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Tannakian group of Galois representations coming from geometry

Let $K$ be a number field. Let $G_K$ be its absolute Galois group.
Let $p$ be a rational prime.

Let $mathcal{R}_{K,p}^g$ be the category of finite-dimensional continuous $p$-adic representations of $G_K$ that come from geometry. Thus, an object $V$ of $RG_{K}^p$ is isomorphic to a sub-quotient of $H^i_{ét}(X, mathbb{Q}_p)(n)$ for some smooth projective variety $X$ and integers $i$ and $n$.

$mathcal{R}_{K,p}^g$ should be a Tannakian category, and therefore, have a Tannakian fundamental group scheme, $mathcal{G}_{K,p}^g$, associated to it.

How is $mathcal{G}_{K,p}^g$ related to $G_K$?

MathOverflow Asked by smn on December 27, 2020

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