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 $X$.



Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them?
I suspect that even if you had a single curve over $\mathbb{F}\_p$, you might not find a lift 
to $\mathbb{Q}$. Below I sketch an argument that works under the assumption that $\mathcal{M}\_g$ does not have Zariski dense set of points. If you believe the conjectures of Lang on rational points, this assumption should be satisfied as soon $\mathcal{M\_g}$ is of general type, e.g. if $g \geq 24$.


**Lemma.**
Suppose that $g$ is an integer such that $\mathcal{M}\_g(\mathbb{Q})$ is not Zariski dense in $\mathcal{M}\_g$. Then for all sufficiently large primes $p$, there are smooth curves $C\_p$ of genus $g$ defined over $\mathbb{F}\_p$ that are not reductions of curves defined over $\mathbb{Q}$.


*Proof.*
By assumption, the set $Z:=\overline{\mathcal{M}\_g(\mathbb{Q})}$ is a proper closed subset of $\mathcal{M}\_g$. In particular, the dimension of $Z$ is smaller than the dimension of $\mathcal{M}\_g$. By the Lang-Weil estimates, the number of points of $\mathcal{M}\_g$ modulo $p$ grows as a polynomial in $p$ of degree $\dim \mathcal{M}\_g$, since $\mathcal{M}\_g$ is irreducible modulo $p$. Similarly, the number of points of $Z$ modulo $p$ grows as a polynomial in $p$ of degree at most $\dim Z$ (the "at most" comes from the fact that $Z$ need not be geometrically integral). Thus, for sufficiently large $p$, there will be points of $\mathcal{M}\_g(\mathbb{F}\_p)$ that are not contained in $Z(\mathbb{F}\_p)$. These points correspond to smooth curves of genus $g$ defined over $\mathbb{F}\_p$ that are not reductions of curves defined over $\mathbb{Q}$, as required.


