Level structure, arithmetic representations, and noncommutative Siegel linearization

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 (Xk̄) 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 (Xk̄) into GLn (ℤ¯), 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.