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