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

## TL;DR

This paper provides a formal classification of two-dimensional neighborhoods of elliptic curves with torsion normal bundle, using formal foliations and holonomy, and explores the infinite-dimensional moduli space for analytic equivalence.

## Contribution

It introduces a formal classification framework based on foliations and holonomy for neighborhoods of elliptic curves with torsion normal bundle, and analyzes the moduli space structure.

## Key findings

- Neighborhoods are classified by their holonomy representations.
- The moduli space for analytic equivalence is infinite dimensional.
- Formal models determine the neighborhoods uniquely up to formal equivalence.

## Abstract

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 discussanalytic equivalence and show, for each formal model, that the corresponding moduli space is infinite dimensional.