Etale Cohomology and the Weil Conjecture

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Etale Cohomology and the Weil Conjecture

Etale Cohomology and the Weil Conjecture

  • Author: Eberhard Freitag,Reinhardt Kiehl
  • Publisher: Springer Science & Business Media
  • ISBN: 3662025418
  • Category: Mathematics
  • Page: 320
  • View: 6143
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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.

The Weil Conjectures

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The Weil Conjectures

The Weil Conjectures

On Math and the Pursuit of the Unknown

  • Author: Karen Olsson
  • Publisher: Farrar, Straus and Giroux
  • ISBN: 0374719632
  • Category: Biography & Autobiography
  • Page: 224
  • View: 2625
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“The Weil Conjectures is a charming meditation on geometry, sacrifice, and adolescent self-discovery, delivered in passionate, impressionistic bursts.” —Jordan Ellenberg, New York Times-bestselling author of How Not to Be Wrong An eloquent blend of memoir and biography exploring the Weil siblings, math, and creative inspiration Karen Olsson’s stirring and unusual third book, The Weil Conjectures, tells the story of the brilliant Weil siblings—Simone, a philosopher, mystic, and social activist, and André, an influential mathematician—while also recalling the years Olsson spent studying math. As she delves into the lives of these two singular French thinkers, she grapples with their intellectual obsessions and rekindles one of her own. For Olsson, as a math major in college and a writer now, it’s the odd detours that lead to discovery, to moments of insight. Thus The Weil Conjectures—an elegant blend of biography and memoir and a meditation on the creative life. Personal, revealing, and approachable, The Weil Conjectures eloquently explores math as it relates to intellectual history, and shows how sometimes the most inexplicable pursuits turn out to be the most rewarding.

Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform

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Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform

Weil Conjectures, Perverse Sheaves and l’adic Fourier Transform

  • Author: Reinhardt Kiehl,Rainer Weissauer
  • Publisher: Springer Science & Business Media
  • ISBN: 3662045761
  • Category: Mathematics
  • Page: 375
  • View: 5113
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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.

Algebraic Geometry

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Algebraic Geometry

Algebraic Geometry

  • Author: Robin Hartshorne
  • Publisher: Springer Science & Business Media
  • ISBN: 1475738498
  • Category: Mathematics
  • Page: 496
  • View: 4411
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An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.

In Search of the Riemann Zeros

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In Search of the Riemann Zeros

In Search of the Riemann Zeros

Strings, Fractal Membranes and Noncommutative Spacetimes

  • Author: Michel Laurent Lapidus
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821842225
  • Category: Mathematics
  • Page: 558
  • View: 7152
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Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line. In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated moduli space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true. Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical or theoretical physics.

Arithmetic of Finite Fields

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Arithmetic of Finite Fields

Arithmetic of Finite Fields

  • Author: Charles Small
  • Publisher: CRC Press
  • ISBN: 9780824785260
  • Category: Mathematics
  • Page: 240
  • View: 2811
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Text for a one-semester course at the advanced undergraduate/beginning graduate level, or reference for algebraists and mathematicians interested in algebra, algebraic geometry, and number theory, examines counting or estimating numbers of solutions of equations in finite fields concentrating on top

Gauss Sums, Kloosterman Sums, and Monodromy Groups

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Gauss Sums, Kloosterman Sums, and Monodromy Groups

Gauss Sums, Kloosterman Sums, and Monodromy Groups

  • Author: Nicholas M. Katz
  • Publisher: Princeton University Press
  • ISBN: 9780691084336
  • Category: Mathematics
  • Page: 246
  • View: 5703
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The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums.

Selected Papers on Number Theory, Algebraic Geometry, and Differential Geometry

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Selected Papers on Number Theory, Algebraic Geometry, and Differential Geometry

Selected Papers on Number Theory, Algebraic Geometry, and Differential Geometry

  • Author: Katsumi Nomizu
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821875117
  • Category: Mathematics
  • Page: 154
  • View: 7099
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This book presents papers that originally appeared in the Japanese journal Sugaku. The papers explore the relationship between number theory, algebraic geometry, and differential geometry.