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I have seen that in Grothendieck's paper "THE COHOMOLOGY THEORY OF ALGEBRAIC VARIETIES", he says "The need of a theory of cohomology for 'abstract' algebraic varieties was first emphasied by Weil, in order to be able to give a precise meaning to his celebratied conjectures in Diophantine Geometry".

Following Grothendieck's reference, I find Weil's paper "Abstract versus classical algebraic geometry". However, it seems that Weil didn't mention such a cohomology theory.

I want ask if there are indeed some papers/letters which show Weil told Serre/Grothendieck about such a theory?

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    $\begingroup$ The paper I would believe is the one that mentions somehow what you ask is: "Numbers of solutions of equations in finite fields", Bull. Amer. Math. Soc. 55(5): 497--508 (May 1949), especially the last paragraph on page 507 where he mentions the conjectural explanation of what he calls Betti numbers for a not yet known (at that time) cohomology theory. $\endgroup$
    – F Zaldivar
    Commented Aug 5, 2022 at 3:35
  • $\begingroup$ This makes sense. Thanks. $\endgroup$
    – Martin Tang
    Commented Aug 5, 2022 at 5:43
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    $\begingroup$ You can't --- he didn't explicitly suggest the existence of a good cohomology theory in characteristic p>0. In fact, it is not clear whether Weil believed there should be such a cohomology theory. He certainly knew that the existence of a "Weil cohomology" would explain much of the "Weil conjectures" (BAMS 1949, and his commentaries in his collected works), and this provided the impetus for people to search for such a cohomology theory. $\endgroup$
    – user483792
    Commented Aug 5, 2022 at 11:37

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As already mentioned in the comments it is very likely that Weil's heuristics appear in "Numbers of solutions of equations in finite fields", Bull. Amer. Math. Soc. 55(5): 497--508 (May 1949). But as Grothendieck says in Récoltes et Semailles:

L’un des plus anciens et des plus cruciaux de ces invariants [topologique], introduits déjà au siècle dernier (par le mathématicien italien Betti), est formé des différents “groupes” (ou “espaces”) dits de “cohomologie”, associés à l’espace. Ce sont eux qui interviennent (surtout “entre les lignes”, il est vrai) dans les conjectures de Weil, qui en font la “raison d’être” profonde et qui (pour moi du moins, “mis dans le bain” par les explications de Serre) leur donnent tout leur sens. Mais la possibilité d’associer de tels invariants aux variétés algébriques “abstraites” qui interviennent dans ces conjectures, de façon à répondre aux desiderata très précis exigés pour les besoins de cette cause-là — c’était là un simple espoir. Je doute qu’en dehors de Serre et de moi-même, personne d’autre (pas même, et surtout, André Weil lui-même !) n’y croyait vraiment...

The earlier reference I found relevant to the discussion is J. P. Serre "Sur la topologie des variétés algébriques en caractéristique $p$", Symposium internacional de topología algebraica, Mexico (1958), pp. 24–53. Just published the same year as Grothendieck's talk at the ICM.

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    $\begingroup$ A rough translation: ‘One of the oldest and most crucial of these [topological] invariants, introduced already in the last century (by the Italian mathematician Betti), consists of the various “groups” (or “spaces”) called “cohomology”, associated to the space. It is those that occur (especially “between the lines”, it’s true) in the Weil conjectures, which provide their deep “rasion d’être” and which (at least for me, “clued in” by the explanations of Serre) give them their whole meaning.’ [cont’d] $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Mar 6, 2023 at 19:26
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    $\begingroup$ [cont’d] ‘But the possibility of associating such invariants to the “abstract” algebraic varieties involved in these conjectures, so as to meet the very specific desiderata required for the needs of that aim — it was just a hope. I suspect that apart from Serre and myself, nobody else (not even, and especially, André Weil himself!) really believed it…’ $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Mar 6, 2023 at 19:26

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