L-series of Artin stacks over finite fields

TL;DR

This paper extends the Lefschetz trace formula to Artin stacks over finite fields, establishes the meromorphic continuation of their L-series, and provides bounds and independence results for their cohomology.

Contribution

It reestablishes the trace formula for stacks, proves meromorphic continuation of L-series, and introduces bounds and l-independence results for cohomology groups.

Findings

Lefschetz trace formula is valid for Artin stacks.

L-series of Artin stacks have meromorphic continuation.

Cohomology groups of stacks have bounded weights and are independent of l.

Abstract

We reprove the Lefschetz trace formula for stacks (in the context of derived categories and the six operations for stacks developed by Laszlo and Olsson), and give the meromorphic continuation of L-series (in particular, zeta functions) of Artin stacks over a finite field. We also give an upper bound for the weights of the cohomology groups of stacks, and an "independence of l" result for a certain class of quotient stacks.