Let $A$ 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 $G_{\mathfrak{p}}$.

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 $G_{\mathfrak{p}}$.