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arXiv · 1203.5261

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

Marjorie Batchelor·Polly Brownlee·William Woods
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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).

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Marjorie Batchelor, Polly Brownlee, William Woods. 2012-03-23. An equivariant covering map from the upper half plane to the complex plane minus a lattice. https://arxiv.org/abs/1203.5261

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