j-invariant
Modular function in mathematics / From Wikipedia, the free encyclopedia
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In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that
Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over , but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine).