Nicholas M. Katz's Books

Moments, Monodromy, and Perversity

A Diophantine Perspective

By: Nicholas M. Katz

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Moments, Monodromy, and Perversity

A Diophantine Perspective

By: Nicholas M. Katz

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.

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492

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Princeton University Press

Published Date

ISBN 10

0691123306

ISBN 13

9780691123301

Moments, Monodromy, and Perversity

A Diophantine Perspective

By: Nicholas M. Katz

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Moments, Monodromy, and Perversity

A Diophantine Perspective

By: Nicholas M. Katz

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492

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Princeton University Press

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ISBN 10

0691123306

ISBN 13

9780691123301

Twisted L-Functions and Monodromy

By: Nicholas M. Katz

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Twisted L-Functions and Monodromy

By: Nicholas M. Katz

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.

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258

Publisher

Princeton University Press

Published Date

ISBN 10

1400824885

ISBN 13

9781400824885

Exponential Sums and Differential Equations. (AM-124), Volume 124

By: Nicholas M. Katz

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Exponential Sums and Differential Equations. (AM-124), Volume 124

By: Nicholas M. Katz

This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.

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445

Publisher

Princeton University Press

Published Date

ISBN 10

1400882435

ISBN 13

9781400882434

Random Matrices, Frobenius Eigenvalues, and Monodromy

By: Nicholas M. Katz,Peter Sarnak

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Random Matrices, Frobenius Eigenvalues, and Monodromy

By: Nicholas M. Katz,Peter Sarnak

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.

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Page Count

441

Publisher

American Mathematical Society

Published Date

ISBN 10

1470475073

ISBN 13

9781470475079

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

By: Nicholas M. Katz

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

By: Nicholas M. Katz

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.

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256

Publisher

Princeton University Press

Published Date

ISBN 10

1400882125

ISBN 13

9781400882120

Rigid Local Systems. (AM-139), Volume 139

By: Nicholas M. Katz

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Rigid Local Systems. (AM-139), Volume 139

By: Nicholas M. Katz

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.

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233

Publisher

Princeton University Press

Published Date

ISBN 10

1400882591

ISBN 13

9781400882595

Gauss Sums, Kloosterman Sums, and Monodromy Groups

By: Nicholas M. Katz

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Gauss Sums, Kloosterman Sums, and Monodromy Groups

By: Nicholas M. Katz

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262

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Princeton University Press

Published Date

ISBN 10

0691084335

ISBN 13

9780691084336

Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108

By: Nicholas M. Katz,Barry Mazur

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Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108

By: Nicholas M. Katz,Barry Mazur

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.

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528

Publisher

Princeton University Press

Published Date

ISBN 10

1400881714

ISBN 13

9781400881710

Convolution and Equidistribution

Sato-Tate Theorems for Finite-Field Mellin Transforms

By: Nicholas M. Katz

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Convolution and Equidistribution

Sato-Tate Theorems for Finite-Field Mellin Transforms

By: Nicholas M. Katz

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.

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212

Publisher

Princeton University Press

Published Date

ISBN 10

0691153310

ISBN 13

9780691153315

Rigid Local Systems

By: Nicholas M. Katz

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Rigid Local Systems

By: Nicholas M. Katz

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232

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Princeton University Press

Published Date

ISBN 10

0691011184

ISBN 13

9780691011189

Twisted L-Functions and Monodromy

By: Nicholas M. Katz

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Twisted L-Functions and Monodromy

By: Nicholas M. Katz

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258

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Princeton University Press

Published Date

ISBN 10

1400824885

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9781400824885

Rigid Local Systems. (AM-139), Volume 139

By: Nicholas M. Katz

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Rigid Local Systems. (AM-139), Volume 139

By: Nicholas M. Katz

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233

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Princeton University Press

Published Date

ISBN 10

1400882591

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9781400882595

Convolution and Equidistribution

Sato-Tate Theorems for Finite-Field Mellin Transforms

By: Nicholas M. Katz

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Convolution and Equidistribution

Sato-Tate Theorems for Finite-Field Mellin Transforms

By: Nicholas M. Katz

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212

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Princeton University Press

Published Date

ISBN 10

0691153310

ISBN 13

9780691153315

The Grothendieck Festschrift

By: Pierre Cartierluc Illusie

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The Grothendieck Festschrift

By: Pierre Cartierluc Illusie

This three-volume work contains articles collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990. The articles were offered as a tribute to one of the world's greatest living mathematicians. Many of the groundbreaking contributions in these volumes contain material that is now considered foundational to the subject. Topics addressed by these top-notch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, K-theory, category theo.

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Page Count

572

Published Date

ISBN 10

1283250772

ISBN 13

9781283250771

Book's by Nicholas M. Katz

Moments, Monodromy, and Perversity

Moments, Monodromy, and Perversity

Moments, Monodromy, and Perversity

Moments, Monodromy, and Perversity

Twisted L-Functions and Monodromy

Twisted L-Functions and Monodromy

Exponential Sums and Differential Equations. (AM-124), Volume 124

Exponential Sums and Differential Equations. (AM-124), Volume 124

Random Matrices, Frobenius Eigenvalues, and Monodromy

Random Matrices, Frobenius Eigenvalues, and Monodromy

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

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

Rigid Local Systems. (AM-139), Volume 139

Rigid Local Systems. (AM-139), Volume 139

Gauss Sums, Kloosterman Sums, and Monodromy Groups

Gauss Sums, Kloosterman Sums, and Monodromy Groups

Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108

Arithmetic Moduli of Elliptic Curves. (AM-108), Volume 108

Convolution and Equidistribution

Convolution and Equidistribution

Rigid Local Systems

Rigid Local Systems

Twisted L-Functions and Monodromy

Twisted L-Functions and Monodromy

Rigid Local Systems. (AM-139), Volume 139

Rigid Local Systems. (AM-139), Volume 139

Convolution and Equidistribution

Convolution and Equidistribution

The Grothendieck Festschrift

The Grothendieck Festschrift

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