Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components in which all the rational points lie only on one component? Concrete examples are really appreciated.

Yes. It is not hard to find an example: Take $$E \colon y^2 = x^3 - 12 x - 1\,.$$ Then $E(\mathbb Q) \cong \mathbb Z$ and $P = (5, 8)$ is a generator (according to Magma). Since $P$ is on the component of the identity, all rational points are on that component.
• For my own understanding, why does the latter statement hold? At least at first glance, it's not clear to me that addition 'respects' components; is it just a continuity argument ($P+0$ is, so $P+(1/n)P$ is, so...)? Jan 23, 2015 at 22:39
• @StevenStadnicki Not sure if this will satisfy you, but $E(\mathbb{R})$ is a compact real Lie group, so it's real-analytically isomorphic to $S^1\times\Phi$, where~$S^1$ is the circle group and $\Phi$ is a finite group. (For elliptic curves, of course, $\Phi$ has to have order 1 or 2.) Anyway, the connected component of $E(\mathbb{R})$ corresponds to the group $S^1$ in this identification. Jan 23, 2015 at 23:09
• In Joe's argument, all you need to know is that $E(\mathbb{R})$ is a topological group; the connected component of the identity is then a subgroup. Jan 24, 2015 at 8:13