# John Torrence Tate

### Quick Info

Born
13 March 1925
Minneapolis, Minnesota, USA
Died
16 October 2019
Lexington, Massachussets, USA

Summary
John Torrence Tate was an American mathematician who won both the Wolf and Abel prizes for his work in Number Theory.

### Biography

First let us clarify that John Torrence Tate should have Jr after his name since his father was also named John Torrence Tate. John Tate Sr (born Lennox, Iowa, 28 July 1889) had a father of Scottish descent and a mother of Irish descent. He obtained a degree in physics from the University of Nebraska and was then awarded a doctorate by the University of Berlin. He married Lois Beatrice Fossler, a high school teacher of English, on 28 December 1917 by which time he was on the staff at the University of Minnesota. John Tate Sr. was a full professor of physics at the University of Minnesota when his son John Torrence Tate Jr was born. John Tate Jr, the subject of this biography, was brought up in Minneapolis.

In 1939 John's mother died. During World War II, John's father served on the National Defense Research Committee, in charge of the Division which researched undersea warfare. John grew up with a fascination for mathematical puzzles, in particular reading books by Henry Dudeney that his father owned. When at high school, he read E T Bell's Men of mathematics from which he learnt about quadratic reciprocity and Dirichlet's theorem on primes in an arithmetic progression. However, despite loving the ideas he had read about, he thought that mathematics was a subject for people who were cleverer than he was, so he decided to study physics at university. He graduated from Harvard University in 1946 and went to Princeton University, still with the intention of undertaking research in physics. However, during his first year of graduate study at Princeton it became clear to him that mathematics was not only the subject he liked best but it was also the subject for which he had the most talent. He was allowed to transfer to graduate study in mathematics and was assigned Emil Artin as his thesis advisor. It was pure coincidence that his thesis advisor had made major contributions to the topics that had most fascinated Tate when he was a schoolboy.

In 1950 Tate was awarded his doctorate for his thesis Fourier Analysis in Number Fields and Hecke's Zeta Functions [5]:-
In his doctoral thesis, Tate introduced harmonic analysis into number theory, paving the way for the adelic approach to automorphic forms and the Langlands programme.
In [4] the authors write:-
In his thesis, which has become a classic, he proved the functional equation for Hecke's L-series by a novel method involving Fourier analysis on idele groups.
The thesis was published in 1967. Returning to 1950, the year Tate was awarded his doctorate, we note that his father died in May of that year.

Tate was appointed as a research assistant and instructor at Princeton in 1950. The Artin-Tate seminar on class field theory given at Princeton University in 1951-1952 covered cohomology theory of groups, the fundamentals of algebraic number theory, a preliminary discussion of class formations, local class field theory, global class field theory, and the abstract theory of class formations and Weil group. Parts of this was written up as the book Class field theory by Artin and Tate and published in 1968. During his three years (1950-53) as a research assistant at Princeton, Tate published papers such as: On the relation between extremal points of convex sets and homomorphisms of algebras (1951); (with Emil Artin) A note on finite ring extensions (1951); Genus change in inseparable extensions of function fields (1952); (with Serge Lang) On Chevalley's proof of Luroth's theorem (1952); and The higher dimensional cohomology groups of class field theory (1952). For this last mentioned paper, Tate received the Frank Nelson Cole Prize in Number Theory from the American Mathematical Society in 1956. He spent the year 1953-54 as a visiting professor at Columbia University then, in 1954, he was appointed to Harvard University. He remained in this position until 1990 when he accepted the Sid Richardson Regents Chair at the University of Texas at Austin.

The London Mathematical Society elected Tate to Honorary Membership in 1999. We quote from the citation in [5] which gives an overview of Tate's remarkable contributions to mathematics:-
His work on class field theory and Galois cohomology over local and global fields, especially his duality theory, underpins much of modern number theory; and the Tate cohomology groups for finite groups, which he invented for use in class field theory, are a standard tool of algebraists. Tate's deep insights have had a crucial impact on the development of arithmetic algebraic geometry from the sixties onwards. Perhaps most celebrated are his conjectures about algebraic cycles on varieties over finite and global fields, formulated 35 years ago but still largely unproved. Equally striking is his seminal 1966 paper 'p-divisible groups', which for the first time recognised the richness of p-adic representations of the absolute Galois group of a p-adic field, as well as indicating the existence of a p-adic analogue of Hodge theory. This is now a key tool in understanding the arithmetic of algebraic varieties. Tate's work on classification of abelian varieties over finite fields is a core part of standard theory, underpinning almost all work on the L-functions of Shimura varieties as well as being the starting point for the study of motives over finite fields. Through his discovery of rigid analytic spaces, he established new foundations for p-adic global analysis which have wide applicability in number theory, algebraic geometry and representation theory. The theory of elliptic curves owes an enormous amount to his contributions, both theoretical and computational; the theory of height functions (Neron-Tate and Mazur-Tate) and descent theory (including his construction of the notorious Shafarevich-Tate group) are of key importance in understanding the arithmetic of elliptic curves, and Tate's algorithm for determining the bad reduction of an elliptic curve plays an equally important role in computation. Other contributions of deep significance include his work with Serre on the deformation theory of abelian varieties, his contributions to algebraic K-theory and its relation with Galois cohomology, his work on the Stark conjectures, and most recently his work in non-commutative ring theory.
Tate received a Sloan Fellowship during 1959-61, and a Guggenheim Fellowship during 1965-66. He was a plenary speaker at the International Congress of Mathematicians held in Nice in 1970 when he gave the lecture Symbols in Arithmetic. In 1972 he was the American Mathematical Society's Colloquium Lecturer and spoke on The arithmetic of elliptic curves. He was a member of the committee that decided on the awards of the Fields Medals in 1974. The committee surprised the mathematical world by only making two awards (to Enrico Bombieri and David Mumford). It was Tate who reported on Mumford's work at the awarding ceremony at the International Congress of Mathematicians in Vancouver.

