# When did people start thinking of elliptic curves as groups?

I have been reading some old papers of Cassels and Selmer from around 1950, and they talk about generators of rational solutions to elliptic curves, in the sense of Mordell–Weil, but do not appear to use the word group. (Edit: Taking another look, at least some of Cassels' papers from this period do use the word group.)

Weil - L'arithmétique sur les courbes algébriques (footnote 1, p. 281) says:

Afin de réserver le mot de groupe au sens qu'il a pris depuis Galois, je parlerai toujours de systèmes de points, bien qu'on air l'habitude en géométrie algebrique de parler de groupes de points sur une courbe.

Question: When did it become common parlance to call elliptic curves groups?

• I think people might have made the distinction at one time (perhaps even now) between the curve and the group of points, calling it the Mordell-Weil group of the curve. I myself don't understand this distinction, since surely an elliptic curve is a group scheme? Mar 12, 2018 at 20:34
• @DavidRoberts: the distinction is important because a curve (over $\mathbf Q$ say) contains more information than just its group of rational points. For example, there are plenty of elliptic curves over $\mathbf Q$ whose group of rational points is trivial.
– Pop
Mar 13, 2018 at 7:21
• @Pop but saying that an elliptic curve is a group scheme is more information yet, and implies the rational points form a group (even if only the trivial group). Mar 13, 2018 at 10:53
• What does Weil mean? What is "le mot de groupe au sens qu'il a pris depuis Galois" ? Why does an elliptic curve not define a group in that sense, but rather a "system of points". I understand French, but I still do not see what he means.
– Joël
Mar 14, 2018 at 2:36
• @Joël, as no expert on mathematical history, I think that a group in the sense of Galois probably means something like: a subgroup of a permutation group, i.e., a group equipped with a specific action on a previously existing set, rather than just a model for the modern collection of axioms. Mar 15, 2018 at 19:00

## 2 Answers

The first mathematician who talked about groups of points on elliptic curves (in the sense of Galois, i.e., in the modern sense of the word group) was Juel [Ueber die Parameterbestimmung von Punkten auf Curven zweiter und dritter Ordnung. Eine geometrische Einleitung in die Theorie der logarithmischen und elliptischen Funktionen, Math. Ann. 47 (1896), 72-104]. Poincare, in his important article referred to by ThiKu, does not use the concept of groups (this is the point of Schappacher's article). Even Mordell proved his theorem, namely that the group of rational points on an elliptic curve is finitely generated, without using the notion of a group: this was only done by Weil.

For a long time, most people interested in elliptic curves regarded them as a variety of diophantine equations, and preferred thinking about secant and tangent methods instead of group operations. Only when it became clear in the 1960s that the rank of an elliptic curve could be computed by applying all kinds of homomorphisms and determining the orders of kernels and images of such homomorphisms, the group theoretic point of view became indispensible.

• Actually, I read recently that Poincaré didn't call abelian groups "groups", only non-abelian groups. So even if he knew the rational points formed a group, he wouldn't have said it in so many words. Mar 29, 2018 at 21:19
• Juel's paper (especially §2, p. 77) can be found here: He explicitely writes the associativity formula. On p. 78, he writes "Alle die betrachten Punkte bilden also unter Zugrundelegung der Addition eine Gruppe deren Punkten [...] geordnet sind.". Mar 29, 2021 at 13:26

H. POINCARÉ

Sur les propriétés arithmétiques des courbes algébriques

Journal de mathématiques pures et appliquées 5e série, tome 7 (1901), p. 161-234.

http://sites.mathdoc.fr/JMPA/PDF/JMPA_1901_5_7_A7_0.pdf

• 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 13, 2018 at 0:58
• This paper explicitly asks the question: what are the possible values for the rank of the group of rational points? Mar 13, 2018 at 5:46
• Though indeed Poincaré does not use the word „groupe“ but calls the generators „un système des points rationelles fundamentaux“. Mar 13, 2018 at 5:51
• 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 6:09
• 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 14:17