*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

The book has been published: The Abel Prize 2008-2012 (Holden and Piene Eds.).
It is also available as an eBook
here.

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.