Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 57–81 | Cite as

Companions on Artin stacks

  • Weizhe ZhengEmail author


Deligne’s conjecture that \(\ell \)-adic sheaves on normal schemes over a finite field admit \(\ell '\)-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld’s theorem to smooth Artin stacks and deduce Deligne’s conjecture for coarse moduli spaces of smooth Artin stacks. We also extend related theorems on Frobenius eigenvalues and traces to Artin stacks.

Mathematics Subject Classification

Primary 14F20 Secondary 14G15 14A20 14D22 



This paper grows out of an answer to Shenghao Sun’s question of extending the theorems of Deligne and Drinfeld to stacks. I thank Yongquan Hu, Yifeng Liu, Martin Olsson, and Shenghao Sun for useful discussions, and Vladimir Drinfeld and Luc Illusie for valuable comments. I am grateful to Ofer Gabber for pointing out a mistake in a draft of this paper. I thank the referee for a careful reading of the manuscript and many helpful comments. Part of this paper was written during a stay at Shanghai Center for Mathematical Sciences and I thank the center for hospitality.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of MathematicsAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina
  2. 2.University of the Chinese Academy of SciencesBeijingChina

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