It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.


My question is: If $S$ is a connected scheme such that has every prime as a residue characteristic, then can there exist an elliptic curve over $S$?



Do there exist elliptic curves over schemes which have all primes as residue characteristics?
Yes, there can. Choose any elliptic curve $E$ over $\mathbb{Q}$ with potential good reduction (for instance, a curve with potential CM) and pass to a number field $K$ over which the reduction is everywhere good. Then $E\_K$ extends to an elliptic curve over the ring of integers $\mathcal{O}\_K$, and the latter has all primes as residue characteristics.


