Hi,
here's there's a construction about elliptic curves that I do not completely understand. Suppose I consider the two following families of elliptic curves over $\mathbb{C}^*$.
The first, which I denote $F_0$ is the trivial family $ E \times \mathbb{C}^*$, where $E$ is an elliptic curve with only $-Id $ as non-trivial automorphism. I will denote by $F_1$ the quotient of $F_0$ via the diagonal action of $-Id$, actiong both on $E$ and $\mathbb{C}^*$.
Remark that the quotient of the base is equal to $\mathbb{C}^*$ itself. Now $F_0$ has trivial monodromy around 0 and $F_1$ has not. The modular maps to $\mathcal{M}_{1,1}$ given by the two $F_i$ are constant, but there exist non-constant lifts depending on the description of the mod space.
The two lifts to the Siegel half space $\mathcal{H}_1$ (seeing the mod space of elliptic curves as a quotient by $SL_2(\mathbb{Z})$) are constant.
Let us now consider the quotient of $\mathbb{C}^2$ minus the discriminant (the Neil-parabola) by the torus $\mathbb{C}^*$ acting with weights $(4,6)$, i.e. let us parametrize the mod space with the coefficients $a$ and $b$ of the Weierstrass equation. The smooth curves are parmetrized by the open subspace $U\subset \mathbb{P}(4,6)$ complement of the point at infinity.
With this presentation, any lift of $F_0$ to $\mathbb{C}^2$ is the constant map to $(a,b)$, but a nonconstant lift of $F_1$ exists, namely and the non-constant map defined by
$$t \mapsto (t^2a, t^3 b).$$
Can you explain me why this happens? Is it related to the monodromy of the family? Or the connectedness of the group acting?
