\(L\)-series of Artin stacks over finite fields. (English) Zbl 1329.14042

Sun, Shenghao

Algebra Number Theory 6, No. 1, 47-122 (2012).

For an \(\mathbb{F}_{q}\)-algebraic stack \(X\) and a stratifiable Weil complex \( K\) of sheaves on \(X\), one has an \(L\)-series \(L\left( X,K,s\right) \) attached to them, i.e., a formal power series of \(s\) constructed by traces of Frobenius maps. Now consider a morphism \(f:X\rightarrow Y\) of \(\mathbb{F}_{q} \)-algebraic stacks.
In this paper the author proves the main result that the two attached \(L\) -series \(L\left( X,K,s\right) =L\left( Y,f_{!}K,s\right) \) are equal. He also obtains the analytic continuation and infinite product for the \(L\) -series \(L\left( X,K,s\right) \).

Reviewer: Feng-Wen An (Hubei)

MSC:

14F20	Étale and other Grothendieck topologies and (co)homologies
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
19F27	Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)

Keywords:

\(\ell\)-adic cohomology; algebraic stack; operations on stack; \(L\)-series of stack; trace formula