Equivariant Poincaré polynomials and counting points over finite fields (Q1348675)

Let \(l\) be a prime and fix an isomorphism \(\overline{\mathbb{Q}}_l \rightarrow \mathbb{C}\). In a first part, the authors prove the following equivariant comparison theorem: Let \(\mathcal{O}\) be the ring of integers of a number field \(K\) and let \(n\) be a positive integer divisible by \(l\). Assume that \(\overline{X}\) is a smooth, proper scheme over \(\mathcal{O}[1/n]\) and that \(X \subset \overline{X}\) is a dense open subscheme, such that \(\overline{X} \setminus X\) is a normal crossings divisor relative to \(\mathcal{O}[1/n]\). Fix a prime \(\mathfrak{p}\) of \(\mathcal{O}[1/n]\) and an embedding \(\mathcal{O}[1/n] \hookrightarrow \mathbb{C}\). If \(k(\mathfrak {p})\) denotes the residue field of \(\mathfrak{p}\) and \(\overline{k (\mathfrak{p})}\) denotes an algebraic closure of \(k(\mathfrak{p})\), then there are canonical isomorphisms \[ H^i(X_{\overline {k({\mathfrak p})}},\overline{\mathbb{Q}}_l) \rightarrow H^i(X_{\mathbb{C}}^{\text{an}},\mathbb{C}) \] and \[ H^i_c(X_{\overline {k({\mathfrak p})}}, \overline {\mathbb{Q}}_l) \rightarrow H^i_c(X_{\mathbb{C}}^{\text{an}},\mathbb{C}) \] where \(X_{\mathbb{C}}^{\text{an}}\) is the complex analytic space attached to \(X_{\mathbb{C}}\). In particular, if a group \(G\) acts on \(X\), then these isomorphisms are \(G\)-equivariant, without assuming that the action of \(G\) extends to \(\overline{X}\). Then, the authors show how this can 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 of 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.