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How did André Weil react when Pierre Deligne finally solved the most important and hardest of the Weil conjectures ?

Is there any written account on this ?

I guess Serre's and Grothendieck's (unfortunate) reaction are well-known, but I have never seen anything written about Weil's own experience on the whole matter.

Mozibur Ullah
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    What do you mean by "I guess Serre's and Grothendieck's (unfortunate) reaction are well-known ..." ? do you have a reference for that statement ? –  Jun 16 '20 at 19:49
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    I don't think G's reaction was 'unfortunate' at all. What is unfortunate is that there aren't more straight-talking mathematicians. – R.P. Jun 16 '20 at 19:54
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    for Grothendieck's reaction, see page 4 of https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Spirituality/Spirituality10.pdf – Carlo Beenakker Jun 16 '20 at 20:21
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    Stop being so stuck up -- you all know what he means. –  Jun 16 '20 at 21:25
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    Quoting from @CarloBeenakker’s link: ‘[Grothendieck] was both fascinated and disappointed. Deligne’s proof was very different from what he had imagined. Deligne had succeeded in avoiding the complete construction of the “right” cohomology, and he had not proven the “standard conjectures”. […] Deligne commented […] “If I had done it using motives, he would have been very interested, because it would have meant that the theory of motives had been developed. Since the proof used a trick, he did not care.”’ (from Part 3 of Winfried Scharlau Who is Alexander Grothendieck, trans. Melissa Schneps) – Peter LeFanu Lumsdaine Jun 17 '20 at 11:13
  • I'm more interested in Deligne's reaction - he must have been delighted! – Mozibur Ullah Sep 26 '21 at 05:29

2 Answers2

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In the commentary on his 1954 paper "Abstract versus classical algebraic geometry" [1954h], p. 539 in volume II of his "Œuvres scientifiques", Weil wrote:

Quant à mes commentaires sur l'hypothèse de Riemann en dimension > 1, cf. [1941]. Mon espoir qu'on la tirerait d'un théorème de positivité valable en géométrie algébrique abstraite ne s'est pas trouvé vérifié, malgré une ingénieuse suggestion de Serre (Ann. of Math. 71 (1960), pp. 392–394); la démonstration de Deligne est toute arithmétique.

I read the last sentence 30 years ago, and never forgot it.

(English translation: As for my comments on the Riemann hypothesis in dimension > 1, cf. [1941]. My hope that it would be obtained from a positivity theorem in abstract algebraic geometry has not been realised, despite an ingenious suggestion of Serre (Ann. of Math. 71 (1960), pp. 392–394); Deligne’s proof is entirely arithmetic.)

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My recollection is that, when Deligne completed the proof of the Weil conjectures, Weil was in Paris and Deligne went to his apartment and explained the proof to him personally. Deligne was on good terms with Weil. Grothendieck was also in Bures around that time, and I remember seeing Messing explaining the proof to him. I think both Grothendieck and Weil reacted positively, although Grothendieck was disappointed that Deligne hadn't proved his standard conjectures, which remain open to this day. As the quote in Merel's answer indicates, Weil also was surprised that Deligne hadn't deduced it from some general positivity statement in algebraic geometry.