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Transactions of the American Mathematical Society

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On the rationality of certain type A Galois representations


Author: Chun Yin Hui
Journal: Trans. Amer. Math. Soc. 370 (2018), 6771-6794
MSC (2010): Primary 11F80, 14F20, 20G30
DOI: https://doi.org/10.1090/tran/7182
Published electronically: April 4, 2018
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Abstract: Let $ X$ be a complete smooth variety defined over a number field $ K$ and let $ i$ be an integer. The absolute Galois group $ \textup {Gal}_K$ of $ K$ acts on the $ i$th étale cohomology group $ H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb{Q}_\ell )$ for all primes $ \ell $, producing a system of $ \ell $-adic representations $ \{\Phi _\ell \}_\ell $. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of $ \Phi _\ell $ admits a reductive $ \mathbb{Q}$-form that is independent of $ \ell $ if $ X$ is projective. Denote by $ \Gamma _\ell $ and $ \mathbf {G}_\ell $ respectively the monodromy group and the algebraic monodromy group of $ \Phi _\ell ^{\textup {ss}}$, the semisimplification of $ \Phi _\ell $. Assuming that $ \mathbf {G}_{\ell _0}$ satisfies some group theoretic conditions for some prime $ \ell _0$, we construct a connected quasi-split $ \mathbb{Q}$-reductive group $ \mathbf {G}_{\mathbb{Q}}$ which is a common $ \mathbb{Q}$-form of $ \mathbf {G}_\ell ^\circ $ for all sufficiently large $ \ell $. Let $ \mathbf {G}_{\mathbb{Q}}^{\textup {sc}}$ be the universal cover of the derived group of $ \mathbf {G}_{\mathbb{Q}}$. As an application, we prove that the monodromy group $ \Gamma _\ell $ is big in the sense that $ \Gamma _\ell ^{\textup {sc}}\cong \mathbf {G}_{\mathbb{Q}}^{\textup {sc}}(\mathbb{Z}_\ell )$ for all sufficiently large $ \ell $.


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Additional Information

Chun Yin Hui
Affiliation: Department of Mathematics, Faculty of Sciences, VU University, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
Address at time of publication: Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, China
Email: pslnfq@tsinghua.edu.cn, pslnfq@gmail.com

DOI: https://doi.org/10.1090/tran/7182
Keywords: Galois representations, the Mumford-Tate conjecture, type A representations
Received by editor(s): May 3, 2016
Received by editor(s) in revised form: January 9, 2017, and January 10, 2017
Published electronically: April 4, 2018
Additional Notes: The present project was supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND)
Article copyright: © Copyright 2018 American Mathematical Society

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