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
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.
I am not sure this question makes sense. Both $e^z$ and $\sin(z)$ are functions of one complex variable. Modular forms like $E_k(\tau)$ are functions of one variable (belonging to the upper half-plane $H$). Elliptic functions like $\wp(z,\tau)$ are either a functions of two variables for variable $\tau$, (i.e., functions on $\mathbb{C} \times H$, or a bi-periodic functions of $z$ on the complex plane for a fixed $\tau_0$; and hence can not be expressed in terms of the modular forms which are functions on the upper half plane.
However, if you take a Laurent series of $\wp(z,\tau)$ in $z$, you will see that the coefficients are modular forms in $\tau$, indeed.