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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the semisimplicity of reductions and adelic openness for $ E$-rational compatible systems over global function fields

Authors: Gebhard Böckle, Wojciech Gajda and Sebastian Petersen
Journal: Trans. Amer. Math. Soc. 372 (2019), 5621-5691
MSC (2010): Primary 11F80; Secondary 20G25
Published electronically: March 25, 2019
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Abstract: Let $ X$ be a normal geometrically connected variety over a finite field $ \kappa $ of characteristic $ p$. Let $ (\rho _\lambda \colon \pi _1(X)\to {\rm GL}_n(E_\lambda ) )_\lambda $ be any semisimple $ E$-rational compatible system, where $ E$ is a number field and $ \lambda $ ranges over the finite places of $ E$ not above $ p$. We derive new properties on the monodromy groups of such systems for almost all $ \lambda $ and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense. Key inputs to our results are automorphic methods and the Langlands correspondence over global function fields proved by L. Lafforgue.

To say more, let $ (\overline {\rho }_\lambda \colon \pi _1(X)\to {\rm GL}_n(k_\lambda ) )_\lambda $ be the corresponding mod-$ \lambda $ system, where for every $ \lambda $ by $ {\mathcal O}_\lambda $ and $ k_\lambda $ we denote the valuation ring and the residue field of $ E_\lambda $, and where the reduction is done with respect to some $ \pi _1(X)$-stable $ {\mathcal O}_\lambda $-lattice $ \Lambda _\lambda $ of $ E_\lambda ^n$. Let also $ G_\lambda ^{\mathrm {geo}}$ be the Zariski closure of $ \rho _\lambda (\pi _1(X_{\overline {\kappa }}))$ in $ {\rm GL}_{n, E}$, and let $ \mathcal {G}_\lambda ^{\mathrm {geo}}$ be its schematic closure in $ {\rm Aut}_{{\mathcal O}_\lambda }(\Lambda _\lambda )$. Assume in the following that the algebraic groups $ G_\lambda ^{\mathrm {geo}}$ are connected.

We prove that for almost all $ \lambda $ the group scheme $ \mathcal {G}_\lambda ^{\mathrm {geo}}$ is semisimple over $ {\mathcal O}_\lambda $, and its special fiber agrees with the Nori envelope of $ \overline {\rho }_\lambda (\pi _1(X_{\overline {\kappa }}))$. A comparable result under different hypotheses was proved by A. Cadoret, C.-Y. Hui, and A. Tamagawa using other methods. As an intermediate result, we show for $ X$ a curve that any potentially tame compatible system of mod-$ \lambda $ representations can be lifted to a compatible system over a number field; this implies for almost all $ \lambda $ the semisimplicity of the restriction $ \overline {\rho }_\lambda \vert _{\pi _1(X_{\overline {\kappa }})}$. Finally, we establish adelic openness for $ (\rho _\lambda \vert _{\pi _1(X_{\overline \kappa })} )_\lambda $ in the sense of C. Y. Hui and M. Larsen, for $ E={\mathbb{Q}}$ in general, and for $ E\supsetneq {\mathbb{Q}}$ under additional hypotheses.

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Additional Information

Gebhard Böckle
Affiliation: IWR, University of Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

Wojciech Gajda
Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61614 Poznań, Poland

Sebastian Petersen
Affiliation: Universität Kassel, Fachbereich 10, Wilhelmshöher Allee 71-73, 34121 Kassel, Germany

Keywords: Compatible system of Galois representations, adelic openness, hyperspecial maximal image for almost all $\lambda$, global function field.
Received by editor(s): June 15, 2017
Received by editor(s) in revised form: May 23, 2018, and October 18, 2018
Published electronically: March 25, 2019
Additional Notes: The first author received support from the DFG within the FG1920 and the SPP1489
The second author was partially supported by NCN grant no. UMO-2014/15/B/ST1/00128 and the Alexander von Humboldt Foundation.
Article copyright: © Copyright 2019 American Mathematical Society