John T. Tate

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John T. Tate (1993)

John Torrence Tate (born March 13, 1925 in Minneapolis , Minnesota - † October 16, 2019 in Lexington , Massachusetts ) was an American mathematician who worked in the fields of algebraic geometry and number theory .

life and work

After three years in the US Navy Tate received in 1946 his BA from Harvard University and a doctorate in 1950 with Emil Artin at Princeton University (Fourier Analysis in Number Fields and Hecke's Zeta Functions). There he was also a professor from 1950 to 1954 before going to Harvard University. In 1990 he went to the University of Texas at Austin .

In his doctoral thesis "Fourier analysis in number fields and Hecke's Zetafunctions" (in Cassels, Fröhlich (Ed.): "Algebraic Number Theory" published in 1966 and commonly known as Tate's Thesis or Tate-Iwasawa theory) he applied harmonic analysis to number fields (Fourier analysis on the Adelering and the Idele group ) and achieved many results by Erich Heckes on L-functions in a different way.

In cooperation with Emil Artin he formulated the class field theory with group cohomology ( Galois cohomology ). In "The higher dimensional cohomology groups of class field theory" (Annals of Mathematics 1952) he introduced the Tate cohomology groups. In his ICM lecture in Stockholm in 1962, “Duality theorems in Galois cohomology over number fields”, he formulated his duality sentences (Tate duality). His Tate Shafarevich groups are fundamental to arithmetic geometry. They measure - roughly speaking - the extent to which the variety deviates from the Hasse principle , according to which one wants to deduce from the p-adic (“local”) and real solvability to the solvability in rational numbers (“global”), which is the case with quadratic forms is possible (Hasse), but in general no longer with cubic curves ( elliptic curves ). Many of the results he found on Galois cohomology were only published in the books of Jean-Pierre Serre .

In 1958, together with Arthur Mattuck , he gave a new proof of the Castelnuovo-Severi inequality in algebraic geometry.

"P-divisible groups" (also called Barsotti-Tate groups) from 1966 (Proc. Conf. Local Fields, Driebergen) deals with p-adic Galois representations, that is, those over local fields with the characteristic p.

In the 1960s he also formulated the Tate conjecture about algebraic cycles, which describes the effect of the absolute Galois group on the L-adic cohomology groups of algebraic varieties ("Algebraic cycles and poles of zeta functions" in Schilling (ed.): "Arithmetical algebraic geometry “1965). In “Endomorphisms of abelian varieties over finite fields” ( Inventiones Mathematicae 1966) he constructs such cycles from cohomological information.

In the 1970s he worked on algebraic K-theory ("Relations between K2 and Galois Cohomology", Inventiones Mathematicae 1976).

In the 1980s he examined the Stark conjectures about zeros of L-functions in the case of function fields. He also examined the Birch-Swinnerton-Dyer conjectures and their analogues in the p-adic case (with Barry Mazur , Teitelbaum, Inv. Math. 1986).

He gave a p-adic uniformization theory of elliptical curves and Abelian varieties ("Tate curve") and introduced "Rigid analytic spaces" (Inventiones Mathematicae 1971).

A conjecture named after him and Mikio Satō postulates a probability distribution of the phases of the coefficients of the Hasse-Weil zeta function of elliptical curves.

The Hodge-Tate theory (as a p-adic analogue of the Hodge theory ) and the Honda- Tate theory (the classification of Abelian varieties over finite fields) also come from him. The Néron-Tate height (also named after André Néron ), Tate cohomology groups, Tate motifs and Tate modules (which are used to classify Abelian varieties except for isogeny in the Tate isogeny theorem) are also named after him .

His students include a. Ken Ribet , Benedict H. Gross , Carl Pomerance , Jonathan Lubin , Joe Buhler, and Joseph Silverman .

In 1956 he received the Cole Prize in number theory. In 1995 he received the Leroy P. Steele Prize of the American Mathematical Society , the Wolf Prize in 2002 and the Abel Prize in 2010 . In 1970 he gave a plenary lecture at the ICM in Nice (Symbols in Arithmetic). He was a fellow of the American Mathematical Society. In 1958 he was elected to the American Academy of Arts and Sciences and in 1969 to the National Academy of Sciences . He was a member of the Académie des Sciences and the Norwegian Academy of Sciences .


Web links

Commons : John T. Tate  - Collection of Images, Videos and Audio Files

Individual evidence

  1. Kenneth Chang: John T. Tate, Familiar Name in the World of Numbers, Dies at 94. In: The New York Times . October 28, 2019, accessed October 29, 2019 .
  2. John T. Tate in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used