Algebraic stacks whose number of points over finite fields is a polynomial

Details Algebraic stacks whose number of points over finite fields is a polynomial Algebraic stacks whose number of points over finite fields is a polynomial

2005-05-10

arxiv.org/abs/math/0505178v3

Number Fields and Function Fields- Two Parallel Worlds, Progress in Mathematics, Vol. 239, Geer, Gerard van der; Moonen, Ben J

Mathematics

Algebraic Geometry

13 pages. Using the first author's results in arXiv:0809.1242, the extra condition that the coarse moduli space is a certain q

Scientific paper

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.

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