Timeline for When did people start thinking of elliptic curves as groups?
Current License: CC BY-SA 3.0
7 events
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| Mar 13, 2018 at 14:20 | comment | added | Kimball | Of course it's hard to measure a conceptual difference, so I meant to ask when people started using the word "group" (regularly) to describe the rational points on an elliptic curve. However, the question you answer is also interesting, and a valid interpretation of the titular question, so I'm happy to see this answer as well. | |
| Mar 13, 2018 at 14:17 | comment | added | Kimball | Sorry for the lack of clarity---I was indeed aware my title might be misleading, but I thought it sounded better than something like "when did people start regularly referring to the $F$-points of an elliptic curve as a group"? I agree that there is not too much difference in practice whether one calls something a group or not, but I think there is a conceptual difference between thinking of $F$-points as solutions that can be geometrically constructed from some "fundamental solutions" versus an instance of this algebraic framework of groups. | |
| Mar 13, 2018 at 6:09 | comment | added | ThiKu | It seems the title of your question „When did people start thinking of elliptic curves as groups“ asks a different question then it is asked in the question‘s body. The only difference I would see between looking at finitely many group generators and looking at a finitely generated group would be that in the second case one can use functoriality, i.e., homomorphisms between different elliptic curves. Is this what you are aiming at? | |
| Mar 13, 2018 at 5:51 | comment | added | ThiKu | Though indeed Poincaré does not use the word „groupe“ but calls the generators „un système des points rationelles fundamentaux“. | |
| Mar 13, 2018 at 5:46 | comment | added | ThiKu | This paper explicitly asks the question: what are the possible values for the rank of the group of rational points? | |
| Mar 13, 2018 at 0:58 | comment | added | Kimball | Are you saying this is the first time an elliptic curve was considered a group or that it became common soon after Poincare's paper? The reason for my question is that I found it strange that even in the 1950's some papers were not explicitly saying things like "the group of rational points." | |
| Mar 12, 2018 at 20:39 | history | answered | ThiKu | CC BY-SA 3.0 |