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

摘要

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