By Nicholas M. Katz
It is now a few thirty years considering Deligne first proved his normal equidistribution theorem, hence constructing the basic consequence governing the statistical houses of definitely "pure" algebro-geometric households of personality sums over finite fields (and in their linked L-functions). approximately conversing, Deligne confirmed that the sort of kinfolk obeys a "generalized Sato-Tate law," and that realizing which generalized Sato-Tate legislations applies to a given relations quantities basically to computing a undeniable complicated semisimple (not unavoidably hooked up) algebraic team, the "geometric monodromy team" connected to that kinfolk.
Up to now, approximately all options for deciding upon geometric monodromy teams have relied, at the least partly, on neighborhood details. In Moments, Monodromy, and Perversity, Nicholas Katz develops new strategies, that are resolutely worldwide in nature. they're in accordance with important components, neither of which existed on the time of Deligne's unique paintings at the topic. the 1st is the idea of perverse sheaves, pioneered by way of Goresky and MacPherson within the topological environment after which brilliantly transposed to algebraic geometry through Beilinson, Bernstein, Deligne, and Gabber. the second one is Larsen's replacement, which practically characterizes classical teams via their fourth moments. those new thoughts, that are of significant curiosity of their personal correct, are first constructed after which used to calculate the geometric monodromy teams connected to a few fairly particular common households of (L-functions connected to) personality sums over finite fields.