# Zeros of the Weierstrass pe-function

This question was prompted by the post here, and I asked this earlier, deleted it, and due to pressure exerted by Ilya Nikokoshev, I am asking it again. Apologies to Pavel Etingof.

Q1. Let $$\Lambda$$ be a lattice in $$\mathbb{C}$$. We look at the behavior of the zero set of the Weierstrass $$\wp$$-function for this lattice. By integration around a unit cell for the lattice, we see that the number of poles and zeros are the same. So there has to be two zeros. We position the fundamental domain to be symmetric about $$0$$, and from the expression for $$\wp$$, we see that the zero should be $$z$$ and $$-z$$ in case they are distinct. Otherwise, it is a double zero, which is one of the $$2$$-torsion points.

Now, in the case that the zero is not a double zero, can anything be said about its location from the knowledge of $$\Lambda$$?

Q2. This is stupid. . But, what is the degree of the branched covering $$\wp: \mathbb{C}/\Lambda \rightarrow \mathbb{P}^1(\mathbb{C})$$? I must confess that I am not good in this stuff.

• I've merged the deleted question into this one (which has the effect of moving Pavel's answer here). The wording of the question is slightly different. – Anton Geraschenko Feb 3 '10 at 21:10
• Ilya. I object to your editing of my personal feelings. I would prefer you adding a line(stating it is your opinion) that you disagree with me. – Anweshi Feb 3 '10 at 21:35
• You're entitled to your opinion and generally have great control over the posts you make; sorry if my editing offended you. I'll move on. The site is a collaborative effort, however, so somebody else could still edit the post. – Ilya Nikokoshev Feb 3 '10 at 21:46
• Yes, I do not question your right to edit my post. You are still free to edit it, of course. I am sorry if I sounded too rude. Everything is okay, since I too have the rights for a rollback .. :) – Anweshi Feb 3 '10 at 21:50

## 3 Answers

There is an explicit formula for the zeroes: [Eichler, M.; Zagier, D. On the zeros of the Weierstrass $\wp$-function. Math. Ann. 258 (1981/82), no. 4, 399--407. MR0650945 (83e:10031)]

• Makes me feel somewhat better .. At least it took a paper. – Anweshi Feb 3 '10 at 21:22
• By Eichler and Zagier no less... – Ben Linowitz Feb 3 '10 at 21:26

The branched cover (defined by the Weierstrass function) has degree 2. To obtain $\Bbb C\Bbb P^1$, we need to quotient the elliptic curve $\Bbb C/\Lambda$ by the transformation $z\to -z$, which has order 2.

• Oh, thanks a lot Pavel. This enables me to answer the question which prompted my question. – Anweshi Feb 3 '10 at 21:14

More recently, Duke and Imamoglu expressed the zeros in terms of hypergeometric functions.