John Tate - Mathematical Work

Mathematical Work

Tate's thesis (1950) on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it; independently and a little earlier, Kenkichi Iwasawa obtained a similar theory. Together with his teacher Emil Artin, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology applied to the idele class group and Galois cohomology. This treatment made more transparent some of algebraic structures in the previous approaches to class field theory which used central division algebras to compute the Brauer group of a global field.

Subsequently Tate introduced what are now known as Tate cohomology groups. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, and relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups, creating the Lubin–Tate local theory of complex multiplication.

He has also made a number of individual and important contributions to p-adic theory; for example, Tate's invention of rigid analytic spaces can be said to have spawned the entire field of rigid analytic geometry. He found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the 'Tate curve' parametrization for certain p-adic elliptic curves and the p-divisible (Tate–Barsotti) groups.

Many of his results were not immediately published and some of them were written up by Serge Lang, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties. The classification of abelian varieties over finite fields was carried out by Taira Honda and Tate (the Honda–Tate theorem).

The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture. They relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of 'Tate cycles' (the fixed cycles for a suitably Tate-twisted action) that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.

Tate has also had a major influence on the development of number theory through his role as a Ph.D. advisor. His students include Joe Buhler, Benedict Gross, Robert Kottwitz, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Ted Chinburg, Joseph H. Silverman, Dinesh Thakur, Jeremy Teitelbaum.