Two-dimensional neighborhoods of elliptic curves: formal classification and foliations

We classify two-dimensional neighborhoods of an elliptic curve $C$ with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having $C$ as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discuss analytic equivalence and for each formal model, we show that the corresponding moduli space is infinite dimensional.