The following question occurred to me while writing lectures last night and distracted me from productive work all morning. The least I can do is distract you all in turn.

Let be an elliptic curve over . Let and be two points of such that is not torsion. It is well known that there are no nonconstant meromorphic functions on which have neither poles nor zeroes in .

Are there any nonconstant holomorphic functions at at all, where we allow essential singularities at and ?

Take a function that’s meromorphic with poles only at these two points, and exponentiate it.

Sorry, I misread the question; now I’m confused. I thought you were asking for only having zeros and poles at these two points.

As stated, there are lots of nonconstant holomorphic functions on E with poles at just one of the two points?

Take a finite map from the elliptic curve to P^1 which is totally ramified at one of the two points. Next, pullback the exponential function.

Any noncompact riemann surface is Stein so it has lots of

holomorphic functions.This is the theorem of Behnke and Stein.

You are all right, of course. I thought of all the above arguments, but I also thought of the following wrong argument: Let be the corresponding periodic function on and let . Integrating around a parallelogram, the residues of at and are negatives of each other, say and for some integer . Then, integrating around the parallelogram, we get for integers and where are the periods, and this is impossible since is non-torsion.

I spent enough time deciding which was right and which was wrong that it seemed worth posting on a Friday, but apparently no one else was sucked into the false line of logic.