An equivariant covering map from the upper half plane to the complex plane minus a lattice

TL;DR

This paper introduces a special covering map from the upper half plane to the complex plane minus a lattice, which generalizes modular functions and relates to Klein's J invariant and root lattices of sl_3(C).

Contribution

It constructs an equivariant covering map with properties linking modular functions, Klein's J invariant, and root lattices, providing new insights into their interrelations.

Findings

The map factorizes Klein's J invariant.

(phi')^6 is a modular function of weight 12.

The map relates to the root lattice of sl_3(C).

Abstract

This paper studies a covering map phi from the upper half plane to the complex plane with a triangular lattice excised. This map is interesting as it factorises Klein's J invariant. Its derivative has properties which are a slight generalisation of modular functions, and (phi')^6 is a modular function of weight 12. There is a homomorphism from the modular group Gamma to the affine transformations of the complex plane which preserve the excised lattice. With respect to this action phi is a map of Gamma-sets. Identification of the excised lattice with the root lattice of sl_3(C) allows functions familiar from the study of modular functions to be expressed in terms of standard constructions on representations of sl_3(C).