LAUSR

Let X be a normal algebraic variety over a finitely generated field k of characteristic zero, and let $$\ell $$ℓ be a prime. Say that a continuous $$\ell $$ℓ-adic representation $$\rho $$ρ of $$\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})$$π1e´t(Xk¯) is arithmetic if there exists a finite extension $$k'$$k′ of k, and a representation $$\tilde{\rho }$$ρ~ of $$\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{k'})$$π1e´t(Xk′), with $$\rho $$ρ a subquotient of $$\tilde{\rho }|_{\pi _1(X_{\bar{k}})}$$ρ~|π1(Xk¯). We show that there exists an integer $$N=N(X, \ell )$$N=N(X,ℓ) such that every nontrivial, semisimple arithmetic representation of $$\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})$$π1e´t(Xk¯) is nontrivial mod $$\ell ^N$$ℓN. As a corollary, we prove that any nontrivial $$\ell $$ℓ-adic representation of $$\pi _1^{\acute{\mathrm{e}}\text {t}}(X_{\bar{k}})$$π1e´t(Xk¯) which arises from geometry is nontrivial mod $$\ell ^N$$ℓN.