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I've been thinking of the relation between elliptic curves of large rank and quadratic imaginary fields with large 3-rank class groups. There are quite a few papers constructing infinitely many quadratic imaginary fields with class groups having torsion subgroups of the form $\mathbb{Z}_p^r$, by using elliptic or hyperelliptic curves with such rational torsion. I have read these papers intently.

In this question I would like to focus on the other way around - constructing elliptic curves with large rank using quadratic imaginary fields with large 3-rank class groups. I do not recall seeing such a construction (excluding trivial examples for rank 2). Let's get into the specifics of this specific question:

Let $D$ be a negative squarefree number, $K$ the associated quadratic field, and to make notation light assume that $O_K = \mathbb{Z}[\sqrt{D}]$. Say the class group has 3-rank $r$ generated by ideals $I_i$, $I_i^3 = (a_i+b_i\sqrt{D})$, $N(I_i)=x_i$, $i=0,...,r-1$. We have the following equations: $$a_i^2+b_i^2|D| = x_i^3$$

If the numbers $b_i$ were all equal, these equations would give $r$ points $(x_i,a_i)$ on the elliptic curve $$y^2=x^3-b^2|D|$$

And if all was relatively normal, these points would be independent because of the independence in the class group. If the $b_i$'s aren't equal, we can try to move around in the ideals' classes. Meaning, we can multiply the equations by cubes of integers in $O_K$.

Before diving into the arbitrary-large-case, let's start with $r=2$. Therefore the question is this:

Let $I_1, I_2$ be two ideals of order 3 in the class group of a quadratic imaginary order $\mathbb{Z}[\sqrt{D}]$, generating a subgroup of order 9. $I_i^3=(a_i+b_i\sqrt{D})$. Do there exist $\alpha, \beta \in \mathbb{Z}[\sqrt{D}]$ such that: $$\text{Im}( (a_1+b_1\sqrt{D})\alpha^3) = \text{Im}((a_2+b_2\sqrt{D})\beta^3)$$

P.S. The question is simple to state and does not immediately concern elliptic curves, even if the motivation does. I am not looking for papers on construction of large rank elliptic curves or quadratic class groups.

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Is it possible that you meant $a_i^2−b_i^2D=x_i^3$ or $a_i^2+b_i^2|D|=x_i^3$, and accordingly for the following formulae? –  Alex B. Sep 24 '10 at 2:42
Yes, thanks! Editing... –  Dror Speiser Sep 24 '10 at 7:24
You have more freedom: if the class group is generated by I_1 and I_2, then it is also generated by I_1^2 and I_2 or by I_1*I_2 and I_2. If you google for "arithmetic of pell surfaces", you will find that the mysterious connection between the class group and the elliptic curve structure has puzzled other people as well -) –  Franz Lemmermeyer Sep 24 '10 at 8:10
@Mahesh: $-Im(c\beta^3) = Im(c(-\beta)^3)$. @Franz: Sounds good, thanks! –  Dror Speiser Sep 24 '10 at 9:11
@Dror.. of course. my bad. –  Mahesh Kakde Sep 24 '10 at 9:14
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