Let $X$ be an Abelian variety defined over a number field $K$, suppose that it has a good reduction over a fine place $\mathfrak{p}$ of $K$. Let $G\_{\mathfrak{p}}$ be the local Galois group for $K\_{\mathfrak{p}}$. Let $s$ be a Hodge cycle in $H^n\_{B}(X^{an}, \mathbb{Q})$, the Betti cohomology. By comparison theorem there is a canonical isomorphism $H^n\_{B}(X^{an}, \mathbb{Q})\otimes \mathbb{Q}\_p\cong H^n\_{et}(X,\mathbb{Q}\_p)$, where the right hand side of the equation is the etale cohomology. So via this isomorphism, $s$ corresponds to a etale cycle $s\_{et}$. Then Does Deligne's absolute Hodge theory implies that $s\_{et}$ is fixed by a finite index subgroup of $G\_{\mathfrak{p}}$.



Absolute Hodge implies Galois invariant?
This is Proposition 2.9(b) of Deligne's 1982 notes.


