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The abelian part of a compatible system and \(\ell \)-independence of the Tate conjecture

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Abstract

Let K be a number field and \(\{V_\ell \}_\ell \) a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let \(\mathbf {G}_\ell \) and \(V_\ell ^{{\text {ab}}}\) be respectively the algebraic monodromy group and the maximal abelian subrepresentation of \(V_\ell \) for all \(\ell \). We prove that the system \(\{V_\ell ^{{\text {ab}}}\}_\ell \) is also a rational strictly compatible system under some group theoretic conditions, e.g., when \(\mathbf {G}_{\ell '}\) is connected and satisfies Hypothesis A for some prime \(\ell '\). As an application, we prove that the Tate conjecture for abelian variety X/K is independent of \(\ell \) if the algebraic monodromy groups of the Galois representations of X satisfy the required conditions.

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Acknowledgements

I would like to thank Gabor Wiese for his interests and comments on the paper. I would like to thank the referee for his/her constructive comments.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084, China

    Chun Yin Hui

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  1. Chun Yin Hui
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Correspondence to Chun Yin Hui.

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The present project was supported partly by the National Research Fund, Luxembourg, cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND) and partly by China’s Thousand Talents Plan: The Recruitment Program for Young Professionals.

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Hui, C.Y. The abelian part of a compatible system and \(\ell \)-independence of the Tate conjecture. manuscripta math. 161, 223–246 (2020). https://doi.org/10.1007/s00229-018-1068-2

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