Timeline for Why is an elliptic curve a group?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jan 8, 2010 at 22:44 | comment | added | Qiaochu Yuan | I meant to say this, but I forgot to: it certainly did not start with the Picard groups. Euler and Gauss both wrote down group laws for specific curves before anyone had written down elliptic functions. | |
| Jan 8, 2010 at 22:05 | comment | added | Qiaochu Yuan | The point is that the group law you get from the Weierstrass function can be written in terms of rational functions. This is an obvious conjecture to make if you think of elliptic functions as analogous to trigonometric functions, and it's also natural to think of the function field of C/Lambda as one-dimensional, so p(nz) should lie in C(p(z), p'(z)) for all n. Once you get rational functions of course the extension to all fields is clear. | |
| Nov 26, 2009 at 12:12 | comment | added | Mariano Suárez-Álvarez | Elliptic functions were originally introduced as the inverse functions to certain integrals, by the same procedure with which you can construct the exponential and trigonometric functions. Everybody knew that these functions have addition laws, so it is quite natural to expect ellitic functions also have one. | |
| Nov 26, 2009 at 9:04 | comment | added | Jose Capco | this is all very nice and cute in the complex number field. But what is amazing is why an elliptic curve over some arbitrary field is a group. Which makes me wonder, how did mathematicians realized elliptic curve is a group, I believe they suspected that way before the weierstrass p-function was used to consider e.c. over the complex. Like the last poster suggested, I think it all started with the picard groups. Though, I think its easier to defined e.c. over the complex fields to a beginner on the subject. | |
| Nov 26, 2009 at 5:11 | history | edited | Harald Hanche-Olsen | CC BY-SA 2.5 |
Typo
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| Nov 26, 2009 at 5:03 | history | answered | Harald Hanche-Olsen | CC BY-SA 2.5 |