElA(c)ments" Grothendieck evolved a new foundation of algebraic geometry with the declared aim to come to a proof of the Weil conjectures by means of a new algebraic cohomology theory. Deligne succeded in proving the Weil conjectures on the basis of Grothendiecks ideas. The aim of this "Ergebnisbericht" is to develop as self-contained as possible and as short as possible Grothendiecks 1-adic cohomology theory including Delignes monodromy theory and to present his original proof of the Weil conjectures.

Weil Conjectures, Perverse Sheaves and l-adic Fourier Transform

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Publisher : Springer Science & Business Media
ISBN 13 : 3662045761
Total Pages : 382 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Weil Conjectures, Perverse Sheaves and l-adic Fourier Transform by : Reinhardt Kiehl

Download or read book Weil Conjectures, Perverse Sheaves and l-adic Fourier Transform written by Reinhardt Kiehl and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors describe the important generalization of the original Weil conjectures, as given by P. Deligne in his fundamental paper "La conjecture de Weil II". The authors follow the important and beautiful methods of Laumon and Brylinski which lead to a simplification of Deligne's theory. Deligne's work is closely related to the sheaf theoretic theory of perverse sheaves. In this framework Deligne's results on global weights and his notion of purity of complexes obtain a satisfactory and final form. Therefore the authors include the complete theory of middle perverse sheaves. In this part, the l-adic Fourier transform is introduced as a technique providing natural and simple proofs. To round things off, there are three chapters with significant applications of these theories.

Weil's Conjecture for Function Fields

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Publisher : Princeton University Press
ISBN 13 : 0691184437
Total Pages : 320 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis Weil's Conjecture for Function Fields by : Dennis Gaitsgory

Download or read book Weil's Conjecture for Function Fields written by Dennis Gaitsgory and published by Princeton University Press. This book was released on 2019-02-19 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil’s conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil’s conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting l-adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil’s conjecture. The proof of the product formula will appear in a sequel volume.

Etale Cohomology (PMS-33)

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Publisher : Princeton University Press
ISBN 13 : 9780691082387
Total Pages : 346 pages
Book Rating : 4.0/5 (823 download)

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Book Synopsis Etale Cohomology (PMS-33) by : J. S. Milne

Download or read book Etale Cohomology (PMS-33) written by J. S. Milne and published by Princeton University Press. This book was released on 1980-04-21 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Introduction to Étale Cohomology

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Publisher : Springer Science & Business Media
ISBN 13 : 3642784216
Total Pages : 192 pages
Book Rating : 4.6/5 (427 download)

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Book Synopsis Introduction to Étale Cohomology by : Günter Tamme

Download or read book Introduction to Étale Cohomology written by Günter Tamme and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: A succinct introduction to etale cohomology. Well-presented and chosen this will be a most welcome addition to the algebraic geometrist's library.

Lecture Notes on Motivic Cohomology

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Publisher : American Mathematical Soc.
ISBN 13 : 9780821838471
Total Pages : 240 pages
Book Rating : 4.8/5 (384 download)

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Book Synopsis Lecture Notes on Motivic Cohomology by : Carlo Mazza

Download or read book Lecture Notes on Motivic Cohomology written by Carlo Mazza and published by American Mathematical Soc.. This book was released on 2006 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of a motive is an elusive one, like its namesake "the motif" of Cezanne's impressionist method of painting. Its existence was first suggested by Grothendieck in 1964 as the underlying structure behind the myriad cohomology theories in Algebraic Geometry. We now know that there is a triangulated theory of motives, discovered by Vladimir Voevodsky, which suffices for the development of a satisfactory Motivic Cohomology theory. However, the existence of motives themselves remains conjectural. This book provides an account of the triangulated theory of motives. Its purpose is to introduce Motivic Cohomology, to develop its main properties, and finally to relate it to other known invariants of algebraic varieties and rings such as Milnor K-theory, etale cohomology, and Chow groups. The book is divided into lectures, grouped in six parts. The first part presents the definition of Motivic Cohomology, based upon the notion of presheaves with transfers. Some elementary comparison theorems are given in this part. The theory of (etale, Nisnevich, and Zariski) sheaves with transfers is developed in parts two, three, and six, respectively. The theoretical core of the book is the fourth part, presenting the triangulated category of motives. Finally, the comparison with higher Chow groups is developed in part five. The lecture notes format is designed for the book to be read by an advanced graduate student or an expert in a related field. The lectures roughly correspond to one-hour lectures given by Voevodsky during the course he gave at the Institute for Advanced Study in Princeton on this subject in 1999-2000. In addition, many of the original proofs have been simplified and improved so that this book will also be a useful tool for research mathematicians. Information for our distributors: Titles in this series are copublished with the Clay Mathematics Institute (Cambridge, MA).

Rational Points on Varieties

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Publisher : American Mathematical Society
ISBN 13 : 1470474581
Total Pages : 357 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rational Points on Varieties by : Bjorn Poonen

Download or read book Rational Points on Varieties written by Bjorn Poonen and published by American Mathematical Society. This book was released on 2023-08-10 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University

Etale Cohomology Theory

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Publisher : World Scientific
ISBN 13 : 9814307726
Total Pages : 622 pages
Book Rating : 4.8/5 (143 download)

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Book Synopsis Etale Cohomology Theory by : Lei Fu

Download or read book Etale Cohomology Theory written by Lei Fu and published by World Scientific. This book was released on 2011 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.

