An open adelic image theorem for motivic representations over function fields

Let $\mathbb{F}$ be a field and $k$ a function field of positive transcendence degree over $\mathbb{F}$. Let $S$ be a smooth, separated, geometrically connected scheme of finite type over $k$. If $\mathbb{F}$ is quasi-finite or algebraically closed we show that for motivic representations of the étale fundamental group $\pi_1 (S)$ of $S$, $\ell$-Galois-generic points are Galois-generic. This is a geometric variant of a previous result of the author for representations of $\pi_1 (S)$ on the adelic Tate module of an abelian scheme $A \to S$ when the base field $k$ is finitely generated of characteristic $0$. The procyclicity of the absolute Galois group of a quasi-finite field allows to reduce the assertion for $\mathbb{F}$ finite to the assertion for $\mathbb{F}$ algebraically closed. The assertion for $\mathbb{F}$ algebraically closed can then be deduced, using basically the same arguments as in the case of abelian schemes, from maximality results for the image of $\pi_1 (S)$ inside the group of $\mathbb{Z}_{\ell}$-points of its Zariski-closure.

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Received 17 March 2017

Accepted 10 July 2017

Published 7 June 2019