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$$L$$-series of Artin stacks over finite fields. (English) Zbl 1329.14042
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 $$L\left( X,K,s\right)$$.

##### 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)
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