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

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

2012-03-23

arxiv.org/abs/1203.5261v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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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