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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?

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3 Answers

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.

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Thanks a lot. I'll try to find that book. If all else fails, I'll look nt up on Amazon. –  potap Nov 13 '12 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 '12 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 '12 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 '12 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 '12 at 4:45
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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.

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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 '12 at 14:53
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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.

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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 '12 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 '12 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 '12 at 14:13
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