nLab Iwasawa-Tate theory

Idea

What has come to be called Iwasawa-Tate theory is a method of expressing zeta functions, L-functions and theta functions as adelic integrals over idele groups and deriving this way their fundamental properties such as their analytic continuation, their functional equation and their Euler product form.

John Tate in his thesis (Tate 50) had generalized the notion of zeta function according to his advisor Emil Artin from “the sum over integral ideals of certain type of ideal character” to

the adelic integral over the idèle group of a rather general weight function times the idèle character which is trivial on field elements. The role of Hecke’s complicated theta-formulas for theta functions formed over a lattice in $n$-dimensional space of classical number theory can be played by a simple Poisson formula for general function of valuation vectors, summed over the discrete subgroup of field elements

Kenkichi Iwasawa in Iwasawa 5x has rediscovered and extended this approach using the wider study of invariant integration? on locally compact groups over ideles. It is distinguished from the related but different subject of Iwasawa theory (cf. wikipedia)

Related entries

• adelic integration, locally compact group, Fourier analysis, zeta function, L-function, ring of adeles, group of ideles

References

The original articles are

• John Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Princeton, May 1950, thesis; reproduced in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 305–347, Acad, Press 1967

  MR0217026

• Kenkichi Iwasawa, A note on functions, Proc. ICM 1950, link MR0044534; On the rings of valuation vectors, Ann. Math. (II) 57:2 (Mar., 1953), pp. 331-356 jstor; Letter to Jean Dieudonne link

Reviews include

• Paul Garrett, Iwasawa-Tate on ζ-functions and L-functions, 2011

  (pdf)

• cyclopaedia.info, Iwasawa-Tate theory

• Wikipedia, Tate’s thesis

  Bibliographic fragments concerning Iwasawa-Tate theory, (pdf)

Further developments include

• David Wright, Twists of the Iwasawa-Tate zeta function, Math. Zeitschrift 200:2, pp 209-231, 1989 (pdf)