John Tate

John Tate
Born	John Torrence Tate, Jr. (1925-03-13) March 13, 1925 Minneapolis, Minnesota, U.S.
Nationality	American
Fields	Mathematics
Institutions	Princeton University (1950–1953) Columbia University (1953–1954) Harvard University (1954–1990) University of Texas at Austin (1990–2009)
Alma mater	Harvard University (A.B., 1946) Princeton University (Ph.D., 1950)
Doctoral advisor	Emil Artin
Doctoral students	Ki-Seng Tan V. Kumar Murty Edward Assmus Theodore Chinburg William C. Waterhouse Benedict Gross Jonathan Lubin Stephen Lichtenbaum Kenneth Alan Ribet Joseph H. Silverman Dinesh Thakur Jerrold Tunnell Robert Warfield Carl Pomerance George Bergman
Known for	Tate conjecture Tate module
Influenced	John H. Coates
Notable awards	Abel Prize (2010) Wolf Prize (2002/03) Steele Prize (1995) Cole Prize in Number Theory (1956)

John Torrence Tate, Jr. (born March 13, 1925) is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University. He was awarded the Abel Prize in 2010.

Biography

Tate was born in Minneapolis. His father, John Tate Sr., was a professor of physics at the University of Minnesota, and a longtime editor of Physical Review. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics from Harvard University, and entered the doctoral program in physics at Princeton University. He later transferred to the mathematics department and received his PhD 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 returned to Harvard as a professor emeritus. He currently resides in Cambridge, Massachusetts with his wife Carol. He has three daughters with his first wife Karin Tate.^[1]

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.^[2] This treatment made more transparent some of the 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.^[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 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).^[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 Benedict Gross, Robert Kottwitz, Jonathan Lubin, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, Dinesh Thakur.

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] In 2012 he became a fellow of the American Mathematical Society.^[5]

In 2010, the Norwegian Academy of Science and Letters, of which he is a member,^[6] 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."^[7]

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]

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, MR 0215665 
• Tate, John (1952), "The higher dimensional cohomology groups of class field theory", Ann. of Math. (2), 56: 294–297, MR 0049950 
• Lang, Serge; Tate, John (1958), "Principal homogeneous spaces over abelian varieties", American Journal of Mathematics, 80: 659–684, MR 0106226, doi:10.2307/2372778 
• Tate, John (1965), "Algebraic cycles and poles of zeta functions", Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), New York: Harper & Row, pp. 93–110, MR 0225778 
• Lubin, Jonathan; Tate, John (1965), "Formal complex multiplication in local fields", Annals of Mathematics, 81: 380–387, MR 0172878, doi:10.2307/1970622 
• Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2: 134–144, MR 0206004, doi:10.1007/bf01404549 
• Tate, John (1967), "p-divisible groups", in Springer, T. A., Proceedings of a Conference on Local Fields, Springer-Verlag, pp. 158–183, MR 0231827 
• Artin, Emil; Tate, John (2009) [1967], Class field theory, AMS Chelsea Publishing, ISBN 978-0-8218-4426-7, MR 2467155 
• Serre, Jean-Pierre; Tate, John (1968), "Good reduction of abelian varieties", Annals of Mathematics, 88: 462–517, MR 0236190, doi:10.2307/1970722 
• Tate, John (1971), "Rigid analytic spaces", Inventiones Mathematicae, 12: 257–289, MR 0306196, doi:10.1007/bf01403307 
• Tate, John (1976), "Relations between K₂ and Galois cohomology", Inventiones Mathematicae, 36: 257–274, MR 0429837, doi:10.1007/bf01390012 
• Tate, John (1984), Les conjectures de Stark sur les fonctions L d'Artin en s=0, Progress in Mathematics, 47, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3188-7, MR 0782485 
• Collected Works of John Tate: Parts I and II, American Mathematical Society, (2016)

See also

• Barsotti–Tate group
• Hodge–Tate module
• Koszul–Tate resolution
• Néron–Tate height
• Sato–Tate conjecture
• Serre–Tate theorem
• Tate duality
• Tate's isogeny theorem
• Tate Lie algebra
• Tate twist
• Tate's algorithm

References

• Milne, J, "The Work of John Tate"

External links

• Tate's home page
• O'Connor, John J.; Robertson, Edmund F., "John Tate", MacTutor History of Mathematics archive, University of St Andrews .
• John Tate at the Mathematics Genealogy Project