A note on images of Galois representations (with an application to a result of Litt)

2018/09/19 by Cadoret, Anna, Moonen, Ben
#11F80 #14F20 #14F35 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.07018

Abstract

Let X be a variety (possibly non-complete or singular) over a finitely generated field k of characteristic 0. For a prime number ℓ, let ρ_ℓ be the Galois representation on the first ℓ-adic cohomology of X. We show that if ℓ varies the image of ρ_ℓ is of bounded index in the group of ℤ_ℓ-points of its Zariski closure. We use this to improve a recent result of Litt about arithmetic representations of geometric fundamental groups. Litt's result says that there exist constants N = N(X,ℓ) such that every arithmetic representation π₁(X_k) → GL_n(ℤ_ℓ) that is trivial modulo ℓ^N is unipotent. We show that these constants can in fact be chosen independently of ℓ.

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