Modular Functions

Abstract

Every elliptic curve E over the complex numbers C corresponds to a complex torus C/L _τ with L _τ = Z τ + Z and τ ∈ ℌ, the upper half plane, where C/L _τ is isomorphic to the complex torus of complex points E(C) on E. In the first section we show easily that C/L _τ and C/L _τ′ are isomorphic if and only if there exists

$$\left( {\begin{array}{*{20}{c}} a & b \\ c & d \\ \end{array} } \right) \in S{L_2}\left( z \right)$$

with

$$ \tau ' = \frac{{a\tau + b}}{{c\tau + d}}.$$

.