Gauss Sums Kloosterman Sums And Monodromy Groups

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Gauss Sums, Kloosterman Sums, and Monodromy Groups

Nicholas M. Katz
Gauss Sums Kloosterman Sums and Monodromy Groups Book PDF

Author : Nicholas M. Katz
Publisher : Princeton University Press
Page : 262 pages
File Size : 44,6 Mb
Release : 1988
Category : Mathematics
ISBN : 0691084335

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Gauss Sums, Kloosterman Sums, and Monodromy Groups by Nicholas M. Katz Pdf

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.

Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116), Volume 116

Nicholas M. Katz
Gauss Sums Kloosterman Sums and Monodromy Groups AM 116 Volume 116 Book PDF

Author : Nicholas M. Katz
Publisher : Princeton University Press
Page : 256 pages
File Size : 46,8 Mb
Release : 2016-03-02
Category : Mathematics
ISBN : 9781400882120

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Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116), Volume 116 by Nicholas M. Katz Pdf

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.

Twisted L-Functions and Monodromy. (AM-150), Volume 150

Nicholas M. Katz
Twisted L Functions and Monodromy AM 150 Volume 150 Book PDF

Author : Nicholas M. Katz
Publisher : Princeton University Press
Page : 264 pages
File Size : 53,7 Mb
Release : 2009-01-10
Category : Mathematics
ISBN : 9781400824885

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Twisted L-Functions and Monodromy. (AM-150), Volume 150 by Nicholas M. Katz Pdf

For hundreds of years, the study of elliptic curves has played a central role in mathematics. The past century in particular has seen huge progress in this study, from Mordell's theorem in 1922 to the work of Wiles and Taylor-Wiles in 1994. Nonetheless, there remain many fundamental questions where we do not even know what sort of answers to expect. This book explores two of them: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? Nicholas Katz answers these questions for families of ''big'' twists of elliptic curves in the function field case (with a growing constant field). The monodromy-theoretic methods he develops turn out to apply, still in the function field case, equally well to families of big twists of objects of all sorts, not just to elliptic curves. The leisurely, lucid introduction gives the reader a clear picture of what is known and what is unknown at present, and situates the problems solved in this book within the broader context of the overall study of elliptic curves. The book's technical core makes use of, and explains, various advanced topics ranging from recent results in finite group theory to the machinery of l-adic cohomology and monodromy. Twisted L-Functions and Monodromy is essential reading for anyone interested in number theory and algebraic geometry.

Featured Reviews in Mathematical Reviews 1997-1999

Donald G. Babbitt,Jane E. Kister
Featured Reviews in Mathematical Reviews 1997 1999 Book PDF

Author : Donald G. Babbitt,Jane E. Kister
Publisher : American Mathematical Soc.
Page : 762 pages
File Size : 51,6 Mb
Release : 2000-05-05
Category : Mathematics
ISBN : 0821896709

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Featured Reviews in Mathematical Reviews 1997-1999 by Donald G. Babbitt,Jane E. Kister Pdf

This second volume of Featured Reviews makes available special detailed reviews of some of the most important mathematical articles and books published from 1997 through 1999. Also included are excellent reviews of several classic books and articles published prior to 1970. Among those reviews, for example, are the following: Homological Algebra by Henri Cartan and Samuel Eilenberg, reviewed by G. Hochschild; Faisceaux algebriques coherents by Jean-Pierre Serre, reviewed by C. Chevalley; and On the Theory of General Partial Differential Operators by Lars Hormander, reviewed by J. L. Lions. In particular, those seeking information on current developments outside their own area of expertise will find the volume very useful. By identifying some of the best publications, papers, and books that have had or are expected to have a significant impact in applied and pure mathematics, this volume will serve as a comprehensive guide to important new research across all fields covered by MR.

