# Modular functions and elliptic functions

Does anybody know of an equation formally equating modular functions and elliptic functions similar to Euler's equation for exponential and trigonometric functions?

Thanks,

-A

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A modular function is a meromorphic function on the upper half-plane $\mathbb{H}$ which is invariant under the action of the modular group $\Gamma = \text{SL}_2(\mathbb{Z})$. The corresponding quotient can be thought of as the moduli space of elliptic curves in a certain way. In other words, a modular function is something like an invariant of elliptic curves.
An elliptic function, on the other hand, is a meromorphic function on the complex plane $\mathbb{C}$ which is invariant under the action of a lattice $\Lambda = \mathbb{Z} \omega_1 \oplus \mathbb{Z} \omega_2$. The corresponding quotient can be thought of as a particular elliptic curve. In other words, an elliptic function is simply a function on a particular elliptic curve.
Your confusion probably stems from the fact that there are elliptic functions, such as the Weierstrass $\wp$ function, which can be uniformly defined for different elliptic curves, and hence which also allow us to define modular functions. This is because the Weierstrass function has more structure than just being an elliptic function; it is also a Jacobi form.