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 $D.$



Analogue of Shafarevich-Ogg's theorem over complex numbers
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 $R^1f\_\*\mathbb{Z} \otimes \mathcal{O}\_{D^\*}$.


