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

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

  4. 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.

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