Welcome to MathOverflow

Question

1

If E is a complex elliptic curve defined as the quotient of C over a lattice generated by w_1 and w_2, then it can be also written in Weierstrass form y^2=4*x^3-g_2*x-g_3. The coefficients g_2 and g_3 can be computed as well-known Eisenstein sums, however, there is a better expression in terms of Jacobi theta-functions of w_1 and w_2 (and as a consequence, an expression for j-invariant via w_2/w_1). Unfortunately, I could not find that expression in the books on my bookshelf, like Koblitz, Milne, Silverman etc, and the only place on the Internet where I found such formulas, were two wiki-articles: http://en.wikipedia.org/wiki/Weierstrass's_elliptic_functions and http://en.wikipedia.org/wiki/Theta_function. These formulas, however, contradict each other, and neither of them, when run on computer, gives rational values for elliptic curves with complex multiplication and class number 1. Can anybody give a reliable link to correct formulas?

flag	asked Nov 13 at 6:22 potap 29●3

Robert Bryant · Answer 1 · 2012-11-13 12:41:06Z UTC

1

I recommend Lester R. Ford's classic book "Automorphic Functions" for this. It gives very explicit formulae for all these quantities, in particular, the $J$-invariant, and it is written with great care in a very readable style.

answered Nov 13 at 12:41

Robert Bryant
21.8k●3●75●97

	Thanks a lot. I'll try to find that book. If all else fails, I'll look nt up on Amazon. – potap Nov 13 at 13:39
	I found a pdf file on the Internet. The book is VERY good, but it does not give any formulas for $j$ in terms of theta-functions of $\tau$. – potap Nov 13 at 14:35
	Oh, sorry. I was working from memory because I don't have a copy of Ford here at home. I thought it was there. Did you try Herb Clemens' book A Scrapbook of Complex Curve Theory? Perhaps the formulae you want might be in there. If I didn't see it in Ford, then it's possible that I saw it in Clemens. – Robert Bryant Nov 13 at 15:32
	@potap: I finally had a chance to look in Clemens' book, and it turns out that the formulae that you want (for the $j$-function in terms of theta functions) are in Chapter III. I'm sorry for mis-directing you to Ford. – Robert Bryant Nov 13 at 23:58
	@Robert: Shame on me! I turned back on my chair and immediately found a paperback copy of that book in Russian translation. How come I did not think about it? (Imagine that I know Herb personally and I have even participated in a party at his home...) Thanks a lot! – potap Nov 14 at 4:45

Alexandre Eremenko · Answer 2 · 2012-11-13 13:39:30Z UTC

1

A good modern source in English is N. Akhiezer, Elements of the theory of elliptic functions. The formulas you are asking are in section 21, chapter IV.

answered Nov 13 at 13:39

Alexandre Eremenko
10.6k●1●24●39

Thanks a lot. I managed to find a dejaview file of that book in Russian. This is exactly what I needed! The best. – potap Nov 13 at 14:53

S. Carnahan · Answer 3 · 2012-11-13 09:41:10Z UTC

0

The formulas at the Wikipedia article on the J-invariant seem to be correct. At least, the formula $g_2 = \frac{2\pi^4}{3}(\theta_{00}^8 + \theta_{01}^8 + \theta_{10}^8)$ yields $g_2 = 60G_4 = \frac{4\pi^4}{3}E_4$, as one would expect from the usual coordinate representation of the $E_8$ lattice.

I don't know what you mean when you say that the expression of $j$ in terms of Jacobi theta functions is "better". Each expression has its advantages.

answered Nov 13 at 9:41

S. Carnahan♦
25.6k●41●90

	Thank you. But I think that $g_2$ cannot be expressed in terms of $\tau$ alone: it depends on both periods $\omega_1$ and $\omega_2$ and can change under homotheties! Do you mean that $\omega_1=1$ and $\omega_2=\tau$? By "better" I mean computational complexity, as I'm doing computer experiments. – potap Nov 13 at 10:06
	The standard convention is $\omega_1 = 1$ and $\omega_2 = \tau$. You can adjust for arbitrary pairs of periods using the fact that weight $k$ forms are homogeneous of degree $-k$ under homothety. – S. Carnahan♦ Nov 13 at 13:43
	Yes I know. However, you can take Weierstrass's $P$ and $P'$ for any lattice and write the relation between them which will depend both on $\omega_1$ and $\omega_2$ (the equivalence class of the curve will be defined by the ratio alone, of course). – potap Nov 13 at 14:13

Express Weierstrass' g_2 and g_3 in terms of theta-functions of the periods

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.