Lifting subgroups of symplectic groups over ℤ/ℓℤ

Creators: Landesman, Aaron; Swaminathan, Ashvin A.; Tao, James; Xu, Yujie

Abstract

For a positive integer g, let Sp_(2g)(R) denote the group of 2g×2g symplectic matrices over a ring R. Assume g≥2. For a prime number ℓ, we give a self-contained proof that any closed subgroup of Sp_(2g)(ℤℓ) which surjects onto Sp2g(ℤ/ℓℤ) must in fact equal all of Sp_(2g)( ℤℓ). The result and the method of proof are both motivated by group-theoretic considerations that arise in the study of Galois representations associated to abelian varieties.

Additional Information

© 2017 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Received: 21 February 2017; Accepted: 26 February 2017; Published: 3 July 2017. This research was supervised by Ken Ono and David Zureick-Brown at the Emory University Mathematics REU and was supported by the National Science Foundation (Grant Number DMS-1557960). We would like to thank David Zureick-Brown for suggesting the problem that led to the present article and for offering us his invaluable advice and guidance. We would also like to thank David Zureick-Brown for providing the intuition behind the proof of Lemma 3. We would like to thank Jackson Morrow, Ken Ono, and David Zureick-Brown for making several helpful comments regarding the composition of this article. We would like to acknowledge Michael Aschbacher and Nick Gill for their helpful advice. We used Magma and Mathematica for explicit calculations. The authors declare that they have no comepting interests.

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