Some elliptic curves arising from the Leech lattice

Abstract

The Leech lattice and its group of all automorphisms (the Conway group SO), are interesting mathematical objects in their own rights. However, a new area has been opened up for the theory of finite simple groups by the discovery in the late 1970’s of a probable relation between the Monster simple group and the theory of modular functions (Thompson [14], Conway and Norton [2]). In this paper, we will show that live elliptic curves defined over Q, all having complex multiplication by the set of algebraic integers of Q(G), arise naturally from an investigation of a set of modular functions defined for each element g of -0. Those five curves are