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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.

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Let $D$ be any C-analytic space. By smoothness, loc. on $D$ there's a section. Hence, answer to the first is "yes": use analytic cohomology and base change (proved like in alg. case, once the basic cohomological formalism is in place, say via "Coherent Analytic Sheaves") and arguments from Ch. 1 and 2 of Katz-Mazur to prove that loc. on $D$ there's a Weierstrass model. For 2nd, answer is "yes" only loc. on the base because $L = {\rm{R}}^1 \pi_{\ast}(\mathbf{Z})$ is a rank-2 local system but may not be globally free. This is related to the univ. property of standard family over $\mathfrak{h}$. – BCnrd Nov 12 '10 at 16:46
By the way, you may enjoy the exercise (perhaps hard, but there are lots of people in Boston whom you can talk with about it) to rigorously prove that the analytified modular curves enjoy the "expected" universal property for analytic families of elliptic curves over any complex-analytic space. (This all generalizes to analytic families of higher-dimensional complex tori over complex-analytic spaces, but that needs a lot more preliminary work.) – BCnrd Nov 12 '10 at 16:52
I'm not assuming that the generic fiber has genus one. Why does this follow from my assumptions? – Lucas Culler Nov 12 '10 at 17:19
Rather, I just don't see how the arguments in Katz-Mazur apply in this situation. – Lucas Culler Nov 12 '10 at 17:30
Dear Lucas: The question says that all fibers have genus 1 (as an aside, pf of local constancy of Euler char. relative to proper flat morphisms as in Mumford's book on abelian varieties in alg. case works verbatim in analytic case), but I didn't notice you didn't assume smoothness (since it doesn't make sense to ask #2 without smoothness). You could have degenerate fibers with many components, etc., so then K-M methods/conclusions don't apply. Will you be happier to know that loc. on the base any proper flat analytic family of curves is pullback of an analytified alg. family? – BCnrd Nov 12 '10 at 17:45

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