Let $D \subset \mathbb{C}$ be the unit disc, and let $E$ be a complex surface with a regular holomorphic map $\pi: E \to D$, whose fibers are all curves of genus one (so $E$ is a nonsingular elliptic surface over $D$). Given a point $z \in D$, let $j(z)$ be the $j$-invariant of the fiber over $z$. Two questions:

1. Is $j$ a holomorphic function of $z$?

2. Is $E$ isomorphic to the quotient of $D \times \mathbb{C}$ by a $\mathbb{Z}^2$-action of the form

$(n,m) \cdot (z,w) = (z, w + n + m\tau(z))$

for some holomorphic map $\tau:D \to \mathbb{H}$?

I assume the answers are yes, but it's not obvious to me. Proving 2 is equivalent to constructing a holomorphic 1-form which is nonzero on each fiber (and this can be reduced to constructing one which is nonzero on a single fiber). This seems like a pretty basic question, so I assume there's a reference which talks about it, but I haven't found one yet.

Let $D \subset \mathbb{C}$ be the unit disc, and let $E$ be a complex surface with a regular holomorphic map $\pi: E \to D$, whose fibers are all curves of genus one (so $E$ is a nonsingular elliptic surface over $D$). Given a point $z \in D$, let $j(z)$ be the $j$-invariant of the fiber over $z$. Two questions:

1. Is $j$ a holomorphic function of $z$?

2. Is $E$ isomorphic to the quotient of $D \times \mathbb{C}$ by a $\mathbb{Z}^2$-action of the form

$(n,m) \cdot (z,w) = (z, w + n + m\tau(z))$

for some holomorphic map $\tau:D \to \mathbb{H}$?

I assume the answers are yes, but it's not obvious to me. This seems like a pretty basic question, so I assume there's a reference which talks about it, but I haven't found one yet.