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Other titles in the Annals of Mathematics Studies series:
Moments, Monodromy, and Perversity: A Diophantine Perspective (Annals of Mathematics Studies)by Nicholas M. Katz
Synopses & ReviewsPublisher 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" algebrogeometric families of character sums over finite fields (and of their associated Lfunctions). Roughly speaking, Deligne showed that any such family obeys a "generalized SatoTate law," and that figuring out which generalized SatoTate 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 (Lfunctions 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" algebrogeometric families of character sums over finite fields (and of their associated Lfunctions). Roughly speaking, Deligne showed that any such family obeys a "generalized SatoTate law," and that figuring out which generalized SatoTate 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 (Lfunctions 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" algebrogeometric families of character sums over finite fields (and of their associated Lfunctions). Roughly speaking, Deligne showed that any such family obeys a "generalized SatoTate law," and that figuring out which generalized SatoTate 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 (Lfunctions attached to) character sums over finite fields. About the AuthorNicholas 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 LFunctions and Monodromy".
Table of ContentsIntroduction 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 A^{n} 111 Chapter 4: Additive character sums on more general X 161 Chapter 5: Multiplicative character sums on A^{n} 185 Chapter 6: Middle addivitve convolution 221 Appendix A6: Swanminimal poles 281 Chapter 7: Pullbacks to curves from A^{1} 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 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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