Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve over $\mathbf{Q}$. When does there exist a flat projective integral normal scheme $X$ over Spec $\mathbf{Z}$ such that the fibre above $p$ is $X_p$ for every prime p and the generic fibre is $X_\mathbf{Q}$?

Suppose that $X$ exists. Then the generic fibre is a smooth projective connected curve over $\mathbf{Q}$. Furthermore, $X_p$ is almost always smooth over $\mathbf{F}_p$. Also, the arithmetic genus is constant in the fibres of $X$. Moreover, the Hilbert polynomial of $X_p$ is independent of $p$.

**Example.** Suppose that $X_p$ is a supersingular elliptic curve for all $p$. Then there does not exist such an $X$.

for allprimes $p$. $\endgroup$ – Pete L. Clark Feb 4 '11 at 15:20