A few days ago I heard an online presentation about elliptic curves and the presenter claimed that the functor which assigns to a scheme $S$ the isomorphism classes of elliptic curves over $S$ is not representable, and it seemed this was taken for granted by everybody.
Can anybody explain to me why this is the case or give a good reference?
I'll elaborate a little on my comment, this answer is not really formal, but it gives the idea which can be made so.
Suppose that your functor were represented by a scheme $X$ over $\mathbb{Q}$, say. Then take two points $p, q \in X(\mathbb{Q})$ corresponding to a pair of elliptic curves $E_p$ and $E_q$ which are nontrivial quadratic twists. Then $p, q$ are distinct, since the twist is nontrivial.
But since $X$ represents our moduli problem, $p$ and $q$ must be the same point in $X(\bar{\mathbb{Q}})$ because they become isomorphic over $\bar{\mathbb{Q}}$. This is absurd, because $X$ is a scheme, and schemes are nice.