## Does Weierstrass function $\wp$ parametrize all solutions? [closed]

Consider the curve $$\mathbb{C}: y^2 = x^3 + Ax + B$$ (s.t. $4A^3 + 27B^2 \neq 0$) over the field of complex numbers. Then there is a Weierstrass function $\wp(z)$ (that depends on $A, B$) such that $$(\wp'(z)/2)^2 = \wp^3(z) + A\wp(z) + B$$ holds (here, $g_2 = 4A, g_3 = 4B$ are fixed). Thus $\wp(z)$ and its derivative parametrize a set of points on the above curve. Are all points obtained in this way? In other words, is the set of pairs $(\wp(z), \wp'(z)/2)$ (when $z$ runs through the complex numnbers) a (1-dimensional) variety?

Thank you.

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You have received several votes to close this question, presumably because the answer to this question is supposed to be covered in any introductory textbook that talks about the Weierstrass function. The answer is yes: the variety you obtain is always a torus, obtained by gluing along the edges of the fundamental lattice. But since the Weierstrass function is actually singular on a whole lattice in the complex plane, your question is not quite as simple as it seems, and those voting to close really ought to have left a comment! – Jacques Carette Nov 1 2011 at 23:42
Some people (not me) are voting to close this, probably because for them it is well-known and/or obvious. (I'm speculating, because no one has given their reasons for closing.) I don't see anything objectionable or inappropriate about your question; but if you can give some idea about how you've come at this question, it might help generate a friendlier response. – Charles Rezk Nov 1 2011 at 23:44
Oops, voted to close before leaving comment why. This material is covered in any basic text (Lang's, mine, probably Knapp's, Ahlfors(?),...). If the poster has checked a bunch of books, he needs to say so before posting, or his questions will be closed. If he's posting before doing some digging himself, he's not using MO properly. As for poles on the lattice, yes, one has to know the basics about meromorphic functions and the projective plane, but again, these are covered in basic courses and books. So I hope this explains why closed. (We need a "closed because covered in basic texts" choice.) – Joe Silverman Nov 2 2011 at 0:03
This is a well known fact and you can find the proof in any introductory textbook on elliptic curves and Riemann surfaces. For instance, see [Jones-Singerman, Complex Functions, Lemma 3.16.1 page 109] – Francesco Polizzi Nov 2 2011 at 0:07
Fair enough. I have come up with an accessible answer in the meanwhile (it becomes obvious once one has that the function is surjective and even). Thank you for your comments. – Albertas Nov 2 2011 at 21:36