*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

- Hecke
*L*-series and the cohomology of number fields
- Abelian varieties and curves
- Rigid analytic spaces
- The Tate conjecture
- Lubin-Tate theory and Barsotti-Tate group schemes
- Elliptic curves
- The
*K*-theory of number fields
- The Stark conjectures
- Noncommutative ring theory
- 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.

23.09.12. (Many minor fixes; thanks to Timo Keller and Matthias Künzer.)

03.12.12. (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.