Function field of a Drinfeld module and product formula I am learning about Drinfeld modules, and I have a few questions. There is an analogue that Drinfeld modules are like elliptic curves, which are projective, or are compact Riemann surfaces over $\mathbb C$. I want to ask if there are analogues of the following:
1) In the algebraic definition of a Drinfeld module, what is the function field of a Drinfeld module? In particular, a product formula for each function in the function field? Or given by lattices over $C_{\infty}$, what is the analogue of the fact that the number of poles are the same as the number of zeros?
 A: Since Drinfeld modules are geometrically the affine line, its naive function field would be simply the rational function field.  Unlike in the elliptic curve case these fields would not distinguish non-isomorphic Drinfeld modules.
However, there are analogues of results from elliptic function theory for Drinfeld modules (of generic characteristic) via their exponential functions.  One such result is due to Pellarin and Perkins in "On certain generating functions in positive characteristic," Monatsh. Math. 180 (2016), 123-144.  They prove the following result for the Carlitz module (see Proposition 12) for $A = \mathbb{F}_q[\theta]$.
Theorem (Pellarin-Perkins, 2016): Any function $f : C_{\infty} \to C_{\infty}$ that is both $A$-periodic and entire in the rigid analytic sense is an element of $C_{\infty}[\exp_C(\tilde{\pi} z)]$, where $\exp_C : C_\infty \to C_\infty$ is the Carlitz exponential and $\tilde{\pi}$ is the Carlitz period.
We could think of this like a Fourier expansion of $f$, but even better the entirety of $f$ implies that it is actually a polynomial in $\exp_C(\tilde{\pi} z)$, rather than a power series.  This should be seen as analogous to the fact that every doubly periodic function on $\mathbb{C}$ can be expressed as a rational function in the Weierstrass $\wp$-function and its derivative for the corresponding lattice.
Although they are dealing with only the rank 1 case, it seems reasonable that their arguments can be applied to a higher rank Drinfeld module in terms of its exponential function.
