Talk:Weierstrass elliptic function

The homogenity relation doesn't seem to work for all c since cτ may not remain in the upper half plane. I don't quite understand how this relation can serve to define doubly periodic functions with arbitrary period pairs.

It would probably be better to choose parameter Λ = periodic lattice initially, rather than τ. Taking the reciprocal of τ would fix up the imaginary part, and that's taking the basis for periods in the other order. That in fact is just a special case of the PSL (2,Z) action that is implicit in choice of general basis, and which means τ can be taken to be in the usual fundamental domain within the upper half plane. But I'd agree this is tough on the reader - does two steps at once. Charles Matthews 19:31, 10 Feb 2004 (UTC)

Also, the statement that all doubly periodic functions with given periods form a field C[P, P'] is not quite clear to me. Does it mean that any such function can be expressed as a rational function in P and P'? AxelBoldt 23:57, 9 Feb 2004 (UTC)

Given a period lattice Λ, what is true is that all meromorphic functions periodic under Λ form a field that is actually C(P, P'), i.e. rational functions in the Weierstrass P and its derivative. The ring notation C[P, P'] would stand for polynomials in P and its derivative; this is enough to generate the functions with singularities only at the points of Λ. To get from there to the general result requires an argument, I guess, like this:

- the general function F will have a finite number of poles, mod Λ

- construct directly a function G_w having a simple pole at 0 and general point w, only;

- given F, subtract off some translates of P and derivative to get a function only with simple poles, and then express that as a linear combination of functions G_w, plus a function with no singularities;

- an elliptic function with no singularity is constant by Liouville's theorem.

The usual construction of a G_w function is as a difference of Weierstrass zeta-functions. This looks like the most serious step.

Charles Matthews 08:34, 10 Feb 2004 (UTC)

I've made some minimal changes to sort out the page. In a sense I think this page should be about Pe-related formulae, and the general elliptic function and elliptic curve theory should live somewhere else.

Charles Matthews 08:57, 10 Feb 2004 (UTC)

The Weierstrass P function should have a double pole at z=0, right? your definition seems to be missing that term. should it read

${\mathcal {P}}(z;\tau )={\frac {1}{z^{2}}}+\sum _{n^{2}+m^{2}\neq 0}{1 \over (z-n-m\tau )^{2}}-{1 \over (n+m\tau )^{2}}$