Equivariant Poincaré Polynomials and Counting Points over Finite Fields

Abstract

Suppose a finite group acts as a group of automorphisms of a smooth complex algebraic variety which is defined over a number field. We show how, in certain circumstances, an equivariant comparison theorem in l-adic cohomology may be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields. This is carried through in three applications: first, for the symmetric group acting on the moduli space of n points on a genus zero curve; second, for a unitary reflection group acting on the complement of its reflecting hyperplanes; and third, for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions.