I'm trying to understand why automorphisms of elliptic curves lead to non-existance of the moduli space (as a variety). Some notes suggest to consider the following example:
Consider the Legendre family $X \to \mathbb{A}^1 \setminus \{0, 1\}$ given by $(x, y, z)$ goes to $z$ where $X$ is the subset of $\mathbb{A}^3$ given by $y^2=x(x-1)(x-z)$. Fibers over $z$ and $1-z$ are isomorphic because they have the same $j$-invariant. Then they say that if such a moduli space exists then this involution should lift to the whole family which is not true. I have two questions: why should it lift and why doesn't it lift?