Level structure, arithmetic representations, and noncommutative Siegel linearization

Borys Kadets; Daniel Litt

doi:10.1515/crelle-2022-0028

Article

Level structure, arithmetic representations, and noncommutative Siegel linearization

Borys Kadets and Daniel Litt

Published/Copyright: May 25, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 788

Abstract

Let ℓ be a prime, k a finitely generated field of characteristic different from ℓ, and X a smooth geometrically connected curve over k. Say a semisimple representation of π1ét⁢(Xk¯) is arithmetic if it extends to a finite index subgroup of π1ét⁢(X). We show that there exists an effective constant N=N⁢(X,ℓ) such that any semisimple arithmetic representation of π1ét⁢(Xk¯) into GLn⁡(ℤℓ¯), which is trivial mod ℓN, is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel’s linearization theorem and the ℓ-adic form of Baker’s theorem on linear forms in logarithms.

Funding statement: This material is partly based upon work supported by the NSF grant DMS-1928930 while the first author participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester. The second author was supported by NSF grant DMS-2001196.

Acknowledgements

We are grateful for useful conversations with Hélène Esnault, Moritz Kerz, Samit Dasgupta, and Simion Filip. We would also like to thank the referee for their many extremely helpful suggestions, and for catching and fixing an error in an earlier version of the paper.

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Received: 2021-09-03

Revised: 2022-03-25

Published Online: 2022-05-25

Published in Print: 2022-07-01

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https://doi.org/10.1515/crelle-2022-0028