Timeline for Why is an elliptic curve a group?
Current License: CC BY-SA 4.0
12 events
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| S Feb 12 at 9:46 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to Wikipedia
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| Feb 12 at 7:52 | review | Suggested edits | |||
| S Feb 12 at 9:46 | |||||
| Apr 7, 2010 at 13:54 | comment | added | Keenan Kidwell | Strictly speaking, an elliptic curve (over $\mathbb{C}$) is not a complex torus. Its set of complex points, which constitute a compact Riemann surface of genus one, is a complex torus. All of this depends on your perspective regarding just what an elliptic curve is. For instance, not every curve of genus one is an elliptic curve, unless you only work over algebraically closed fields, the point being that a genus one curve over a field which is not algebraically closed may fail to possess a rational point. | |
| Jan 8, 2010 at 23:52 | comment | added | Anweshi | Also I have referred to the simpler books of Miranda and Narasimhan, with the questioner in mind. These books use only complex analysis of one variable, and the definition of genus from topology. | |
| Jan 8, 2010 at 23:22 | comment | added | Anweshi | Not to mention that Riemann-Roch would give the group law over any field/ring/scheme. You have done some handwaving with the the Lefschetz principle for transporting to over other fields. This is not quite enough. Riemann-Roch is much neater. | |
| Jan 8, 2010 at 23:20 | comment | added | Anweshi | Riemann-Roch would tell you why there is a group law precisely on curves of genus 1. Also, Riemann-Roch is obvious more profound than just looking at a torus or Weierstrass elliptic functions on the complex plane. | |
| Jan 8, 2010 at 22:58 | history | edited | Ilya Nikokoshev | CC BY-SA 2.5 |
reworded
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| Jan 8, 2010 at 22:46 | comment | added | Ilya Nikokoshev | Well, if you are sure that the poster is familiar with elliptic curves, line bundles and sheaf cohomology (which I doubt), then, yes, Riemann-Roch theorem would give the group law. But why exactly this is more profound than curve being a complex torus? | |
| Jan 8, 2010 at 22:42 | history | edited | Ilya Nikokoshev | CC BY-SA 2.5 |
expanded
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| Jan 8, 2010 at 22:33 | history | edited | Ilya Nikokoshev | CC BY-SA 2.5 |
expanded
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| Jan 8, 2010 at 22:30 | comment | added | Anweshi | You missed the whole central idea which the the deepest: The Riemann-Roch theorem gives the group law. | |
| Jan 8, 2010 at 22:27 | history | answered | Ilya Nikokoshev | CC BY-SA 2.5 |