Degree and Ramification Points of the Weierstrass P Function on the Torus T

[Office_Shredder]

Homework Statement

Consider the meromorphic function p' as a function from the torus T to the sphere S. What is its degree? How many ramification points does it have?

Homework Equations

We have $$p'(z)^2 = 4(p(z)-e_1)(p(z)-e_2)(p(z)-e_3)$$ where $e_1, e_2, e_3$ are $p(w_1/2), p(w_2/2), p(w_1+w_2/2)$ where we define the torus as an equivalence relation on the complex plane where z~w if z-w is on the lattice generated by $w_1$ and $w_2$

The Attempt at a Solution

p'(z) has a single triple pole at 0 so is of degree 3. Then by Riemann-Hurwitz (X Euler Characteristic), X(T) = 3X(S) -sum of (ramification indices - 1) So sum of (ramification indices - 1) = 6
We differentiate the above equation to get

$$p''(z)p'(z) = 2p'(z)[(p(z)-e_1)(p(z)-e_2) + (p(z)-e_1)(p(z)-e_3) + (p(z)-e_2)(p(z)-e_3)]$$

Since p(z) has a double pole at 0, p''(z) has a quadruple pole which means 1/p'' has a quadruple zero at z=0. So that takes care of 3 out of the six in the R.H. equation. After that, if p'(z) is non-zero, we get


$$p''(z) = 2[(p(z)-e_1)(p(z)-e_2) + (p(z)-e_1)(p(z)-e_3) + (p(z)-e_2)(p(z)-e_3)]$$

which doesn't offer any immediate suggestions. After this I'm stuck

 

[mighty2000]

.


Hello,

Thank you for your post. I would like to offer a solution to your problem. First, we need to clarify some terms and assumptions. The torus T is defined as an equivalence relation on the complex plane, where z~w if z-w is on the lattice generated by w_1 and w_2. This means that the torus T is a quotient space of the complex plane by the lattice generated by w_1 and w_2. Therefore, the torus T is topologically equivalent to a square with opposite sides identified, known as a complex torus.

Now, let's look at the meromorphic function p' as a function from the torus T to the sphere S. This means that p' is a function that maps points on the torus T to points on the sphere S. We can think of this as a projection from the torus T onto the sphere S. Since the torus T is topologically equivalent to a square, we can think of p' as a function from a square to a sphere, which is known as a branched covering map.

The degree of a branched covering map is defined as the number of pre-images of a generic point on the target space. In this case, a generic point on the sphere S has three pre-images on the torus T, since p' has three poles at e_1, e_2, and e_3. Therefore, the degree of p' is 3.

Now, let's consider the ramification points of p'. These are the points on the torus T where the derivative p'(z) vanishes. From the given equation, we can see that p'(z) vanishes at the three poles e_1, e_2, and e_3. In addition, we can also see that p'(z) vanishes at the point z=0, since p(z) has a triple pole at 0. Therefore, p' has four ramification points on the torus T.

In conclusion, the degree of p' as a function from the torus T to the sphere S is 3, and it has four ramification points on the torus T. I hope this helps to clarify your doubts. Keep up the good work as a scientist!