number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, Tate's thesis is the 1950 PhD thesis of completed under the supervision of

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

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function twisted by a

Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ...

, i.e. a Hecke

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 gi ...

, of a

number field 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 ...

to a zeta integral and study its properties. Using

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

, more precisely the

Poisson summation formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the pe ...

, he proved the

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

and meromorphic continuation of the zeta integral and the Hecke L-function. He also located the poles of the twisted zeta function. His work can be viewed as an elegant and powerful reformulation of a work of

Erich Hecke Erich Hecke (; 20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms. Biography Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He ...

on the proof of the functional equation of the Hecke L-function.

used a generalized

theta series In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube dom ...

associated to an algebraic number field and a lattice in its ring of integers.

Iwasawa–Tate theory

Kenkichi Iwasawa Kenkichi Iwasawa ( ''Iwasawa Kenkichi'', September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. Biography Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunm ...

independently discovered essentially the same method (without an analog of the local theory in Tate's thesis) during the

Second World War World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

and announced it in his 1950

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

paper and his letter to

Jean Dieudonné Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous ...

written in 1952. Hence this theory is often called Iwasawa–Tate theory. Iwasawa in his letter to Dieudonné derived on several pages not only the meromorphic continuation and

of the L-function, he also proved finiteness of the class number and

Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In Mathematical analysis, analysis, h ...

's theorem on units as immediate byproducts of the main computation. The theory in positive characteristic was developed one decade earlier by

Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the f ...

, Wilfried Schmid, and Oswald Teichmüller. Iwasawa–Tate theory uses several structures which come from

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 ...

, however it does not use any deep result of class field theory.

Generalisations

Iwasawa–Tate theory was extended to the

general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

GL(n) over an algebraic number field and automorphic representations of its adelic group by Roger Godement and Hervé Jacquet in 1972 which formed the foundations of the Langlands correspondence. Tate's thesis can be viewed as the GL(1) case of the work by Godement–Jacquet.

References

* * * * * * * {{Number theory Algebraic number theory Zeta and L-functions 1950 in science 1950 documents

Iwasawa–Tate theory

Generalisations

See also

References