TITLE: Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
QUESTION [13 upvotes]: Is there an elliptic curve over $\mathbb{C}[t, t^{-1}]$ that has a nonconstant $j$-invariant? What is an equation for such a curve, if it exists?

REPLY [6 votes]: It is an older topic, and very nice answers have been given. But I am new in MO and let me give another answer. 
The reason for nonexistence of such an elliptic curve is the same as for nonexistence of such an elliptic curve over $S:=$Spec$(\mathbb{C}[t])$. In order to have a universal elliptic curve one must remove the points from $S$ which correspond to elliptic curves with automorphism group larger than $\{ \pm1\}$, and these fibers have $j$-invariants 0 and 1728. This is explained in the book Arithmetic Moduli of Elliptic Curves by Katz and Mazur. So we must remove at least two points from $S$, i.e. inverting $t$ in $\mathbb{C}[t]$ is not enough. 
This argument works if we replace $\mathbb{C}$ by any algebraically closed field of characteristic $p \geq 5$. But for $p=2$ or 3 we have $0 = 1728$, so it is enough to remove only one point from $S$ in these cases and this explains the existence of examples given by Elkies above.