This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.

Question. Is there a smooth proper scheme $X\to\operatorname{Spec}(\mathbb{Z})$ such that $X(\mathbb{Q}_v)\neq\emptyset$ for every place $v$ of $\ \mathbb{Q}$ (including $v=\infty$), and yet $X(\mathbb{Q})=\emptyset$ ?

I believe the answer is No. The evidence is flimsy : $X$ cannot be a curve or a torsor under an abelian variety (Abrashkin-Fontaine) or a twisted form of $\mathbb{P}_n$.

I haven't gone through a list of smooth projective $\mathbb{Q}$-varieties which contradict the Hasse principle to check if any of them has good reduction everywhere, but the chances of such a thing are slim.

Colliot-Thélène and Xu give a systematic treatement of many known examples of quasi-projective $\mathbb{Z}$-schemes $Y$ such that $Y(\mathbb{Z}_p)\neq\emptyset$ for every prime $p$ and $Y(\mathbb{R})\neq\emptyset$, but $Y(\mathbb{Z})=\emptyset$. Some of these schemes might even be smooth over $\mathbb{Z}$, but none of them is proper.

Fontaine's letter to Messing (MR1274493) might be of some relevance here.


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