Complex analysis poser

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 $E$ be an elliptic curve over $\mathbb{C}$. Let $x_1$ and $x_2$ be two points of $E$ such that $x_1 - x_2$ is not torsion. It is well known that there are no nonconstant meromorphic functions on $E$ which have neither poles nor zeroes in $E \setminus \{ x_1, x_2 \}$.

Are there any nonconstant holomorphic functions at $E \setminus \{ x_1, x_2 \}$ at all, where we allow essential singularities at $z_1$ and $z_2$?

5 thoughts on “Complex analysis poser”

1. Tyler Lawson says:

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

2. Tyler Lawson says:

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?

3. 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.

4. Mohan Ramachandran says:

Any noncompact riemann surface is Stein so it has lots of
holomorphic functions.This is the theorem of Behnke and Stein.

5. David Speyer says:

You are all right, of course. I thought of all the above arguments, but I also thought of the following wrong argument: Let $f(z)$ be the corresponding periodic function on $\mathbb{C}$ and let $g(z) =f'(z)/f(z)$. Integrating around a parallelogram, the residues of $g$ at $x_1$ and $x_2$ are negatives of each other, say $r$ and $-r$ for some integer $r$. Then, integrating $z g(z)$ around the parallelogram, we get $r(x_2 - x_1) = p \omega_1 + q \omega_2$ for integers $p$ and $q$ where $\omega_i$ are the periods, and this is impossible since $x_1 - x_2$ 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.