In Lawrence Washington's book Elliptic Curves: Number Theory and Criptography I read that if $E$ is an elliptic curve defined over the real numbers $\mathbb{R}$ then the set of real points $E(\mathbb{R})$ can be obtained as the intersection of the torus of complex points $E(\mathbb{C})$ and a plane. The following is the relevant page of the book.
Basically it says that if the plane passes through the hole in the torus, then the real locus of the elliptic curve looks like the following
and if not, it looks like this one
So passing through the hole in the middle seems to determine whether the graph of the elliptic curve has two real components or just one. I have never seen such a claim before in other books about elliptic curves, and I'm really curious about it. Thus I have a couple of questions about this.
- Can this claim about the real locus being the intersection of a torus with a plane be made precise somehow, and if so can anyone please provide an explanation about it?
- If this is possible, can an explicit example be given?
Notes:
- I have to say that I'm really confused about this because the correspondence of an elliptic curve with a torus (when considering the complex points $E(\mathbb{C})$) is given by an isomorphism with the complex modulo a lattice, so even thinking about intersecting "the torus" with a plane seems rather odd to me.
- The reference to section 9.3 in the book does not seem to clarify this, it basically deals with the identification of the elliptic curve with the torus.
Thank you very much for any help with these questions.