Moments, Monodromy and Perversity

A Diophantine Perspective

Nicholas M. Katz - Collection Annals of Mathematic Studies

480 pages, parution le 04/10/2005

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Résumé

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.

L'auteur - 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.

Sommaire

Basic results on perversity and higher moments
How to apply the results of Chapter 2
Additive character sums on An
Additive character sums on more general X
Multiplicative character sums on An
Middle addivitve convolution
Appendix A6: Swan-minimal poles
Pullbacks to curves from A1
One variable twists on curves
Weierstrass sheaves as inputs
Weirstrass families
FJTwist families and variants
Uniformity results
Average analytic rank and large N limits

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	PAPIER
Éditeur(s)	Princeton University Press
Auteur(s)	Nicholas M. Katz
Collection	Annals of Mathematic Studies
Parution	04/10/2005
Nb. de pages	480
Format	18 x 25
Couverture	Broché
Poids	825g
Intérieur	Noir et Blanc
EAN13	9780691123301
ISBN13	978-0-691-12330-1