A Map Between Moduli Spaces of Connections

TL;DR

This paper explores a correspondence between moduli spaces of rank 2 logarithmic connections on elliptic curves and the Riemann sphere, providing explicit descriptions of the transformation map between these spaces.

Contribution

It introduces a detailed construction of a map between moduli spaces of connections on elliptic curves and the sphere, including explicit formulas and normal forms.

Findings

Explicit description of the moduli space of connections on elliptic curves.

Construction of a transformation map between moduli spaces.

Normal form and birational modifications for connections.

Abstract

We are interested in studying moduli spaces of rank 2 logarithmic connections on elliptic curves having two poles. To do so, we investigate certain logarithmic rank 2 connections defined on the Riemann sphere and a transformation rule to lift such connections to an elliptic curve. The transformation is as follows: given an elliptic curve $C$ with elliptic quotient $\pi\colon C\to{\mathbb P}^1$, and the logarithmic connection $(E,\nabla)$ on ${\mathbb P}^1$, we may pullback the connection to the elliptic curve to obtain a new connection $(\pi^*E, \pi^*\nabla)$ on $C$. After suitable birational modifications we bring the connection to a particular normal form. The whole transformation is equivariant with respect to bundle automorphisms and therefore defines a map between the corresponding moduli spaces of connections. The aim of this paper is to describe the moduli spaces involved and compute explicit expressions for the above map in the case where the target space is the moduli space of rank 2 logarithmic connections on an elliptic curve $C$ with two simple poles and trivial determinant.