The Work of John Tate --- J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate
Errata
Tate helped shape the great reformulation of arithmetic and geometry which has taken place since the 1950s
         Andrew Wiles (Introduction to Tate's talk at the conference on the Millenium Prizes, 2000).

Abstract

This is my article on Tate's work for the second volume in the book series on the Abel Prize winners. True to the epigraph, I have attempted to explain it in the context of the "great reformulation".

Contents

  1. Hecke L-series and the cohomology of number fields
  2. Abelian varieties and curves
  3. Rigid analytic spaces
  4. The Tate conjecture
  5. Lubin-Tate theory and Barsotti-Tate group schemes
  6. Elliptic curves
  7. The K-theory of number fields
  8. The Stark conjectures
  9. Noncommutative ring theory
  10. Miscellaneous articles

pdf file for my manuscript (First posted 18.03.12; with minor fixes 03.12.12.)

Published as: J.S. Milne. The Work of John Tate. In Helge Holden and Ragni Piene, editors, The Abel Prize 2008--2012, pages 259--340. Springer, Heidelberg, 2014. It is also available as an eBook here.

The AMS should have included a shortened simplified version of this article, which I would have been happy to provide, in the Tate issue of the Bulletin (October 2017). Instead, they stupidly just reprinted the table of contents.

Don't bother searching for the article in Math. Reviews or MR Lookup --- it isn't listed there. Nor is Gowers's article on the work of Szemerédi, or the article of Siebenmann on the work of Milnor, or the article of Lyons and Guralnik on the work of Thompson....

One article of Tate I missed:
Tate, J., Correspondance Grothendieck-Serre (book review). Nw. Archief Wiskunde 5 (2004), 42--44.

This book gives us intimate views of the collaboration which took place between A. Grothendieck and J-P. Serre during the period 1955--1969, when they played such key roles in revolutionary developments in algebraic geometry. The volume contains about eighty essentially unedited letters from that period plus a few from the mid eighties. It exists thanks to Serre who preserved the letters, to Pierre Colmez who obtained Grothendieck's permission to publish them through an intermediary Jean Malgoire and organized an army of students to TeX the letters, to the efforts of those students, and to the French Mathematical Society which created a new book series, Documents Mathématiques to accomodate the publication of the letters. The book is attractively and readably printed, with reproductions of two of Grothendieck's letters, one handwritten, one typed, which show that TeXing the letters was not always a routine job. Serre has added very helpful notes explaining unclear references, noting assertions which turned out to be false, and indicating as far as possible the subsequent fate of conjectures made in the letters.

Unfortunately the historical record provided by the letters is full of gaps because the letters were written for the most part only when Serre and Grothendieck were not both in the Paris region. When together there they spent hours on the phone. One can only regret that those conversations were not taped. But if not furnishing a complete historical record, the letters do afford a real understanding of how these men thought and worked together. Their collaboration was especially fruitful because the two think so differently: Grothendieck rushes ahead on his own, full of optimism, guided by the big picture, putting everything in the most general context, sometimes overlooking details in his hurry; Serre, with an amazing knowledge of the literature and special cases, very careful with details, working with a good balance between special and general, and always ready to construct an example or counterexample to keep Grothendieck on track.

Faisceaux Algébriques Cohérents
The collaboration by letter begins in January 1955. Serre has seen that sheaf theory which was so useful in the Cartan seminars on complex analytic varieties serves also in abstract algebraic geometry despite the weirdness of the Zariski topology. He is finishing his paper "Faisceaux Algébriques Cohérents" (FAC) on the subject. Grothendieck has decided to abandon topological vector spaces for algebraic geometry. In the early letters he bombards Serre with questions, some naive, some not so, which Serre answers patiently. Much of the discussion concerns sheaves.

Grothendieck is developing his own view of abelian categories, derived functors, the various theories of sheaf cohomology, spectral sequences,. . .as only he could. His summary was later published as "Sur quelques points d'algèbre homologique" in Tohoku Math. J. (1957). By the end of 1955 Grothendieck explains to Serre a general formulation of Serre duality, saying it is almost evident, and implicit in FAC. Serre is excited by this formulation, saying it is the good way to state duality in both the algebraic case and the analytic case. FAC finished, Serre is following it with GAGA, "Géométrie Algébrique et Géométrie Analytique", proving that for projective varieties, the analytic coherent sheaves are the "same" as algebraic ones.

