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Set $K$ to be a number field, denote by $\mathcal{O}_K$ the integer ring of $K$. My question is the following:

Is there a smooth proper family $X \to \mathcal{O}_K$ whose fibers are not Mazur-Ogus?

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    $\begingroup$ What is Mazur-Ogus? $\endgroup$
    – Yosemite Stan
    Commented May 13, 2021 at 3:20
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    $\begingroup$ This will hold for all primes except for finitely many. $\endgroup$
    – Piotr Achinger
    Commented May 13, 2021 at 6:04
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    $\begingroup$ So if "whose fibers" means "all of whose fibers", then the answer is no: $H^*_{\rm dR}(X/\mathcal{O}_K[1/N])$ as well as the relative Hodge groups will be locally free with formation commuting with base change for some $N$, and then you get torsion-free crystalline cohomology for primes not dividing $N$ by Berthelot's comparison, and degenerate spectral sequence because it degenerates in char. 0 (or indeed by Deligne-Illusie in char. bigger than dimension). If you mean "some fibers", then constructing interesting smooth proper $X/\mathcal{O}_K$ is hard (often impossible for $K=\mathbf{Q}$)... $\endgroup$
    – Piotr Achinger
    Commented May 14, 2021 at 9:36
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    $\begingroup$ (continued) ... But I guess that it is known that there exist Enriques surfaces over $\mathcal{O}_K$ (not for $K=\mathbf{Q}$ by work of Schroer). That would likely give you something which is not Mazur-Ogus at a prime above $2$. $\endgroup$
    – Piotr Achinger
    Commented May 14, 2021 at 9:37
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    $\begingroup$ @PiotrAchinger There are indeed number fields $K$ such that there exist Enriques surfaces over $O_K$; this follows from work of Moret-Bailly on Skolem's problem (applied to the moduli of Enriques surfaces). $\endgroup$
    – Ariyan Javanpeykar
    Commented May 15, 2021 at 8:02

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