Algebraic Stacks Whose Number of Points over Finite Fields is a Polynomial

Abstract

The aim of this article is to investigate the cohomology (l-adic as well as Betti) of schemes, and more generally of certain algebraic stacks, that are proper and smooth over the integers and have the property that there exists a polynomial P with rational coefficients such that for all prime powers q the number of points over the field with q elements is P(q). We prove that for all prime numbers l the l-adic etale cohomology is a direct sum of Tate twists of the trivial representation. Our main tools here are Behrend's Lefschetz trace formula and l-adic Hodge theory. In the last section we investigate the Hodge structure on the Betti cohomology. The motivation for this article comes from applications to certain moduli stacks of curves.