Documents -- J.S. Milne
Documents - J.S. Milne, Top

Grothendieck and me

In 1985, when Grothendieck wrote Récoltes et Semailles, he had been out of touch with the mathematical community for more than 10 years. In particular, he didn't realize how dominant his persona and work still were. When he tried to reconstruct things from a somewhat paranoid perspective, he came up with a conspiracy theory that can only be described as delusional. In 1986, he sent me part of the work, and I wrote to him asking for the rest. Here is our exchange of letters.

Deligne, The Weil Conjecture, I.

This is a translation of
Deligne, Pierre, La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. No. 43 (1974), 273--307.
pdf file October 24, 2021.

Deligne, Étale cohomology: starting points

This is a translation of
Deligne, P., Cohomologie étale: les points de départ. Cohomologie étale, 4--75, Lecture Notes in Math., 569, Springer, Berlin, 1977.
At the AMS Summer Institute in Algebraic Geometry in 1974, Deligne gave a series of lectures "Inputs of étale cohomology" intended to explain the étale cohomology that his recent proof of the Weil conjectures was based on. The lectures were written up by Boutot (in French) and published in SGA 4 1/2. There is a TeXed version of the whole of SGA 4 1/2 here.
pdf file Last updated October 2022.

Deligne, Shimura varieties: modular interpretation and construction techniques for canonical models

This is a footnoted translation of
Deligne, Pierre. Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, pp. 247--289, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.
pdf file Last updated September 6, 2020 (many small corrections).

Deligne, The Fundamental Group of the Projective Line Minus Three Points

This is a translation of two sections (5. Algebraic geometry in a tannakian category. 6. The fundamental group of a tannakian category) from the classic article
Deligne, P. Le groupe fondamental de la droite projective moins trois points. Galois groups over Q (Berkeley, CA, 1987), 79--297, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989.
pdf file Last updated June 30, 2024.

Translation of a letter from Langlands to Saint-Aubin, 2020-02-21.

To create a geometric Langlands theory in the style of Hecke, that is to say with eigenvalues and eigenfunctions, we need a theory other than that of the Russians.
Caution: This is an unofficial translation, not checked or approved by anyone.
pdf file Last updated May 4, 2021.

Langlands: On the analytic form of the geometric theory of automorphic forms

Translation of the first five sections of Langlands 2018 into googlish. This may help readers gain some idea of what the manuscript is about until there is an official translation.
pdf file Last updated May 31, 2018.

Langlands, Functoriality in the theory of automorphic forms: its discovery and aims.

This is a translation of
Robert P. Langlands, Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele, in Emil Artin and beyond --- class field theory and L-functions, European Math. Soc., 2015, pp. 175--209.
This is a very fine essay. In translating it my goal was to produce a readable text while staying as close to the original as possible.
Caution: This is an unofficial translation, not checked or approved by anyone. Indeed, Langlands disapproves of the two split infinitives.
pdf file Last updated January 4, 2021.

Notes from the 1964 Algebraic Geometry Institute at Woods Hole.

"The summer institute gave us the idea that algebraic geometry was now a real subject, and no longer simply a mass of iffy results with a few valiant people like Zariski and Weil struggling to make order..." (Mumford)
In the summer of 1964, the AMS held a conference on Algebraic Geometry at Woods Hole, and distributed the informal mimeographed proceedings of the conference to a select few. Despite their great historical importance --- this was where Tate stated his conjectures on cycles, Artin and Verdier stated their duality theorem, and Serre and Tate stated their lifting theorem --- the AMS has ignored all requests to make the proceedings more widely available. More... Here (courtesy of Roy Smith), I've posted the complete notes. File, bookmarked, 10MB.

Proceedings of the 1955 Tokyo-Nikko Conference on Algebraic Number Theory

This was perhaps the first major mathematics conference held in Japan. It was attended by most of the leading Western algebraic number theorists (Artin, Brauer, Chevalley, Deuring, Néron, Serre, Weil) as well as the new generation of young Japanese mathematicians (Iwasawa, Kubota, Nakayama, Satake, Shimura, Taniyama, ...). The talks by Shimura, Taniyama, and Weil marked the birth of the theory of complex multiplication for abelian varieties of dimension greater than 1.
Proceedings
Photo with key (76 men, 1 woman, Keiko Satake)

Letter of Deligne November 30, 2011

Shows that a Tannakian category over an algebraically closed field has a fiber functor (handwritten, 6 pages). Posted with permission (via Drinfeld). pdf

Deligne: Hodge cycles on abelian varieties (Notes by J.S. Milne)

These are the notes for Deligne's seminar, "Periodes des Integrales Abeliennes", I.H.E.S., 1978--79 as written by J.S. Milne. They were published as pp9--100 of Deligne, Pierre; Milne, James S.; Ogus, Arthur; Shih, Kuang-yen. Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982. The notes were written by Milne (in English) based on the seminar (in French) of Deligne, and read by Deligne before they were published.
1982 Original authentic typed version.
2003 TeXed, corrected, endnotes added. Original numbering retained.
2011 Minor revision. Original numbering retained.
2018 Minor revision. Original numbering retained.

Some old notes of Mike Artin

Grothendieck Topologies, Notes of a Seminar by M. Artin, Spring, 1962, Harvard University.
Artin1962

Autoduality of the Jacobian, Lectures by Michael Artin, Advanced Science Seminar in Algebraic Geometry, Bowdoin College, Summer 1967, Notes by Bostwick Wyman
Artin1967

FAC: Serre, Jean-Pierre. Faisceaux algébriques cohérents. Ann. of Math. (2) 61, (1955). 197--278.

This is a translation of Serre's famous article. Except for the notion of a sheaf, this may still be the best short treatment of algebraic varieties over an algebraically closed field and their cohomology.
fac.pdf