Etale Cohomology Theory

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Publisher : World Scientific
ISBN 13 : 9814464805
Total Pages : 624 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Etale Cohomology Theory by : Lei Fu

Download or read book Etale Cohomology Theory written by Lei Fu and published by World Scientific. This book was released on 2011-01-31 with total page 624 pages. Available in PDF, EPUB and Kindle. Book excerpt: New Edition available hereEtale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.

Noncommutative Motives

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Publisher : American Mathematical Soc.
ISBN 13 : 1470423979
Total Pages : 114 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Noncommutative Motives by : Gonçalo Tabuada

Download or read book Noncommutative Motives written by Gonçalo Tabuada and published by American Mathematical Soc.. This book was released on 2015-09-21 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of motives began in the early 1960s when Grothendieck envisioned the existence of a "universal cohomology theory of algebraic varieties". The theory of noncommutative motives is more recent. It began in the 1980s when the Moscow school (Beilinson, Bondal, Kapranov, Manin, and others) began the study of algebraic varieties via their derived categories of coherent sheaves, and continued in the 2000s when Kontsevich conjectured the existence of a "universal invariant of noncommutative algebraic varieties". This book, prefaced by Yuri I. Manin, gives a rigorous overview of some of the main advances in the theory of noncommutative motives. It is divided into three main parts. The first part, which is of independent interest, is devoted to the study of DG categories from a homotopical viewpoint. The second part, written with an emphasis on examples and applications, covers the theory of noncommutative pure motives, noncommutative standard conjectures, noncommutative motivic Galois groups, and also the relations between these notions and their commutative counterparts. The last part is devoted to the theory of noncommutative mixed motives. The rigorous formalization of this latter theory requires the language of Grothendieck derivators, which, for the reader's convenience, is revised in a brief appendix.

The Brauer–Grothendieck Group

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Publisher : Springer Nature
ISBN 13 : 3030742482
Total Pages : 450 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis The Brauer–Grothendieck Group by : Jean-Louis Colliot-Thélène

Download or read book The Brauer–Grothendieck Group written by Jean-Louis Colliot-Thélène and published by Springer Nature. This book was released on 2021-07-30 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer–Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications. Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available in book form elsewhere; notably, de Jong’s proof of Gabber’s theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer–Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples. Over half a century after Grothendieck's foundational seminars on the topic, The Brauer–Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry.

Etale Homotopy of Simplical Schemes

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Publisher : Princeton University Press
ISBN 13 : 9780691083179
Total Pages : 196 pages
Book Rating : 4.0/5 (831 download)

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Book Synopsis Etale Homotopy of Simplical Schemes by : Eric M. Friedlander

Download or read book Etale Homotopy of Simplical Schemes written by Eric M. Friedlander and published by Princeton University Press. This book was released on 1982-12-21 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions. One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Motives

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Publisher : American Mathematical Soc.
ISBN 13 : 0821827987
Total Pages : 694 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Motives by :

Download or read book Motives written by and published by American Mathematical Soc.. This book was released on 1994-02-28 with total page 694 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'Motives' were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, and to play the role of the missing rational cohomology. This work contains the texts of the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991.

Rigid Local Systems

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Publisher : Princeton University Press
ISBN 13 : 9780691011189
Total Pages : 236 pages
Book Rating : 4.0/5 (111 download)

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Book Synopsis Rigid Local Systems by : Nicholas M. Katz

Download or read book Rigid Local Systems written by Nicholas M. Katz and published by Princeton University Press. This book was released on 1996 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: Riemann introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study nth order linear differential equations by studying the rank n local systems (of local holomorphic solutions) to which they gave rise. His first application was to study the classical Gauss hypergeometric function, which he did by studying rank-two local systems on P1- {0,1,infinity}. His investigation was successful, largely because any such (irreducible) local system is rigid in the sense that it is globally determined as soon as one knows separately each of its local monodromies. It became clear that luck played a role in Riemann's success: most local systems are not rigid. Yet many classical functions are solutions of differential equations whose local systems are rigid, including both of the standard nth order generalizations of the hypergeometric function, n F n-1's, and the Pochhammer hypergeometric functions. This book is devoted to constructing all (irreducible) rigid local systems on P1-{a finite set of points} and recognizing which collections of independently given local monodromies arise as the local monodromies of irreducible rigid local systems. Although the problems addressed here go back to Riemann, and seem to be problems in complex analysis, their solutions depend essentially on a great deal of very recent arithmetic algebraic geometry, including Grothendieck's etale cohomology theory, Deligne's proof of his far-reaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the l-adic Fourier Transform.

Arithmetic Duality Theorems

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Publisher :
ISBN 13 :
Total Pages : 440 pages
Book Rating : 4.3/5 (97 download)

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Book Synopsis Arithmetic Duality Theorems by : J. S. Milne

Download or read book Arithmetic Duality Theorems written by J. S. Milne and published by . This book was released on 1986 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, ,tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.