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Type A images of Galois representations and maximality

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Abstract

Given a compatible subsystem \(\{\rho _\ell \}_\ell \) of n-dimensional \(\ell \)-adic Galois representations arising from étale cohomology of any complete, non-singular variety over a number field K, we define \(\Gamma _\ell := \rho _\ell ({\text {Gal}}(\bar{K}/K))\) and let \(\mathbf {G}_\ell \) denote the Zariski closure of \(\Gamma _\ell \) in \(\mathrm {GL}_n\). If \(\mathbf {G}_\ell \) is of Type A in the sense that all simple composition factors are of type A in the Cartan-Killing classification, then \(\Gamma _\ell \) is, in a suitable sense, maximal in \(\mathbf {G}_\ell \) for all \(\ell \gg 0\). As a corollary, if \(\rho _\ell \) is semisimple and \(\ell \) is sufficiently large, then \(\mathbf {G}_\ell \) is unramified.

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Acknowledgments

We would like to thank Frank Calegari for his comments on an earlier draft of this paper.

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Authors and Affiliations

  1. Mathematics Research Unit, University of Luxembourg, 6, rue Coudenhove-Kalergi, 1359, Luxembourg-Kirchberg, Luxembourg

    Chun Yin Hui

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Michael Larsen

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  1. Chun Yin Hui
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  2. Michael Larsen
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Correspondence to Chun Yin Hui.

Additional information

Chun Yin Hui is supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND). Michael Larsen was partially supported by a grant from the National Science Foundation.

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Hui, C.Y., Larsen, M. Type A images of Galois representations and maximality. Math. Z. 284, 989–1003 (2016). https://doi.org/10.1007/s00209-016-1683-0

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