Question

Let $f:E\to D^*$ be a family of complex elliptic curves parametrized by the punctured open disk $D^*.$ Assume that the monodromy on $H^1$ is trivial (i.e. $R^1f_*\mathbb Z$ is a constant sheaf on $D^*$). Does this imply that $f$ extends to a family of elliptic curves over the full disk $D?$

Here's an attempt, which I'm not sure if it works: maybe there is an equivalence (Riemann-Deligne?) between families of elliptic curves over a (smooth) base over $\mathbb C$ and variations of $\mathbb Z$-Hodge structures satisfying Griffiths transversality, of rank 2 and weight 1 on the same base. The constant local system certainly extends to $D.$

Answer 1

The answer is, as you expected, yes:

Choosing a symplectic isomorphism of $R^1f_* \mathbb{Z}$ with the constant sheaf $\mathbb{Z}^2$ (i.e. full level structure) gives an analytic map $c:D^* \to \mathfrak{h}$ where $\mathfrak{h}$ is the upper half plane (viewed as the moduli space of elliptic curves with full level structure). Since $\mathfrak{h}$ is analytically isomorphic to a bounded domain it follows from the removable singularity theorem that $c$ extends to an analytic map $D \to \mathfrak{h}$. Pulling back the universal family over $\mathfrak{h}$ -- the fibre over $\tau$ is the elliptic curve $\mathbb{C}/\mathbb{Z} + \mathbb{Z}\tau$ with the level structure given by $1 \mapsto (1,0)$, $\tau \mapsto (0,1)$ -- gives the required extension.

The main point in the above is the analyticity of $c$. This is a consequence of the fact that $R^1f_*\mathcal{O}_E$ is a holomorphic sub-bundle of $R^1f_*\mathbb{Z} \otimes \mathcal{O}_{D^*}$.

Answer 2

Let me give another proof using the classical Picard-Lefschetz theory.

We first take the flat limit of your family $f \colon E \to D^*$, obtaining a family $g \colon S \to D$. Now we have to understand what is the central fiber $g^{-1}(0)$.

I will refer to the paper by Durfee "The monodromy of a degenerate family of curves", Inventiones Mathematicae 28 (1975). By Theorem $2$, the monodromy of $g$ is of finite order if and only if there exists a finite base change $D \to D$, $z \to z^n$ such that the pull-back family $g' \colon S' \to D$ has a most nodes as singularities and contains no vanishing cycles in its central fibre. This means that $g' \colon S' \to D$ is a smooth family.

If $n \geq 2$ then the central fibre of $g \colon S \to D$ would be of type ${}_nI_0$ in Kodaira's classification, that is a smooth elliptic curve with multiplicity $n$. But then the monodromy of $g \colon S \to D$ would be of order $n$, contradiction.

Then $n=1$, that is $g=g'$ and this shows that $g$ is a smooth family.