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This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, automorphic forms, the Langland's program and so forth was because my appeal is to things that have the feel of SGA1, and those things do not.

I was wondering, therefore, if Grothendieck had devoted thought to this, and if so where it can be found, and how it is treated in the field at the moment.

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    $\begingroup$ You could try reading Deligne. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 11, 2010 at 12:08
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    $\begingroup$ ...or Katz.$$ $$ $\endgroup$
    – Felipe Voloch
    Commented Dec 11, 2010 at 16:59
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    $\begingroup$ In "R&S" G' says "II’est vrai que les “formes modulaires” représentent un trou regrettable (parmi bien d’autres) dans ma culture mathématique, tout comme la théorie analytique des nombres, sur laquelle je n’ai encore jamais “accroché”. Mais je suis quand même suffisamment informé pour savoir qu’un compréhension des formes modulaires n’est guère pensable sans les idées provenant de la géométrie algébrique, qui donne à la théorie son contenu “géométrique”,... $\endgroup$
    – user234212323
    Commented Nov 12, 2021 at 15:26
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    $\begingroup$ ...et que les questions les plus profondes de la théorie des formes modulaires sont intimement liées à la présence (pendant longtemps tacite) des motifs." $\endgroup$
    – user234212323
    Commented Nov 12, 2021 at 15:26

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No.

That is perhaps a little too categorical, but a mathscinet search with Grothendieck as author and "modular form" or "forme modulaire" as "anywhere" gives no result. I don't remember him mentionning modular forms in "Recoltes et Semailles" either.

More to the point, it is a commonplace in the field of modular and automorphic forms to wish that Grothendieck had given some time to the subject -- and made it a little more "Grothendieck-style". Pierre Cartier gave a talk at the IHES in January 2009 where he deplored that "Grothendieck and Langlands never met".

Also, the correspondence between Serre and Grothendieck contains several letters where Serre tries to attract Grothendieck to the subject of modular forms, and where Grothendieck doesn't conceal his disinterest (to say the least).

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    $\begingroup$ FWIW, I also did a MathSciNet search along the same lines, with the same negative results. The closest I found was Grothendieck's work on vector bundles. $\endgroup$
    – Pete L. Clark
    Commented Dec 11, 2010 at 8:10
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    $\begingroup$ I wanted to add that this doesn't mean that Grothendieck's works didn't infuence the theory of modular forms enormously. It did, of course, in so mamy ways that it is not possible to describe them all. One obvious thing is his work on the Weil's conjecture, with when sompleted by Deligne leads to a proof of the Ramanujan's conjecture. An other, perhaps even more important thing, is G.'s theory of moduli scheme, used in the theory and study of modular curves (first in the very influential Deligne-Rapoport's paper) and more generally Shimura's varieties. $\endgroup$
    – Joël
    Commented Dec 11, 2010 at 13:54
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In « Récoltes et Semailles » Grothendieck has a reflexion about the article of Langlands : « Automorphic representations, Shimura varieties, and motives » where he sees the influence of his ideas on the « motivic Galois group » that Langlands has may be found in the thesis of a student of Grothendieck : Neantro Saavedra Rivano or from Deligne. Grothendieck sees also his influence in the references made by Langlands on the so called « Catégories Tannakiennes » which were parts (not under this name) of his reflexions on motives.

Grothendieck writes that « la théorie à la Langlands des formes automorphes » is intimately linked to motives. He adds that he is regrettably ignorant of the theory of automorphic functions and that he doesn't know if he will have the occasion to change this fact.

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    $\begingroup$ Good quote! This is the way Grothendieck could have become interested in the Langlands program : not by the modular forms, too special, too peculiar, to computational for his taste, but by the great vision of Langlands using Motives, Tannakian Categories. How wonderful would have it been if Grothendieck had learnt this theory and worked on it. $\endgroup$
    – Joël
    Commented Mar 24, 2015 at 14:51
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I don't know any published work by Grothendieck specifically on modular forms.

It seems however that Grothendieck has spent some time thinking about moduli spaces of elliptic curves, for example. He has written a very long manuscript "La longue marche à travers la théorie de Galois", which is about what is now called Grothendieck-Teichmüller theory, if I'm not mistaken. At some point, he even seems to do some "explicit" computations... I don't know this text very well though, so I would appreciate the opinion of other people about it (note that the text isn't available anymore on the Grothendieck Circle website).

I would also say that there are at present many approaches of modular forms in the "Grothendieck style". Here I would mention the whole theory of automorphic forms, by Deligne, Langlands and many others, which is at the same time very abstract and very powerful.

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  • $\begingroup$ Can you give explicit references to accompany the last sentence (“the whole theory of automorphic forms, by Deligne, Langlands and many others”)? $\endgroup$
    – jmc
    Commented Dec 20, 2014 at 20:48
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Grothendieck was aware (at least) of modular forms and their relations to motives by J P Serre in a letter dated December 31, 1986.

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Grothendieck (also) develop on "La longue Marche" a theory of "topos modulaire" (now completely available online!).

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    $\begingroup$ Just a note, I don't think his "topos modulaire de Teichmüller" have anything to do with modular forms. $\endgroup$
    – Myshkin
    Commented May 28, 2017 at 16:55

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