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

arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

and related areas in algebraic geometry. 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 Pri ...

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 list of lakes in Minneapolis, thirteen lakes, wetlands, the Mississippi River, creeks and waterfalls. ...

, 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 Soc ...

''. 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 high ...

, and entered the doctoral program in physics at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...

. 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 ...

. 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

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 ...

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 cred ...

, 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 cohomolog ...

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 In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a nat ...

. 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 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 Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...

. With

Jonathan Lubin Jonathan Darby Lubin (born August 10, 1936, in Staten Island, New York) is a professor emeritus of mathematics at Brown University. He received an A.B. from Columbia College in 1957 and a Ph.D. from Harvard University in 1963 under the direc ...

, he recast local class field theory by the use of formal groups, creating the Lubin–Tate local theory of

complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...

. 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 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 coh ...

, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. 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 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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...

Joseph H. Silverman Joseph Hillel Silverman (born March 27, 1955, New York City) is a professor of mathematics at Brown University working in arithmetic geometry, arithmetic dynamics, and cryptography. Biography Joseph Silverman received an Sc.B. from Brown Unive ...

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 ...

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...

. 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, subt ...

s was carried out by

Taira Honda was a Japanese mathematician working on number theory who proved the Honda–Tate theorem classifying abelian varieties over finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that ...

and Tate (the Honda–Tate theorem). The Tate conjectures are the equivalent for

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...

of the Hodge conjecture. They relate to the Galois action on the ℓ-adic cohomology of an algebraic variety, identifying a space of "

Tate cycle Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the U ...

s" (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 George Bergman,

Ted Chinburg TED may refer to: Economics and finance * TED spread between U.S. Treasuries and Eurodollar Education * ''Türk Eğitim Derneği'', the Turkish Education Association ** TED Ankara College Foundation Schools, Turkey ** Transvaal Education Depart ...

, Bernard Dwork, Benedict Gross,

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 ...

, Stephen Lichtenbaum, 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 Carl Bernard Pomerance (born 1944 in Joplin, Missouri) is an American number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number ...

, Ken Ribet,

, 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 algebraic number theory. 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 Unive ...

, 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 algebraic number theory and

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"