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.
