 right|thumb|200px|klein's -invariant in the complex plane in mathematics, klein's j-invariant, regarded as a function of a complex variable&nbsp;Ï, is a modular function of weight zero for  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 :j\left(e^{\frac{2}{3}\pi i}\right) = 0, \quad j(i) = 1728. rational functions of  are modular, and in fact give all modular functions. classically, the -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). ==definition==  thumb|200px|phase of the -invariant as a function of the nome q on the unit disk  while the -invariant can be defined purely in terms of certain infinite sums (see  below), these can be motivated by considering isomorphism classes of elliptic curves. every elliptic curve  over  is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of . this is done by identifying opposite edges of each parallelogram in the lattice. it turns out that multiplying the lattice by complex numbers, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, and thus we can consider the lattice generated by  and some  in  (where  is the upper half-plane). conversely, if we define :g_2 = 60\sum_{(m,n) \neq (0,0)} (m + n\tau)^{-4}, :g_3 = 140\sum_{(m,n) \neq (0,0)} (m + n\tau)^{-6}, then this lattice corresponds to the elliptic curve over  defined by  via the weierstrass elliptic functions. then the -invariant is defined as :j(\tau) = 1728 \frac{g_2^3}{\delta} where the modular discriminant  is :\delta = g_2^3 - 27g_3^2 it can be shown that  is a modular form of weight twelve, and  one of weight four, so that its third power is also of weight twelve. thus their quotient, and therefore , is a modular function of weight zero, in particular a meromorphic function  invariant under the action of . as explained below,  is surjective, which means that it gives a bijection between
Elliptic_functions Modular_forms Moonshine_theory