The best category of commutative rings
Schemes were already in the air, though always with restrictions on the rings involved. In February 1955 Serre mentions that the theory of coherent sheaves works on the spectrum of commutative rings in which every prime ideal is an intersection of maximal ideals. A year later, Grothendieck tells of Cartier's introducing the technically useful notion of quasicoherent sheaf on arithmetic varieties made by gluing together the spectra of noetherian rings. But it would take two more years before Grothendieck realized that the noetherian condition should be dropped and one should include all prime ideals in the spectrum, that in the end, as Serre puts it in the notes, the best category of commutative rings is the category of all commutative rings.

So far, we have given a very sketchy and incomplete description of the mathematics involved in the first two years of the collaboration, as revealed by 27 letters covering 55 pages. The problems under discussion were largely foundational ones. The next seven years saw the exchange of 26 letters, about 85 pages. Some of the topics discussed during these years are: Gothendieck's proof of Riemann-Roch, the progress of his writing of Éléments de Géométrie Algébrique(EGA), begun in 1958, Serre's geometric class field theory, Grothendieck's theory of the fundamental group (one of the first big successes of the theory of schemes), Weil's restriction of scalars functor, the astonishing news of Dwork's proof of the rationality of the zeta function of a variety over a finite field (1959), generalized Jacobians and local symbols, purity, the (sad?) situation of mathematics in Paris in 1961, p-adic Lie algebras, M. Artin's proof that every non-singular variety is covered by "good" opens, the local invariants in the formula expressing the Euler characteristic of an étale sheaf on a curve in terms of its singularities in the case of higher ramification which were requested by Grothendieck, and supplied by Serre, the Swan representation.

An outburst of correspondence
Suddenly, in 1964, there are 22 letters in one year. A big theme is algebraic cycles and the topics are becoming more arithmetical. Serre writes to Grothendieck an extensive report on the Woods Hole Summer Institute: Woods Hole fixed point formula, conjectures on l-adic cohomology of a variety over a local field inspired by Ogg's work on elliptic curves, my conjectures on algebraic cycles and poles of zeta functions, formal groups, the Serre-Tate theorem and Serre's canonical lift. Much of the discussion in the following months is stimulated by this report and by a letter from Serre to A. Ogg announcing the criterion of Ogg-Néron-Shafarevich for good reduction of an abelian variety and its agreeable corollaries, together with a question which Serre calls "naive": is it true that after a finite extension of the ground field, the connected component of the special fiber of the Néron model of an abelian variety is an extension of an abelian variety by a torus. Grothendieck has no sentiment at first, but later proves it. There is much discussion of zeta and L-functions. Grothendieck discusses vanishing cycles and introduces a new concept which he calls a "motive", which is to become one of his greatest contributions. Soon after, we have motivic Galois groups, and in early 1965 he formulates the "standard conjectures".

Farewell to Grothendieck
The collaboration is coming to an end. There are three more letters exchanging mathematical ideas. The last, in January 1969 from Grothendieck contains his thoughts on what Serre has told him about Steinberg's theorem. Not long after that Grothendieck abandoned the writing of EGA and dropped out of the mathematical world.

The rest of the book, six letters from the mid 1980s is sad to read. Grothendieck has isolated himself and begun writing Récoltes et Semailles, a rambling account of his bitter somewhat paranoid reflections on his mathematical life and on the behaviour of his former students and his colleagues, which he distributes to them. Serre gets the first chapters and writes Grothendieck his comments. Grothendieck finds that Serre accepts the nice things in his account but rejects the disagreeable ones, and says that since he's known him Serre has always shown a horror of serious self-examination. In his reply, Serre asks why Grothendieck abandoned his mathematical program and hazards two guesses. Had the writing of EGA and the seminars simply become too great a burden? Or was the problem a deeper one, that Grothendieck realized that his method of doing mathematics, building big general theories, which worked so well for topological vector spaces and algebraic geometry was not as effective for the problems of number theory and modular forms which were looming in the late sixties when Grothendieck abandoned his program?

At the end of 1986, Serre writes a last mathematical letter, explaining to Grothendieck his conjectures on modular Galois representations of degree 2, and sending best wishes for 1987. Grothendieck replies, thanking Serre for his trouble, but explaining that it is not worth the effort. He has his own projects and is not interested, but wishes Serre a happy 1987 and all success in his work.

Of course the book will be a great resource for future historians of 20th century mathematics, but it is much more than that. It gives today's reader a feel for a very different mathematical era and a unique opportunity to be present at the exchange of mathematical ideas at the highest level, in complete comradeship, between two masters.