


Moments, Monodromy, and Perversity. (AM159): 
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. Introduction 1 Other Princeton books authored or coauthored by Nicholas M. Katz:
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