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Other titles in the Annals of Mathematics Studies series:

Moments, Monodromy, and Perversity: A Diophantine Perspective (Annals of Mathematics Studies)

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Synopses & Reviews

Publisher Comments:

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.

Synopsis:

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.

Synopsis:

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.

About the Author

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".

Table of Contents

Introduction 1

Chapter 1: Basic results on perversity and higher moments 9

Chapter 2: How to apply the results of Chapter 2 93

Chapter 3: Additive character sums on An 111

Chapter 4: Additive character sums on more general X 161

Chapter 5: Multiplicative character sums on An 185

Chapter 6: Middle addivitve convolution 221

Appendix A6: Swan-minimal poles 281

Chapter 7: Pullbacks to curves from A1 295

Chapter 8: One variable twists on curves 321

Chapter 9: Weierstrass sheaves as inputs 327

Chapter 10: Weirstrass families 349

Chapter 11: FJTwist families and variants 371

Chapter 12: Uniformity results 407

Chapter 13: Average analytic rank and large N limits 443

References 455

Notation Index 461

Subject Index 467

Product Details

ISBN:
9780691123301
Editor:
Griffiths, Phillip A.
Editor:
Mather, John N.
Editor:
Griffiths, Phillip A.
Editor:
Mather, John N.
Author:
Katz, Nicholas M.
Publisher:
Princeton University Press
Location:
Princeton
Subject:
Number Theory
Subject:
Sheaf theory.
Subject:
L-functions.
Subject:
General Mathematics
Subject:
Mathematics
Subject:
Mathematics-Number Theory
Copyright:
Edition Description:
Trade paper
Series:
Annals of Mathematics Studies
Publication Date:
September 2005
Binding:
TRADE PAPER
Grade Level:
College/higher education:
Language:
English
Pages:
488
Dimensions:
10 x 7 in 29 oz

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Related Subjects

Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Number Theory
Science and Mathematics » Physics » General

Moments, Monodromy, and Perversity: A Diophantine Perspective (Annals of Mathematics Studies) New Trade Paper
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Product details 488 pages Princeton University Press - English 9780691123301 Reviews:
"Synopsis" by , 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.

"Synopsis" by , 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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