Is there a variety $X$ over $\mathbb{Q}$ and a line bundle $L$ over $X$ (other than the trivial line bundle $\mathcal{O}\_X$ ) such that $L\_v$ is the trivial line bundle over $X\_v=X\times\_{\mathbb{Q}}\mathbb{Q}\_v$ for every place $v$ of $\mathbb{Q}$ ?


(Answer known. There is a pun on "locally trivial" in the title.)



An everywhere locally trivial line bundle
The following example was provided to me by Colliot-Thélène some years ago : Let $X$ be the complement in $\mathbb{P}\_{1,\mathbb{Q}}$ of the three closed points defined by $x^2=13$, $x^2=17$, $x^2=221$. Then $\operatorname{Pic}(X)=\mathbb{Z}/2\mathbb{Z}$ but $\operatorname{Pic}(X\_v)=0$ for every place $v$ of $\mathbb{Q}$.


