Tate's thesis

In number theory, Tate's thesis is the 1950 thesis of John Tate (1950) under supervision of Emil Artin. In it, he used a translation invariant integration on the locally compact group of ideles to lift the zeta function of a number field, twisted by a Hecke character, to a zeta integral and study its properties. Using harmonic analysis, more precisely the summation formula, he proved the functional equation and meromorphic continuation of the zeta integral and the twisted zeta 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 on the proof of the functional equation of the twisted zeta function (L-function). Hecke used a generalized theta series associated to an algebraic number field and a lattice in its ring of integers.

Kenkichi Iwasawa independently discovered during the war essentially the same method (without an analog of the local theory in Tate's thesis) and announced it in his 1950 ICM paper and his letter to Dieudonné 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 functional equation of the L-function, he also proved finiteness of the class number and Dirichlet's theorem on units as immediate byproducts of the main computation. The theory in positive characteristic was developed one decade earlier by Witt, Schmid and Teichmuller.

Iwasawa-Tate theory uses several structures which come from class field theory, however it does not use any deep result of class field theory.

Generalisations

A noncommutative generalisation: Iwasawa-Tate theory was extended to a general linear group over an algebraic number field and automorphic representations of its adelic group by Roger Godement and Hervé Jacquet in 1972. This work is part of activities in the Langlands correspondence.

A higher-dimensional generalisation: unramified Iwasawa-Tate theory was extended to a regular model of an elliptic curve over an algebraic number field and the function field of a curve over a finite field by Ivan Fesenko in 2010. This work is part of activities in the study of the arithmetic zeta functions of arithmetic schemes using complex analytic and higher adelic methods. It uses K-theoretical structures which are involved in higher class field theory but does not use deep results of the latter.

References

• Fesenko, Ivan (2010), "Analysis on arithmetic schemes. II", J. K-theory (Cambridge University Press) 5: 437–557 
• Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lect. Notes Math. 260, Springer 
• Goldfeld, Dorian; Hundley, Joseph (2011), Automorphic representations of L-functions for the general linear group, Cambridge University Press 
• Iwasawa, Kenkichi (1952), "A note on functions", Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950 1, Providence, R.I.: American Mathematical Society, p. 322, MR 0044534 
• Iwasawa, Kenkichi (1992) [1952], "Letter to J. Dieudonné", in Kurokawa, Nobushige; Sunada., T., Zeta functions in geometry (Tokyo, 1990), Adv. Stud. Pure Math. 21, Tokyo: Kinokuniya, pp. 445–450, ISBN 978-4-314-10078-6, MR 1210798 
• Kudla, Stephen S. (2003), "Tate's thesis", in Bernstein, Joseph; Gelbart, Stephen, An introduction to the Langlands program (Jerusalem, 2001), Boston, MA: Birkhäuser Boston, pp. 109–131, ISBN 978-0-8176-3211-3, MR 1990377 
• Ramakrishnan, Dinakar; Valenza, Robert J. (1999). Fourier analysis on number fields. Graduate Texts in Mathematics 186. New York: Springer-Verlag. doi:10.1007/978-1-4757-3085-2. ISBN 0-387-98436-4. MR 1680912. 
• Tate, John T. (1950), "Fourier analysis in number fields, and Hecke's zeta-functions", Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–347, ISBN 978-0-9502734-2-6, MR 0217026