So I've been reading about elliptic curves with CM recently. I am aware of the following theorem:
Let $E/\mathbb{C}$ be an elliptic curve and let $\Lambda=\mathbb{Z}\oplus\mathbb{Z} \tau$ the corresponding lattice. We have two possible cases:
- $End(E)\cong\mathbb{Z}$
- $\mathbb{Q}(\tau)$ is a quadratic imaginary extension of $\mathbb{Q},$ and $End(E)$ is isomorphic to an order of $\mathbb{Q}(\tau)$
I understand the theorem completely, if we have an elliptic curve over $\mathbb{C}$ with CM, then $End(E)$ is isomorphic to an order of $\mathbb{Q}(\tau).$
My question is, what happens when we have an elliptic curve defined over an algebraic number field? As I understood, the same thing is true, $End(E)$ is isomorphic to an order of $\mathbb{Q}(\tau)$ (where $\tau$ is from the corresponding lattice as we look at the same elliptic curve defined over $\mathbb{C}$), I just can't seem to find results about it. Is it even true? Can somebody help me with that?