### John Torrence Tate

Born 13 March 1925, Minneapolis, Minnesota, USA

Abel Prize (2010) “for his vast and lasting impact on the theory of numbers”

John Torrence Tate’s family was an academic one: his father was a professor of Physics at the University of Minnesota, and his mother taught English at a high school. Tate studied Physics and graduated from Harvard University in 1946. He then went to Princeton University, initially to continue with physics, but in his first year he changed to Mathematics and did his doctorate with Emil Artin on ‘Fourier Analysis in Number Fields and Hecke’s Zeta Functions’ (Ph.D. 1950). Subsequently Tate worked as a research assistant and taught at Princeton University (1950-1953) and as a visiting professor at Columbia University (1953-1954). In 1954 he took up a post as a professor at Harvard University, where he taught and conducted research for the next 36 years. 1990 he went to the University of Texas at Austin where he held a Sid W. Richardson Chair in mathematics until his retirement in 2009.

Tate was a guest professor at the University of California at Berkeley, at the Institut des Hautes Études Scientifiques and at the Université de Paris, at Princeton University and at the École Normale Supérieure in Paris.

He is married and has three daughters from his first marriage.

Tate has won many awards, including the American Mathematical Society’s Frank Nelson Cole Prize in Number Theory (1956) and Leroy P. Steele Prize for Lifetime Achievement (1995), as well as the Wolf Prize (2003). He was awarded a Sloan Fellowship (1959-1961) and a Guggenheim Fellowship (1965-1966). He is a member of the U.S. National Academy of Sciences (1969) and the French Académie des Sciences (1992), and since 1999 has been an honorary member of the London Mathematical Society.

In his thesis, Tate gave a new simple proof of the basic properties of Hecke’s L-functions, which are a generalization of the functions introduced by Dirichlet in order to prove his theorem that if a and b are two positive integers with no common factor, then the arithmetic progression a, a+b, a+2b, … contains an infinite number of primes. His methods were later important in the study of automorphic forms and L-functions – special functions that connect geometry, algebra and number theory in modern algebraic geometry. Tate’s vital preliminary work made possible the famous Langlands program in the algebraic geometry of Robert Langlands (*1936).

As a postdoc, he collaborated with his supervisor Emil Artin on a seminar, the notes of which, taken by Serge Lang, eventually became the classic book ‘Class Field Theory’ by Artin and Tate. In the next half century Tate worked on many different problems in arithmetic algebraic geometry, often involving elliptic curves and abelian varieties. The “Tate conjecture”, roughly that a necessary condition for a cohomology class to be algebraic is also sufficient, is a famous open-ended problem. To recall two other of Tate’s ideas, with his Ph.D. student Jonathan Lubin, he found a completely new approach to local class field theory via formal groups, and he also generalized Hodge theory to p-adic numbers. Hodge theory, named after William Hodge (1903-1975) studies special differential forms of continuous manifolds. p-adic numbers (the p stands for a prime number) are extensions of rational fractions, which exhibit many interesting properties, including that they can ‘break up’ many statements and check them modularly for each p. Today, the p-adic Hodge theory is one of the most active areas of mathematics and mathematical physics.

For an in depth description of John Tate’s work and its place in the development of arithmetic and geometry since 1950, see J. S. Milne’s account, in “The Work of John Tate”, pp. 249-340, in “The Abel Prize 2008-2012”, Helge Holden and Ragni Piene Eds., Springer Verlag, Berlin Heidelberg 2014.