Equivariant Poincaré polynomials and counting points over finite fields
type: Item
instance of: scholarly article
title: Equivariant Poincaré polynomials and counting points over finite fields
zbMATH Open document ID: 1039.14005
published in: Journal of Algebra
publication date: 2002-01-01T00:00:00Z
review text: 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.
reviewed by: Christophe Ritzenthaler
Mathematics Subject Classification ID: 14F43
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 1740512
zbMATH Keywords: \(\ell\)-adic cohomology
zbMATH Keywords: smooth scheme
zbMATH Keywords: action of group
zbMATH Keywords: equivariant comparison
zbMATH Keywords: action on cohomology
zbMATH Keywords: numbers of rational points
zbMATH Keywords: Poincaré polynomials
author: Mark Kisin
author: Gus I. Lehrer
MaRDI profile type: Publication
full work available at URL: jabr.2001.9029
OpenAlex ID: W1974160549
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DOI: 10.1006/JABR.2001.9029
description: scientific article; zbMATH DE number 1740512