Complex analysis poser March 25, 2011
Posted by David Speyer in Uncategorized.trackback
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 
Are there any nonconstant holomorphic functions at 
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.