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In the paper Sur les représentations modulaire de degré 2 de Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$, Serre makes the following comment:

Remarque. La relation existant entre "solutions de l'équation de Fermat" et "points de $p$-division de certaines courbes elliptiques" figure déjà dans un travail de Hurwitz ([20]) datant de 1886.

(Trans.: The connection between “Solutions of Fermat’s equation” and “points of order $p$ on certain elliptic curves” is already in work of Hurwitz from 1886)

Hurwitz’s paper is Über endliche Gruppen linearer Substitutionen, welche in der Theorie der elliptischen Transcendenten auftreten. I’m unable to read it myself. Can anyone shed light on Serre’s comment? How did Hurwitz connect rational points on high-genus curves to points of finite order on elliptic curves?

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On page 322 of

Serre, Jean-Pierre, The works of Wiles (and Taylor,(\dots)). I., Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 319-332, Exp. No. 803 (1996). ZBL0957.11027.

one reads:

Je profite quand même de l’occasion pour rectifier une assertion de [30], fin du n° 4.2: “La relation existant entre solutions de l’équation de Fermat... figure déjà dans un travail de Hurwitz...”. C’est faux, comme me l’a signalé N. Schappacher: il n’y a rien de tel dans Hurwitz. Mea culpa.

(Trans: I will take this opportunity to correct an assertion in [30], end of no. 4.2: “The relationship between solutions of Fermat’s equation... already appears in a work by Hurwitz...”. This is false, as N. Schappacher pointed out to me: there is nothing like that in Hurwitz. Mea culpa.)

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    $\begingroup$ Oh, wow! That was... unexpected. One has to wonder what sequence of events has led to an expert such as Schappacher having to point out to no less than Serre that something like that isn't in Hurwitz. I can't even begin to portray a story... $\endgroup$
    – Alon Amit
    Commented Jul 27 at 19:49
  • $\begingroup$ Anyway, @Francois, thank you so much for clearing this up for me! $\endgroup$
    – Alon Amit
    Commented Jul 27 at 20:41
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Serre's Bourbaki article cites:

  • Joseph Oesterlé, Nouvelles approches du « théorème » de Fermat, Séminaire Bourbaki : volume 1987/88, exposés 686-699, Astérisque, no. 161-162 (1988), Talk no. 694, 22 p. (Numdam)

Oesterlé credits Hellegouarch with the first connection between Fermat and elliptic curves.

In fact, Hellegouarch's own book contains a fascinating and detailed appendix titled "The Origin of the Elliptic Approach to Fermat’s Last Theorem":

  • Yves Hellegouarch, Invitation to the Mathematics of Fermat-Wiles, Academic Press, 2001.
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  • $\begingroup$ Wow, I've never seen a 'withdrawn' book on Science Direct before: sciencedirect.com/book/9780123392510/… I guess Hellegouarch declined to make it electronically available? And I can't even buy a hardcopy from the Elsevier shop... $\endgroup$
    – David Roberts ♦
    Commented Aug 29 at 8:57
  • $\begingroup$ I don't know if you read French, but the 2nd edition of the French version (ed. : Dunod) is quite easy to get. $\endgroup$
    – PseudoNeo
    Commented Aug 29 at 11:41
  • $\begingroup$ I borrowed it from the Internet Archive library. $\endgroup$
    – David Espinosa
    Commented Aug 30 at 9:12

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