Moments, Monodromy, and Perversity. (AM-159)

Nicholas M. Katz
Moments Monodromy and Perversity AM 159 Book PDF

Author : Nicholas M. Katz
Publisher : Princeton University Press
Page : 484 pages
File Size : 40,9 Mb
Release : 2005-09-12
Category : Mathematics
ISBN : 9781400826957

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Moments, Monodromy, and Perversity. (AM-159) by Nicholas M. Katz Pdf

It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions). Roughly speaking, Deligne showed that any such family obeys a "generalized Sato-Tate law," and that figuring out which generalized Sato-Tate law applies to a given family amounts essentially to computing a certain complex semisimple (not necessarily connected) algebraic group, the "geometric monodromy group" attached to that family. Up to now, nearly all techniques for determining geometric monodromy groups have relied, at least in part, on local information. In Moments, Monodromy, and Perversity, Nicholas Katz develops new techniques, which are resolutely global in nature. They are based on two vital ingredients, neither of which existed at the time of Deligne's original work on the subject. The first is the theory of perverse sheaves, pioneered by Goresky and MacPherson in the topological setting and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber. The second is Larsen's Alternative, which very nearly characterizes classical groups by their fourth moments. These new techniques, which are of great interest in their own right, are first developed and then used to calculate the geometric monodromy groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields.

Random Matrices, Frobenius Eigenvalues, and Monodromy

Nicholas M. Katz,Peter Sarnak
Random Matrices Frobenius Eigenvalues and Monodromy Book PDF

Author : Nicholas M. Katz,Peter Sarnak
Publisher : American Mathematical Society
Page : 441 pages
File Size : 46,7 Mb
Release : 2023-11-13
Category : Mathematics
ISBN : 9781470475079

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Random Matrices, Frobenius Eigenvalues, and Monodromy by Nicholas M. Katz,Peter Sarnak Pdf

The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.

Analysis at Large

Artur Avila,Michael Th. Rassias,Yakov Sinai
Analysis at Large Book PDF

Author : Artur Avila,Michael Th. Rassias,Yakov Sinai
Publisher : Springer Nature
Page : 388 pages
File Size : 50,9 Mb
Release : 2022-11-01
Category : Mathematics
ISBN : 9783031053313

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Analysis at Large by Artur Avila,Michael Th. Rassias,Yakov Sinai Pdf

​Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples of topics discussed in this book include Bourgain’s discretized sum-product theorem, his work in nonlinear dispersive equations, the slicing problem by Bourgain, harmonious sets, the joint spectral radius, equidistribution of affine random walks, Cartan covers and doubling Bernstein type inequalities, a weighted Prékopa-Leindler inequality and sumsets with quasicubes, the fractal uncertainty principle for the Walsh-Fourier transform, the continuous formulation of shallow neural networks as Wasserstein-type gradient flows, logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrödinger operators, polynomial equations in subgroups, trace sets of restricted continued fraction semigroups, exponential sums, twisted multiplicativity and moments, the ternary Goldbach problem, as well as the multiplicative group generated by two primes in Z/QZ. It is hoped that this volume will inspire further research in the areas of analysis treated in this book and also provide direction and guidance for upcoming developments in this essential subject of mathematics.

Handbook of Number Theory I

József Sándor,Dragoslav S. Mitrinovic,Borislav Crstici
Handbook of Number Theory I Book PDF

Author : József Sándor,Dragoslav S. Mitrinovic,Borislav Crstici
Publisher : Springer Science & Business Media
Page : 638 pages
File Size : 44,7 Mb
Release : 2005-11-17
Category : Mathematics
ISBN : 9781402042157

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Handbook of Number Theory I by József Sándor,Dragoslav S. Mitrinovic,Borislav Crstici Pdf

This handbook covers a wealth of topics from number theory, special attention being given to estimates and inequalities. As a rule, the most important results are presented, together with their refinements, extensions or generalisations. These may be applied to other aspects of number theory, or to a wide range of mathematical disciplines. Cross-references provide new insight into fundamental research. Audience: This is an indispensable reference work for specialists in number theory and other mathematicians who need access to some of these results in their own fields of research.

$p$-Adic Methods in Number Theory and Algebraic Geometry

Alan Adolphson,Steven Sperber
p Adic Methods in Number Theory and Algebraic Geometry Book PDF

Author : Alan Adolphson,Steven Sperber
Publisher : American Mathematical Soc.
Page : 241 pages
File Size : 53,5 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821851456

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$p$-Adic Methods in Number Theory and Algebraic Geometry by Alan Adolphson,Steven Sperber Pdf

Two meetings of the AMS in the fall of 1989--one at the Stevens Institute of Technology and the other at Ball State University--included Special Sessions on the role of $p$-adic methods in number theory and algebraic geometry. This volume grew out of these Special Sessions. Drawn from a wide area of mathematics, the articles presented here provide an excellent sampling of the broad range of trends and applications in $p$-adic methods.

