John Torrence Tate Jr. (March 13, 1925 – October 16, 2019) was an American mathematician, distinguished for many fundamental contributions in

algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

, arithmetic geometry and related areas in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

. He was awarded the

Abel Prize The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...

in 2010.

Biography

Tate was born in

Minneapolis Minneapolis () is the largest city in Minnesota, United States, and the county seat of Hennepin County. The city is abundant in water, with thirteen lakes, wetlands, the Mississippi River, creeks and waterfalls. Minneapolis has its origins ...

, Minnesota. His father, John Tate Sr., was a professor of physics at the

University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...

, and a longtime editor of ''

Physical Review ''Physical Review'' is a peer-reviewed scientific journal established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the American Physical S ...

''. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics in 1946 from

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

, and entered the doctoral program in physics at

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

. He later transferred to the mathematics department and received his PhD in mathematics in 1950 after completing a doctoral dissertation, titled "Fourier analysis in number fields and Hecke's zeta functions", under the supervision of

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...

. Tate taught at Harvard for 36 years before joining the

University of Texas The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 ...

in 1990 as a Sid W. Richardson Foundation Regents Chair. He retired from the Texas mathematics department in 2009, and returned to Harvard as a professor emeritus. Tate died at his home in

Lexington, Massachusetts Lexington is a suburban town in Middlesex County, Massachusetts, United States. It is 10 miles (16 km) from Downtown Boston. The population was 34,454 as of the 2020 census. The area was originally inhabited by Native Americans, and was firs ...

, on October 16, 2019, at the age of 94.

Mathematical work

Tate's thesis (1950) on

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...

number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

has become one of the ingredients for the modern theory of

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

s and their

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...

s, 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

, Tate gave a cohomological treatment of

global class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

, using techniques of

group cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...

applied to the

idele class group In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...

and Galois cohomology. 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 Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

of a global field. Subsequently, Tate introduced what are now known as

Tate cohomology group In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by , and are used in class field theory. Defin ...

s. In the decades following that discovery he extended the reach of Galois cohomology with the

Poitou–Tate duality In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and . Local Tate duality For a ''p''-adic local f ...

, the

Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of ...

, and relations with algebraic K-theory. With Jonathan Lubin, he recast

local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...

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 In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing ''p''-adic elliptic curves with bad redu ...

. He found a ''p''-adic analogue of

Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...

, now called Hodge–Tate theory, which has blossomed into another central technique of modern

. Other innovations of his include the "

Tate curve In mathematics, the Tate curve is a curve defined over the ring of formal power series \mathbb q with integer coefficients. Over the open subscheme where ''q'' is invertible, the Tate curve is an elliptic curve. The Tate curve can also be defined fo ...

" parametrization for certain ''p''-adic

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

s 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 Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...

, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on

good reduction This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of pro ...

of abelian varieties. The classification of abelian varieties over

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...

s was carried out by Taira Honda and Tate (the

Honda–Tate theorem In mathematics, the Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order ''q'' correspond to algebraic integers all of whose c ...

). The

Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conje ...

s 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 Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction. Educati ...

Gerd Faltings Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry. Education From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathema ...

. Tate has also had a major influence on the development of number theory through his role as a Ph.D. advisor. His students include

George Bergman George Mark Bergman, born on 22 July 1943 in Brooklyn, New York, is an American mathematician. He attended Stuyvesant High School in New York City and received his Ph.D. from Harvard University in 1968, under the direction of John Tate. The year ...

, Ted Chinburg, Bernard Dwork,

Benedict Gross Benedict Hyman Gross is an American mathematician who is a professor at the University of California San Diego, the George Vasmer Leverett Professor of Mathematics Emeritus at Harvard University, and former Dean of Harvard College.Robert Kottwitz Robert Edward Kottwitz (born 1950 in Lynn, Massachusetts) is an American mathematician. Kottwitz studied at the University of Washington (B.A.) and then went to Harvard University, where he received his Ph.D. in 1977 under the supervision of Phil ...

, Jonathan Lubin,

Stephen Lichtenbaum Stephen Lichtenbaum (1939 in Brooklyn) is an American mathematician who is working in the fields of algebraic geometry, algebraic number theory and algebraic K-theory. Lichtenbaum was an undergraduate at Harvard University (bachelor's degree "su ...

, James Milne,

V. Kumar Murty Vijaya Kumar Murty (born 20 May 1956) is an Indo-Canadian mathematician working primarily in number theory. He is a professor at the University of Toronto and is the Director of the Fields Institute. Early life and education V. Kumar Murty is th ...

, Carl Pomerance,

Ken Ribet Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's ...

, Joseph H. Silverman, and Dinesh Thakur.

Awards and honors

In 1956 Tate was awarded the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number ...

for outstanding contributions to number theory. In 1992 he was elected as Foreign Member of the French Academie des Sciences. 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 The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...

in 2002/03 for his creation of fundamental concepts in

. In 2012 he became a fellow of the American Mathematical Society. In 2010, the

Norwegian Academy of Science and Letters The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick Univer ...

, of which he was a member, awarded him the

, 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

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." 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 at Austin The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 ...

Personal life

Tate married twice. His first wife was Karin Artin, his doctoral advisor's daughter. Together they had three daughters, six grandchildren, and one great-grandson. One of his grandchildren, Dustin Clausen, currently works as a mathematician in

University of Copenhagen The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in ...

. After Tate divorced, he married Carol MacPherson.

Selected publications

Ph.D. thesis under

. Reprinted in * * * * * * * * * * *
''Collected Works of John Tate: Parts I and II''
American Mathematical Society, (2016)

References

*Milne, J
"The Work of John Tate"