Tate's thesis
In
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 on the proof of the functional equation of the Hecke L-function. Erich Hecke used a generalized theta series
associated to an algebraic number field and a lattice in its ring of integers.
Iwasawa–Tate theory
Second World War and announced it in his 1950 International Congress of Mathematicians paper and his letter to Jean 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 Ernst Witt, Wilfried Schmid, and Oswald Teichmüller
.
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
Iwasawa–Tate theory was extended to the
Langlands correspondence
. Tate's thesis can be viewed as the GL(1) case of the work by Godement–Jacquet.
See also
References
- Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lect. Notes Math., vol. 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, vol. 1, Providence, R.I.: MR 0044534, archived from the originalon 2011-10-03
- Iwasawa, Kenkichi (1992) [1952], "Letter to J. Dieudonné", in Kurokawa, Nobushige; Sunada., T. (eds.), Zeta functions in geometry (Tokyo, 1990), Adv. Stud. Pure Math., vol. 21, Tokyo: Kinokuniya, pp. 445–450, MR 1210798
- Kudla, Stephen S. (2003), "Tate's thesis", in MR 1990377
- Ramakrishnan, Dinakar; Valenza, Robert J. (1999). Fourier analysis on number fields. Graduate Texts in Mathematics. Vol. 186. New York: Springer-Verlag. 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, MR 0217026
