John Tate

John Torrence Tate ( born March 13, 1925 in Minneapolis, Minnesota) is an American mathematician who works in the fields of algebraic geometry and number theory.

Life and work

After three years in the U.S. Navy, he received his BA from Harvard University in 1946 and his doctorate in 1950 with Emil Artin at Princeton University. He was also from 1950 to 1954 professor before he went to Harvard University. In 1990 he went to the University of Texas at Austin.

In his thesis, " Fourier analysis in number fields and Hecke's Zeta Functions" ( in Cassels, Fröhlich ( ed.), " Algebraic Number Theory " published in 1966 ), he turned the harmonic analysis in number fields on (Fourier analysis on adele group) and achieved many results Erich Hecke L- functions in a different way.

In collaboration with Emil Artin, he formulated the class field theory with Gruppenkohomologie ( Galoiskohomologie of ideal classes). In "The higher dimensional cohomology groups of class field theory " ( Annals of Mathematics 1952) he introduced the Tate cohomology groups. On his ICM lecture in Stockholm in 1962 "Duality theorems in Galois cohomology over number fields", he formulated his duality theorems ( Tate - duality ). His Tate - Shafarevich groups are of fundamental importance for the arithmetic geometry. They measure roughly speaking, to what extent the variety of Hasse principle differs, after which one of the p- adic ( "local" ) and real solvability on the solvability in rational numbers ( "global" ) wants to close what is possible with quadratic forms ( Hasse ), in cubic curves ( elliptic curves), but have generally not more. Many results found by him to Galoiskohomologie have been published only in the books of Jean -Pierre Serre.

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

" P- ​​divisible groups" (also called Barsotti - Tate groups) of 1966 ( Proc.Conf. Local Fields, Driebergen ) treated p- adic Galois representations, ie those over local fields of characteristic p.

In the 1960s, he also articulated the Tate conjecture on algebraic cycles, which describes the action of the Galois group on the l -adic cohomology 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 constructed such cycles of cohomological information.

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

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

He gave a p- adic uniformization of elliptic curves and abelian varieties ( " Tate curve " ) and led "Rigid analytic spaces" a ( Inventiones Mathematicae 1971).

An assumption which is named after him and Mikio Sato, postulates a probability distribution of the phases of the coefficients of the Hasse -Weil zeta function of elliptic curves.

He also created the Hodge - Tate theory comes from ( as p- adisches analogue of Hodge theory) and the Honda -Tate theory ( the classification of abelian varieties over finite fields ). Also named after him, the Neron -Tate height ( also named after André Néron ), Tate cohomology, Tate motives and Tate modules ( the abelian varieties are used for the classification up to isogeny in the Tate - isogeny theorem).

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

He received the 1956 Cole Price in number theory. In 1995 he received the Leroy P. Steele Prize of the American Mathematical Society, the 2002 Wolf Prize in 2010 and the Abel Prize. In 1970 he gave a plenary lecture at the ICM in Nice ( Symbols in Arithmetic ). He is a Fellow of the American Mathematical Society.

Writings

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