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Weil Conjectures, Perverse Sheaves and l-adic Fourier Transform

2013-03-14 Mathematics
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Release : 2013-03-14
Genre : Mathematics
Kind : eBook
Book Rating : 761/5 ( reviews)

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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. This book was released on 2013-03-14. 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.

Etale Cohomology and the Weil Conjecture

2013-03-14 Mathematics
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Release : 2013-03-14
Genre : Mathematics
Kind : eBook
Book Rating : 418/5 ( reviews)

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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. This book was released on 2013-03-14. 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.

Weil's Conjecture for Function Fields

2019-02-19 Mathematics
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Release : 2019-02-19
Genre : Mathematics
Kind : eBook
Book Rating : 437/5 ( reviews)

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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. This book was released on 2019-02-19. 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.

Zeta and L-Functions of Varieties and Motives

2020-05-07 Mathematics
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Release : 2020-05-07
Genre : Mathematics
Kind : eBook
Book Rating : 912/5 ( reviews)

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Book Synopsis Zeta and L-Functions of Varieties and Motives by : Bruno Kahn

Download or read book Zeta and L-Functions of Varieties and Motives written by Bruno Kahn. This book was released on 2020-05-07. Available in PDF, EPUB and Kindle. Book excerpt: The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta and L-functions sit at the meeting point of all these theories and have played a profound role in shaping the evolution of number theory. This book presents a big picture of zeta and L-functions and the complex theories surrounding them, combining standard material with results and perspectives that are not made explicit elsewhere in the literature. Particular attention is paid to the development of the ideas surrounding zeta and L-functions, using quotes from original sources and comments throughout the book, pointing the reader towards the relevant history. Based on an advanced course given at Jussieu in 2013, it is an ideal introduction for graduate students and researchers to this fascinating story.

The Local Langlands Conjecture for GL(2)

2006-08-29 Mathematics
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Release : 2006-08-29
Genre : Mathematics
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Book Rating : 11X/5 ( reviews)

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Book Synopsis The Local Langlands Conjecture for GL(2) by : Colin J. Bushnell

Download or read book The Local Langlands Conjecture for GL(2) written by Colin J. Bushnell. This book was released on 2006-08-29. Available in PDF, EPUB and Kindle. Book excerpt: The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

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