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

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 table of contents was reprinted in the Tate issue of the Bull. AMS (October 2017). But note that they managed to get the title wrong.

Don't bother searching for this 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....


First posted 18.03.12, 72 pages.
03.12.12. Many minor fixes).

One correction: "Nakayama (1957)" is a reference to T. Nakayama, Cohomology of class field theory and tensor product modules I, Ann. of Math., 65 (1957), pp. 255-267.

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

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…
Suddenly, in 1964, there are 22 letters in one year. A big theme is algebraic cycles and the topics are becoming more arithmetical … 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"…
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…
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.