# Level structure, arithmetic representations, and noncommutative Siegel linearization

@article{Kadets2022LevelSA,
title={Level structure, arithmetic representations, and noncommutative Siegel linearization},
author={Borys Kadets and Daniel Litt},
journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
year={2022},
volume={2022},
pages={219 - 238}
}
• Published 5 July 2021
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)
Abstract Let ℓ{\ell} be a prime, k a finitely generated field of characteristic different from ℓ{\ell}, and X a smooth geometrically connected curve over k. Say a semisimple representation of π1ét⁢(Xk¯){\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} is arithmetic if it extends to a finite index subgroup of π1ét⁢(X){\pi_{1}^{{\text{\'{e}t}}}(X)}. We show that there exists an effective constant N=N⁢(X,ℓ){N=N(X,\ell)} such that any semisimple arithmetic representation of π1ét⁢(Xk¯){\pi_{1}^{{\text{\'{e}t…

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