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

Any advice much appreciated.

Thanks,

-A

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1 Answer 1

I don't really understand what this means; modular functions and elliptic functions don't even have the same domain, although they are closely related.

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.

The analogy to trigonometric functions seems to fall short: elliptic functions are analogous to trigonometric functions, but there isn't a good notion of "modular function" here since the moduli space of circles is a point.

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Maybe what's going on here is this: elliptic curves can be parametrized by elliptic functions. They can also be parametrized by modular functions. So maybe the question is asking for a relation between the elliptic functions parametrizing a given elliptic curve and the modular functions parametrizing that curve. –  Gerry Myerson May 9 '11 at 23:59

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