Tate was honoured with election to the U.S. National Academy of Sciences in 1969 and to the Académie de Sciences in Paris in 1992. In 1995 he received the Leroy P Steele Prize For Lifetime Achievement from the American Mathematical Society [6]:-
... for scientific accomplishments spanning four and a half decades. He has been deeply influential in many of the important developments in algebra, algebraic geometry, and number theory during this time.
In 2003 he received the Wolf prize:-
... for his creation of fundamental concepts in algebraic number theory.
Although the citation is similar to that of the London Mathematical Society which we quoted above, we give the following extract:-
For over a quarter of a century, Professor John Tate's ideas have dominated the development of arithmetic algebraic geometry. Tate has introduced path breaking techniques and concepts, that initiated many theories which are very much alive today. These include Fourier analysis on local fields and adele rings, Galois cohomology, the theory of rigid analytic varieties, and p-divisible groups and p-adic Hodge decompositions, to name but a few. Tate has been an inspiration to all those working on number theory. Numerous notions bear his name: Tate cohomology of a finite group, Tate module of an abelian variety, Tate-Shafarevich group, Lubin-Tate groups, Neron-Tate heights, Tate motives, the Sato-Tate conjecture, Tate twist, Tate elliptic curve, and others. John Tate is a revered name in algebraic number theory.
In the first semester of the academic year 1980-81 Tate gave a course of lectures on Stark's conjectures at Université de Paris-Sud (Orsay). This was published in 1984 as Les conjectures de Stark sur les fonctions L d'Artin en s = 0. This is not the only book by Tate based on a lecture course he had given previously. In 1992 he published Rational points on elliptic curves coauthored with Joseph H Silverman. This book was based on a course Tate had given over 30 years earlier in 1961 at Haverford College. Andrew Bremner begins a review as follows:-
The authors' goal has been to write a textbook in a technically difficult field which is accessible to the average undergraduate mathematics major, and it seems that they have succeeded admirably. The book is quite delightful. ... The most obvious drawback to a text for undergraduates in a field such as this is that it is not possible to be entirely rigorous, and so, as the authors declare, "much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than rigorously prove." An appendix does develop the necessary algebraic geometry, but throughout the book the approach to the underlying geometry is informal, allowing a more rapid and intuitive access to the number theory.
On 24 May 2000, Atiyah and Tate presented the Clay Mathematics Institute Millennium Prize Problems in Paris. Tate's lecture covers the Riemann hypothesis, the Birch-Swinnerton-Dyer conjecture and the $P = NP$ problem. He explained the problems and put them into their historical context.

On 24 March 2010 the President of the Norwegian Academy of Science and Letters announced that Tate would be presented with the Abel Prize in Oslo on 25 May:-
... for his vast and lasting impact on the theory of numbers.
The theory of numbers stretches from the mysteries of prime numbers to the ways in which we store, transmit, and secure information in modern computers. Over the past century it has developed into one of the most elaborate and sophisticated branches of mathematics, interacting profoundly with other key areas. John Tate is a prime architect of this development. John Tate's scientific accomplishments span six decades. A wealth of essential mathematical ideas and constructions were initiated by Tate and later named after him, such as the Tate module, Tate curve, Tate cycle, Hodge-Tate decompositions, Tate cohomology, Serre-Tate parameter, Lubin-Tate group, Tate trace, Shafarevich-Tate group, Néron-Tate height, to mention just a few. According to the Abel 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."
We should end with one final note. Tate was one of the younger members of the Bourbaki team and almost unique in that team in that he was not French.

### References (show)

1. A Chanbert-Loir, Le prix Wolf 2003 attribué à John Tate, Gaz. Math. 98 (2003), 49-55.
2. R Cuculière, Histoire de la loi de réciprocité quadratique: Gauss et Tate, Study Group on Ultrametric Analysis. 7th-8th years: 1979-1981 36 (Secrétariat Math., Paris, 1981).
3. A Jackson, Sato and Tate receive 2002-2003 Wolf Prize, Notices Amer. Math. Soc. 50 (5) (2003), 569-570.
4. L Ji, S-T Yau and J-K Yu, Preface [Special issue: in honor of John Torrence Tate], Pure Appl. Math. Q. 5 (4) (2009), 1199.
5. John Tate : Honorary Member 1999, Bull. London Math. Soc. 32 (2000) 759-768.
6. 1995 Steele Prizes, Notices Amer. Math. Soc. 42 (11) (1995), 1288-1292.

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Honours awarded to John Torrence Tate

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Written by J J O'Connor and E F Robertson
Last Update March 2010