Level structure, arithmetic representations, and noncommutative Siegel linearization

Abstract

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 π 1 ét ⁢ ( X k ¯ ) {\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} is arithmetic if it extends to a finite index subgroup of π 1 ét ⁢ ( 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 π 1 ét ⁢ ( X k ¯ ) {\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} into GL n ⁡ ( ℤ ℓ ¯ ) {\operatorname{GL}_{n}(\overline{\mathbb{Z}_{\ell}})} , which is trivial mod ℓ N {\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.