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.
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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.
answered Jan 23, 2015 at 21:33
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$\begingroup$ 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...)? $\endgroup$ Commented Jan 23, 2015 at 22:39
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9$\begingroup$ @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. $\endgroup$ Commented Jan 23, 2015 at 23:09
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7$\begingroup$ @StevenStadnicki A line that intersects the finite component must intersect it twice, so (by the geometric interpretation of the group law) if P and Q are on the infinite (identity) component then so are P+Q, -P. For E,P above (we're told that) P generates all of E(Q). $\endgroup$ Commented Jan 24, 2015 at 1:05
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$\begingroup$ @JoeSilverman, David: Those are both excellent explanations - it's great to get both the group-theoretic and geometric versions; thank you! $\endgroup$ Commented Jan 24, 2015 at 1:21
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8$\begingroup$ 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. $\endgroup$ Commented Jan 24, 2015 at 8:13