TITLE: Do there exist elliptic curves over schemes which have all primes as residue characteristics?
QUESTION [6 upvotes]: 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 $S$?

REPLY [7 votes]: Another perspective: Let $K$ be any number field and let $E$ be any elliptic curve over $K$ with $j$-invariant in $\mathcal{O}_K$. Then there is an extension $L$ of $K$ such that $E \times_K L$ extends to an elliptic curve over $\mathcal{O}_L$. 
Proof: By properness of the stack $\overline{M}_{1,1}$, there is some $L$ so that we get a stable genus one curve with one marked point over $\mathrm{Spec}\ \mathcal{O}_L$. We just need to rule out the possibility that some fibers are nodal. If the fiber over a prime $\pi$ of $\mathcal{O}_L$ is nodal, than a computation with Tate curves shows that $v_{\pi}(j)<0$, contradicting that $j \in \mathcal{O}_L$. $\square$
I think it is worth bringing this up, as well as the CM solutions above, because there are many more integral $j$-invariants than there are $j$-invariants with CM.