An equivariant covering map from the upper half plane to the complex plane minus a lattice
Classical Analysis and ODEs
2012-03-26 v1 Complex Variables
Number Theory
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).
Keywords
Cite
@article{arxiv.1203.5261,
title = {An equivariant covering map from the upper half plane to the complex plane minus a lattice},
author = {Marjorie Batchelor and Polly Brownlee and William Woods},
journal= {arXiv preprint arXiv:1203.5261},
year = {2012}
}