Independence of ℓ-adic representations of geometric Galois groups

Gebhard Böckle; Wojciech Gajda; Sebastian Petersen

doi:10.1515/crelle-2015-0024

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Independence of ℓ-adic representations of geometric Galois groups

Gebhard Böckle , Wojciech Gajda und Sebastian Petersen

Veröffentlicht/Copyright: 8. Juli 2015

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2018 Heft 736

Abstract

Let k be an algebraically closed field of arbitrary characteristic, let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ℓ, the absolute Galois group of K acts on the ℓ-adic étale cohomology modules of X. We prove that this family of representations varying over ℓ is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ℓ become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: FG 1920

Award Identifier / Grant number: SPP 1489

Funding statement: Gebhard Böckle is supported by the DFG in the FG 1920 and by the DFG/FNR within the SPP 1489. Wojciech Gajda was supported by the Alexander von Humboldt Foundation and by the National Centre of Sciences of Poland under research grant UMO-2012/07/B/ST1/03541.

Acknowledgements

Gebhard Böckle thanks the Fields Institute for a research stay in the spring of 2012 during which part of this work was written. He also thanks Adam Mickiewicz University in Poznań for making possible a joint visit of the three authors in the fall of 2012. Wojciech Gajda thanks the Interdisciplinary Center for Scientific Computing (IWR) at Heidelberg University for hospitality during research visits in January 2012 and in January 2014. Sebastian Petersen thanks the Mathematics Department at Adam Mickiewicz University for hospitality and support during several research visits. We thank F. Orgogozo and L. Illusie for interesting correspondence concerning this project. In addition, the authors thank the anonymous referee for a thorough review of the paper and many helpful comments and suggestions, and in particular for pointing us to [Modular functions of one variable. II (Antwerp 1972), Lecture Notes in Math. 349, Springer, Berlin (1973), 501–597, Théorème 9.8] that replaced earlier arguments involving the global Langlands correspondence proven by L. Lafforgue.

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Received: 2013-10-15

Revised: 2015-2-16

Published Online: 2015-7-8

Published in Print: 2018-3-1

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https://doi.org/10.1515/crelle-2015-0024