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
Academie 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 Universite 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
=
N
P
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 press release reads:-
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, Neron-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.
- A Chanbert-Loir, Le prix Wolf
2003
attribue a John Tate,
Gaz. Math.
98
(2003)
,
49
-
55
.
- R Cuculiere, Histoire de la loi de reciprocite quadratique: Gauss et Tate,
Study Group on Ultrametric Analysis.
7
th-
8
th years:
1979
-
1981
36
(
Secretariat Math., Paris,
1981)
.
- A Jackson, Sato and Tate receive
2002
-
2003
Wolf Prize,
Notices Amer. Math. Soc.
50
(5)
(2003)
,
569
-
570
.
- 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
.
- John Tate : Honorary Member
1999
,
Bull. London Math. Soc.
32
(2000)
759
-
768
.
- 1995
Steele Prizes,
Notices Amer. Math. Soc.
42
(11)
(1995)
,
1288
-
1292
.
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show
)
Cross-references
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)
Written by
J J O'Connor and E F Robertson
Last Update March 2010