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John Torrence Tate Jr. (born March 13, 1925) is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. In 2010, he won the Abel Prize, one of the most prestigious awards in mathematics. Tate has been described as "one of the seminal mathematicians for the past half-century" by William Beckner, Chairman of the Department of Mathematics at the University of Texas.[1]

Biography

Tate was born in Minneapolis. He received his bachelor's degree in mathematics from Harvard University, and his doctoral degree from Princeton University in 1950 as a student of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990. He retired from the Texas mathematics department in 2009, and currently resides in Cambridge, Massachusetts with his wife Carol. He has three daughters.[1]

Mathematical work

Tate's 1950 thesis on Fourier analysis in number fields paved the way for the modern theory of automorphic forms and their L-functions by its use of the adele ring. He gave a new treatment of global class field theory with Emil Artin, using techniques of group cohomology applied to the idele class group.[2]

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.[2] 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 were written up by Jean-Pierre Serre. They collaborated on a major published 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).[3]

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, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, Dinesh Thakur and Jeremy Teitelbaum.

Awards and honors

In 1956 Tate was awarded the American Mathematical Society's Cole Prize for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society. He was awarded a Wolf Prize in Mathematics in 2002/03 for his creation of fundamental concepts in algebraic number theory.[4]
“ I got a phone call at 7 in the morning from a guy with a very strong Norwegian accent. That was the first I heard of it. I feel very fortunate. I realize that there is any number of people they could have chosen. ”

—John Tate, [1]

In 2010, the Norwegian Academy of Science and Letters awarded him the Abel Prize, citing "his vast and lasting impact on the theory of numbers". According to a release by the Abel Prize committee "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate. He has truly left a conspicuous imprint on modern mathematics."[5] The Abel Prize award ceremony is planned for May 25 in Oslo.[6] The prize carries a cash award of NOK 6,000,000 (close to € 730,000 or just under US $1m as of the date of the award.)[5]

Selected publications

* Tate, John (1950), Fourier analysis in number fields and Hecke's zeta functions , Princeton University Ph.D. thesis under Emil Artin. Reprinted in Cassels, J. W. S.; Fröhlich, Albrecht, eds. (1967), Algebraic number theory, London: Academic Press, pp. 305–347, MR0215665
* Tate, John (1952), " The higher dimensional cohomology groups of class field theory ", Annals of Mathematics 56 : 294-297, MR 0049950.
* Lang, Serge; Tate, John (1958), "Principal homogeneous spaces over abelian varieties", American Journal of Mathematics 80: 659–684, MR0106226
* Tate, John (1963), " Algebraic cycles and poles of zeta functions", in Arithmetical Algebraic Geometry, Harper and Row: 93-110, MR0225778.
* Lubin, Jonathan; Tate, John (1965), "Formal complex multiplication in local fields", Annals of Mathematics 81: 380–387, MR0172878
* Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae 2: 134–144, MR0206004
* Tate, John (1967), "p-divisible groups", in Springer, T. A., Proceedings of a Conference on Local Fields, Springer-Verlag, pp. 158–183, MR0231827
* Artin, Emil; Tate, John (2009) [1967], Class field theory, AMS Chelsea Publishing, MR2467155, ISBN 978-0-821-84426-7
* Serre, Jean-Pierre; Tate, John (1968), "Good reduction of abelian varieties", Annals of Mathematics 88: 462–517, MR0236190
* Tate, John (1971), "Rigid analytic spaces", Inventiones Mathematicae 12: 257–289, MR0306196
* Tate, John (1976), "Relations between K2 and Galois cohomology", Inventiones Mathematicae 36: 257–274, MR0429837
* Tate, John (1984), " Les conjectures de Stark sur les fonctions L d'Artin en s=0 ", Progress in Math. 47, Birkhäuser Boston, MR0782485.