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

Paperback | 2005 | $120.00 | £88.95 | ISBN: 9780691123301
488 pp. | 7 x 10
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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.

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

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