Meromorphic differential

[sylar]

This semester I'm taking "Introduction to Algebraic Curves" course. Up to now, the only problems i have with this course are the notion of meromorphic(and holomorphic) differentials, and coordinate charts. I'm good with the algebraic ideas. Here is one question from the book we are studying:

Let w be a meromorphic differential on a Riemann surface C. Show that we can choose an appropriate coordinate chart so that w = (z^u)dz, u is an integer, in some neighborhood of a pole. Prove also that this integer u is independent of the coordinate chart selected.

Actually i could prove the uniqueness of the number u for the function version of the statement(in a nbd. of a pole, a meromorphic function f on a Riemann surface can be expressed as f = (z^u')g(z), where u' is an integer, g(z) is a holomorphic function and z(q)=0 for this pole p). But i have trouble with proving the statement for the meromorphic differentials.

I know that a meromorphic differential f dg is represented on the coordinate chart
phi_i :U_i -> V_i by the meromorphic function f((phi_i)^(-1)) g((phi_i)^(-1)) but don't know how to use it for this question.

Instead i tried to prove the result as follows:

Let w be a meromorphic differential which have the local representation

w = p(z)dz = ((a_n) z^n + (a_(n+1)) z^(n+1) + ...)dz, where a_n is non-zero and n is a negative integer. Then w has a pole of order n at the point q. Let z = f(w) be a mapping such that f(0)=0 and f'(0) is nonzero.Then we get gbar(w) = g(z)dz = g(f(w))(df/dw). So,

lim (w->0) (w^(-n))*gbar(w) = lim (w/f(w))^(-n)*[f(w)^(-n) g(f(w))(df/dw)
= f'(0)* lim (z->0) (z^(-n)) p(z).

Here f'(0) is nonzero by construction, and thus the last expression is finite and non-zero when so is lim (z^(-n)) p(z) (this can be proved easily). So this number n, aka u in the original quesion is independent of the local chart.

Does this prove the statement? Is there another good way of showing the desired result? Thanks!

 

[jvicens]

Hello, thank you for sharing your question and progress with us. It seems like you have made good progress in understanding the uniqueness of the integer u for meromorphic functions and are now trying to extend that to meromorphic differentials. Your approach of using a mapping f to relate the two is a good idea.

To prove the statement for meromorphic differentials, we can use a similar approach as you did for meromorphic functions. Let w be a meromorphic differential with a pole of order n at a point q. Then, as you mentioned, it has a local representation of the form w = p(z)dz where p(z) is a meromorphic function with a pole of order n at z=0.

Now, let z=f(w) be a mapping such that f(0)=q and f'(0) is nonzero. Then, we can express w in terms of f as w = f'(0)g(f(w))(df/dw) where g(z) is a holomorphic function. This is similar to the expression you used for gbar(w) in your approach.

Now, we can use the fact that f'(0) is nonzero to show that the order of the pole of w at q is equal to the order of the pole of g(f(w)) at z=0. This can be done by considering the Laurent series expansion of g(f(w)) around z=0 and using the fact that f'(0) is nonzero.

Therefore, we have shown that the integer u, which represents the order of the pole of w, is independent of the coordinate chart chosen. This proves the statement for meromorphic differentials.

I hope this helps. Let me know if you have any further questions or if you would like me to clarify anything. Good luck with your course!