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

• Superior document: Title is part of eBook package: De Gruyter Princeton Annals of Mathematics eBook-Package 1940-2020
• VerfasserIn: Katz, Nicholas M.,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2005]; ©2006
• Year of Publication: 2005
• Language: English
• Series: Annals of Mathematics Studies ; 159
• Physical Description: 1 online resource (488 p.)

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

Frontmatter -- Contents -- Introduction -- Chapter 1: Basic results on perversity and higher moments -- Chapter 2: How to apply the results of Chapter 1 -- Chapter 3: Additive character sums on An -- Chapter 4: Additive character sums on more general X -- Chapter 5: Multiplicative character sums on An -- Chapter 6: Middle additive convolution -- Appendix A6: Swan-minimal poles -- Chapter 7: Pullbacks to curves from A1 -- Chapter 8: One variable twists on curves -- Chapter 9: Weierstrass sheaves as inputs -- Chapter 10: Weierstrass families -- Chapter 11: FJTwist families and variants -- Chapter 12: Uniformity results -- Chapter 13: Average analytic rank and large N limits -- References -- Notation Index -- Subject Index