1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Weierstrass p function

  1. Jun 11, 2009 #1

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    1. The problem statement, all variables and given/known data
    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?


    2. Relevant equations
    We have [tex] p'(z)^2 = 4(p(z)-e_1)(p(z)-e_2)(p(z)-e_3)[/tex] where [itex]e_1, e_2, e_3[/itex] are [itex]p(w_1/2), p(w_2/2), p(w_1+w_2/2)[/itex] 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 [itex]w_1[/itex] and [itex]w_2[/itex]


    3. 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

    [tex] 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)][/tex]

    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


    [tex] 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)][/tex]

    which doesn't offer any immediate suggestions. After this I'm stuck
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted