Arithmetic representations of fundamental groups I

Let X be a normal algebraic variety over a finitely generated field k of characteristic zero, and let ℓ be a prime. Say that a continuous ℓ -adic representation ρ of π 1 e ´ t ( X k ¯ ) is arithmetic if there exists a finite extension k ′ of k , and a representation ρ ~ of π 1 e ´ t ( X k ′ ) , with ρ a subquotient of ρ ~ | π 1 ( X k ¯ ) . We show that there exists an integer N = N ( X , ℓ ) such that every nontrivial, semisimple arithmetic representation of π 1 e ´ t ( X k ¯ ) is nontrivial mod ℓ N . As a corollary, we prove that any nontrivial ℓ -adic representation of π 1 e ´ t ( X k ¯ ) which arises from geometry is nontrivial mod ℓ N .