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Let $p\neq q$ be two primes. For a given integer $g>0$ choose a smooth proper geometrically connected curve of genus $g$ over $\mathbb{F}_p$ and similarly for $q$. Is there a proper flat morphism $X\to\mathrm{Spec}\:\mathbb{Z}$ such that the fibers over $p$ and $q$ are isomorphic to the chosen curves?

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    $\begingroup$ This question appears close to weak approximation on the moduli space of curves. Thus, while it may be true for low genus (e.g. if the moduli space of curves is (uni-)rational),it is hard to imagine that this is true for large genus (e.g. when the moduli space is of general type). $\endgroup$
    – damiano
    Jun 23, 2020 at 8:49
  • $\begingroup$ Let $E$ be an elliptic curve defined over the rational numbers, and let $p$, $q$ be two primes such that the reductions $E_p$ and $E_q$ are both supersingular. Let $E'_q$ be a non-supersingular elliptic curve defined over $\mathbb{F}_q$. Is it possible that $E_p$ and $E'_q$ are fibres of a proper, flat morphism over $\mathrm{Spec} \, \mathbb{Z}$? $\endgroup$ Jun 23, 2020 at 10:28
  • $\begingroup$ Well, probably yes since the curve $E$ with the given reductions at the two primes is not is not unique... $\endgroup$ Jun 23, 2020 at 10:35

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