Elliptic curve over a ring (reviewed)

An elliptic curve over a commutative ring $R$ is an elliptic scheme $E \to \operatorname{Spec} R$.

For example, an elliptic curve over $\Z[1/N]$ is the same as an elliptic curve over $\Q$ with good reduction at all primes not dividing $N$. (More precisely, the latter is the generic fiber of the former.)