Say we have $X$ a smooth projective curves over $\mathbb{Q}$, then I know there is an isomorphism $H_{ét}^1(X\times_\mathbb{Q}\overline{\mathbb{Q}},\mathbb{G}_m)\cong Pic(X\times_\mathbb{Q}\overline{\mathbb{Q}})$. I am wondering about its properties.
First, we have a $G_\mathbb{Q}$-action on both sides, are they compatible?
And second, is it functorial in $X$?
I would appreciate references.
Thank you.