Moonshine: The First Quarter Century and Beyond: Borcherds' Proof of the Conway-Norton Conjecture

We give a summary of R. Borcherds' solution (with some modifications) to the following part of the Conway-Norton conjectures: Given the Monster simple group and Frenkel-Lepowsky-Meurman's moonshine module for the group, prove the equality between the graded characters of the elements of the Monster group acting on the module (i.e., the McKay-Thompson series) and the modular functions provided by Conway and Norton. The equality is established using the homology of a certain subalgebra of the monster Lie algebra, and the Euler-Poincare identity.

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