By Nicholas M. Katz
It is now a few thirty years in view that Deligne first proved his common equidistribution theorem, therefore setting up the basic outcome governing the statistical homes of certainly "pure" algebro-geometric households of personality sums over finite fields (and in their linked L-functions). approximately conversing, Deligne confirmed that this sort of relatives obeys a "generalized Sato-Tate law," and that knowing which generalized Sato-Tate legislation applies to a given relatives quantities primarily to computing a definite complicated semisimple (not inevitably hooked up) algebraic workforce, the "geometric monodromy staff" connected to that relatives.
Up to now, approximately all options for picking out geometric monodromy teams have relied, no less than partially, on neighborhood info. In Moments, Monodromy, and Perversity, Nicholas Katz develops new innovations, that are resolutely worldwide in nature. they're in line with important elements, neither of which existed on the time of Deligne's unique paintings at the topic. the 1st is the speculation of perverse sheaves, pioneered via Goresky and MacPherson within the topological surroundings after which brilliantly transposed to algebraic geometry by way of Beilinson, Bernstein, Deligne, and Gabber. the second one is Larsen's replacement, which practically characterizes classical teams through their fourth moments. those new recommendations, that are of significant curiosity of their personal correct, are first built after which used to calculate the geometric monodromy teams connected to a few particularly particular common households of (L-functions hooked up to) personality sums over finite fields.
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