Moments, Monodromy, and Perversity. (AM-159): A Diophantine Perspective. (AM-159)

Nicholas M. Katz

Published by Princeton University Press 2006

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

A Diophantine Perspective. (AM-159)

Nicholas M. Katz

Volume 159 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400826957

Overview
Contents

About this book

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.

Author / Editor information

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.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 1: Basic results on perversity and higher moments Requires Authentication Unlicensed Licensed Download PDF	9
Chapter 2: How to apply the results of Chapter 1 Requires Authentication Unlicensed Licensed Download PDF	93
Chapter 3: Additive character sums on Aⁿ Requires Authentication Unlicensed Licensed Download PDF	111
Chapter 4: Additive character sums on more general X Requires Authentication Unlicensed Licensed Download PDF	161
Chapter 5: Multiplicative character sums on Aⁿ Requires Authentication Unlicensed Licensed Download PDF	185
Chapter 6: Middle additive convolution Requires Authentication Unlicensed Licensed Download PDF	221
Appendix A6: Swan-minimal poles Requires Authentication Unlicensed Licensed Download PDF	281
Chapter 7: Pullbacks to curves from A¹ Requires Authentication Unlicensed Licensed Download PDF	295
Chapter 8: One variable twists on curves Requires Authentication Unlicensed Licensed Download PDF	321
Chapter 9: Weierstrass sheaves as inputs Requires Authentication Unlicensed Licensed Download PDF	327
Chapter 10: Weierstrass families Requires Authentication Unlicensed Licensed Download PDF	349
Chapter 11: FJTwist families and variants Requires Authentication Unlicensed Licensed Download PDF	371
Chapter 12: Uniformity results Requires Authentication Unlicensed Licensed Download PDF	407
Chapter 13: Average analytic rank and large N limits Requires Authentication Unlicensed Licensed Download PDF	443
References Requires Authentication Unlicensed Licensed Download PDF	455
Notation Index Requires Authentication Unlicensed Licensed Download PDF	461
Subject Index Requires Authentication Unlicensed Licensed Download PDF	467