Let $E$ be an elliptic curve which has CM and $K$ the imaginary quadratic field in which $End(E)$ is an order. I have 3 questions, which are somehow related to each other:
- If one considers $E(K)$ as a module over $End(E)$ instead of $\mathbb{Z}$, and $E(K)$ has rank $n$ over $End(E)$, will it have rank $2n$ over $\mathbb{Z}$?
- Is it true that the rank of the Mordell-Weil group (i.e. rank of $E(K)$ as a $\mathbb{Z}$ module) of such a curve is even? (of course, second question is clearly true if the answer to the first one is affirmative)
- If questions 1) and 2) fail and $E(K)$ has rank $2$ as a $\mathbb{Z}$-module, how can I test if it has rank $1$ as an $End(E)$ module?
All the examples that I have come accross in Cremona's tables and another sample of $500$ curves tested with Magma satisfy the second assertion (it is maybe worth pointing out that almost all of these examples were curves which can be defined over $\mathbb{Q}$)
Clearly $A=E(K)/E(K)_{\text{tors}}$ is a free Abelian group, and also a module over an order $\mathfrak O$ in a the quadratic field $K$. Then $A$ must have even rank: $A\otimes_{\Bbb Z}\Bbb Q=A\otimes_{\mathfrak O}K$ has dimension over $\Bbb Q$ twice the dimension over $K$.