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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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You could try reading Deligne. –  S. Carnahan Dec 11 '10 at 12:08
...or Katz.$$ $$ –  Felipe Voloch Dec 11 '10 at 16:59

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up vote 18 down vote accepted


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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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. –  Pete L. Clark Dec 11 '10 at 8:10
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. –  Joël Dec 11 '10 at 13:54

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 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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