Zeta Functions in Algebra and Geometry

Antonio Campillo
Zeta Functions in Algebra and Geometry Book PDF

Author : Antonio Campillo
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 55,7 Mb
Release : 2012
Category : Algebraic varieties
ISBN : 9780821869000

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Zeta Functions in Algebra and Geometry by Antonio Campillo Pdf

The volume contains the proceedings of the ``Second International Workshop on Zeta Functions in Algebra and Geometry'' held May 3-7, 2010 at the Universitat de les Illes Balears, Palma de Mallorca, Spain. Zeta functions can be naturally attached to several mathematical objects, including fields, groups, and algebras. The conference focused on the following topics: arithmetic and geometric aspects of local, topological, and motivic zeta functions, Poincare series of valuations, zeta functions of groups, rings, and representations, prehomogeneous vector spaces and their zeta functions, and height zeta functions. This book is published in cooperation with Real Sociedad Matematica Espanola (RSME).

Surveys in Number Theory

Bruce Berndt,M.A. Bennett,N. Boston,H.G. Diamond,A.J. Hildebrand,W. Philipp
Surveys in Number Theory Book PDF

Author : Bruce Berndt,M.A. Bennett,N. Boston,H.G. Diamond,A.J. Hildebrand,W. Philipp
Publisher : CRC Press
Page : 368 pages
File Size : 49,6 Mb
Release : 2002-11-20
Category : Mathematics
ISBN : 9781000065282

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Surveys in Number Theory by Bruce Berndt,M.A. Bennett,N. Boston,H.G. Diamond,A.J. Hildebrand,W. Philipp Pdf

This volume, based on fourteen papers from the Millennial Conference on Number Theory, represents surveys of topics in number theory and provides an outlook into the future of number theory research. It serves as an inspiration to graduate students and as a reference for research mathematicians.

Handbook of Finite Fields

Gary L. Mullen,Daniel Panario
Handbook of Finite Fields Book PDF

Author : Gary L. Mullen,Daniel Panario
Publisher : CRC Press
Page : 1048 pages
File Size : 44,9 Mb
Release : 2013-06-17
Category : Computers
ISBN : 9781439873823

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Handbook of Finite Fields by Gary L. Mullen,Daniel Panario Pdf

Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and

Convolution and Equidistribution

Nicholas M. Katz
Convolution and Equidistribution Book PDF

Author : Nicholas M. Katz
Publisher : Princeton University Press
Page : 213 pages
File Size : 53,9 Mb
Release : 2012-01-24
Category : Mathematics
ISBN : 9781400842704

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Convolution and Equidistribution by Nicholas M. Katz Pdf

Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.

Many Rational Points

N.E. Hurt
Many Rational Points Book PDF

Author : N.E. Hurt
Publisher : Springer Science & Business Media
Page : 368 pages
File Size : 49,7 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9789401702515

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Many Rational Points by N.E. Hurt Pdf

This volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators." "The book will be of use to mathematicians, physicists and engineers interested in the mathematical methods of algebraic geometry as they apply to coding theory and cryptography."--Jacket

Colloquium De Giorgi 2013 and 2014

Umberto Zannier
Colloquium De Giorgi 2013 and 2014 Book PDF

Author : Umberto Zannier
Publisher : Springer
Page : 142 pages
File Size : 50,9 Mb
Release : 2015-03-19
Category : Mathematics
ISBN : 9788876425158

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Colloquium De Giorgi 2013 and 2014 by Umberto Zannier Pdf

​Since 2001 the Scuola Normale Superiore di Pisa has organized the "Colloquio De Giorgi", a series of colloquium talks named after Ennio De Giorgi. The Colloquio is addressed to a general mathematical audience, and especially meant to attract graduate students and advanced undergraduate students. The lectures are intended to be not too technical, in fields of wide interest. They must provide an overview of the general topic, possibly in a historical perspective, together with a description of more recent progress. The idea of collecting the materials from these lectures and publishing them in annual volumes came out recently, as a recognition of their intrinsic mathematical interest, and also with the aim of preserving memory of these events.