# Constructing large rank elliptic curves by multiplying quadratic imaginaries by cubes so that all have same imaginary part

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