Is the real locus of an elliptic curve the intersection of a torus with a plane?

In Lawrence Washington is publication Elliptic Curves: Number Theory and Criptography I read that if $E$ is an elliptic contour specified over the actual numbers $\mathbb{R}$ after that the set of actual factors $E(\mathbb{R})$ can be gotten as the junction of the torus of intricate factors $E(\mathbb{C})$ and also an aircraft. The following is the pertinent web page of guide.

Basically it claims that if the aircraft travels through the opening in the torus, after that the actual locus of the elliptic contour resembles the adhering to

and also otherwise, it resembles this set

So travelling through the opening between appears to establish whether the chart of the elliptic contour has 2 actual parts or simply one. I have actually never ever seen such a case prior to in various other publications concerning elliptic curves, and also I'm actually interested concerning it. Hence I have a number of inquiries concerning this.

  1. Can this case concerning the actual locus being the junction of a torus with an aircraft be made specific in some way, and also if so can any person please give a description concerning it?
  2. If this is feasible, can a specific instance be offered?

Notes:

  1. I need to claim that I'm actually perplexed concerning this due to the fact that the document of an elliptic contour with a torus (when taking into consideration the facility aims $E(\mathbb{C})$) is offered by an isomorphism with the intricate modulo a latticework, so also thinking of converging "the torus" with an aircraft appears instead weird to me
  2. The reference to area 9.3 in guide does not appear to clarify this, it primarily manages the recognition of the elliptic contour with the torus

Thank you significantly for any kind of aid with these inquiries.

17
2022-06-07 14:30:22
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Answers: 2

It appears to me that the flow in Washington is misdirecting in one regard. If you have a torus $T$ in $\mathbb{R}^3$ and also converge it with an aircraft which misses out on the opening, the resulting junction will certainly be contractible in $T$. Nonetheless, the actual locus of an elliptic contour is NOT contractible within that contour.

In case where the actual locus has one part, the elliptic contour resembles $\mathbb{C}/\Lambda$ where $\Lambda$ is the latticework created by $1$ and also $1/2 + i \tau$ for some actual $\tau$. A basic domain name for this latticework is a rhombus, with vertices $0$, $1/2 + i \tau$, $1$ and also $1/2 - i \tau$. The actual locus of the contour is the diagonal of this rhombus ranging from $0$ to $1$. Specifically, it is not contractible.

Apart from that, I certainly concur with Matt E.'s solution.

20
2022-06-08 04:37:13
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The facility factors of the elliptic contour hinge on the projective aircraft over $\mathbb C$. Intending that the coefficients of the elliptic contour remain in $\mathbb R$, it after that makes excellent feeling to converge the contour with the projective aircraft over $\mathbb R$. If we neglect the factor at infinity, after that we can simply assume that we are converging a particular pierced torus in $\mathbb C^2$ with $\mathbb R^2$.

Certainly, the torus is not installed as nicely as aware you uploaded (or,. extra properly, it is not a lot an inquiry of "neatness" of the embedding, yet instead the reality that the torus is ingrained holomorphically in $\mathbb C^2$,. which as an actual manifold is of measurement 4 ),. yet the summary of the junction is however proper.

Included: David Speyer is solution below makes a vital factor concerning the geography of the junction, which I really did not resolve in my solution.

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2022-06-07 14:44:41
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