Level structure, arithmetic representations, and noncommutative Siegel linearization

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}}}(X_{\bar{k}})} into GLn⁡(ℤℓ¯){\operatorname{GL}_{n}(\overline{\mathbb{Z}_{\ell}})}, which is trivial mod ℓN{\ell^{N}}, 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{\ell}-adic form of Baker’s theorem on linear forms in logarithms.