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?



Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
Let me give an analytic argument, to complement Noam's algebraic one. Suppose $E\to \mathbb{C}^\*$ is an elliptic curve. Then pulling back $E$ along the universal covering map (also known as the exponential map) $\mathbb{C}\to \mathbb{C}^\*$, one obtains an elliptic curve $\tilde E\to \mathbb{C}$. Choosing a basis for the first homology of $\tilde E$ induces a holomorphic map $$\mathbb{C}\to \mathbb{H},$$ where $\mathbb{H}$ is the upper half-plane, viewed as the moduli space of elliptic curves with a homology basis for $H\_1$. But any such map must be constant by Liouville's theorem. $\blacksquare$


**Added Later.** Since the OP enjoyed the sketch algebraic version in the comments (corrected by user74230), let me give some more details (and in particular, say what happens in characteristic $p>3$). WLOG $k$ is algebraically closed. Suppose $E\to \mathbb{G}\_m/k$ is an elliptic curve. If we can find some $n$ so that the pullback of $E$ along $\mathbb{G}\_m\overset{[n]}\longrightarrow \mathbb{G}\_m$ has trivial $\ell$-torsion, with $(\ell, p)=1$ and $\ell \gg 0$, we're done, because choosing a trivialization we get a map from $\mathbb{G}\_m$ to a high genus modular curve, which must be constant as $\mathbb{G}\_m$ is rational.


To do this, we must show that for infinitely many $\ell$, the map $E[\ell]\to \mathbb{G}\_m$ has tame ramification at $0, \infty$. It suffices to find $\ell$ so that $GL\_2(\mathbb{Z}/\ell\mathbb{Z})$ has order prime to $p$. But this order is $(\ell^2-1)(\ell^2-\ell)=\ell(\ell-1)^2(\ell+1).$ But by Dirichlet's theorem on primes in arithmetic progressions, there are infinitely many $\ell$ so that $\ell(\ell-1)^2(\ell+1)$ is prime to $p$, if $p>3$. $\blacksquare$


As Noam observes in the comment, the result is *false* in characteristic $2$ and $3$; he gives examples of non-isotrivial families in these characteristics.


