Moments, Monodromy, and Perversity by Nicholas M. Katz
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Moments, Monodromy, and Perversity
By Nicholas M. Katz

Moments, Monodromy, and Perversity

A Diophantine Perspective

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Format: Paperback

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Moments, Monodromy, and Perversity by Nicholas M. Katz

Book Description

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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Book Details

ISBN: 9780691123301
ISBN-10: 0691123306
Format: Paperback
(254mm x 178mm x 16mm)
Pages: 488
Imprint: Princeton University Press
Publisher: Princeton University Press
Publish Date: 12-Sep-2005
Country of Publication: United States

Books By Author Nicholas M. Katz

Convolution and Equidistribution by Nicholas M. Katz Convolution and Equidistribution, Paperback (January 2012)

Explores an important aspect of number theory - the theory of exponential sums over finite fields and their Mellin transforms - from a categorical point of view. This book presents fundamentally important results and a plethora of examples, opening up new directions in the subject.

Twisted L-Functions and Monodromy by Nicholas M. Katz Twisted L-Functions and Monodromy, Paperback (February 2002)

For hundreds of years, the study of elliptic curves has played a central role in mathematics. This book explores: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? It is suitable for those interested in number theory and algebraic geometry.

Rigid Local Systems by Nicholas M. Katz Rigid Local Systems, Paperback (December 1995)

The author 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.

Exponential Sums and Differential Equations by Nicholas M. Katz Exponential Sums and Differential Equations, Paperback (September 1990)

Deals with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. This 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).

» View all books by Nicholas M. Katz

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Author Biography - Nicholas M. Katz

Nicholas M. Katz is Professor of Mathematics at Princeton University. He is the author of five previous books in this series: "Arithmetic Moduli of Elliptic Curves" (with Barry Mazur); "Gauss Sums, Kloosterman Sums, and Monodromy Groups"; "Exponential Sums and Differential Equations"; "Rigid Local Systems"; and "Twisted L-Functions and Monodromy".

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Recent books by Nicholas M. Katz close
Convolution and Equidistribution by Nicholas M. Katz
Twisted L-Functions and Monodromy by Nicholas M. Katz
Rigid Local Systems by Nicholas M. Katz
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Gauss Sums, Kloosterman Sums, and Monodromy Groups by Nicholas M. Katz
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»
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