nLab adelic integral

Idea

The concept of adelic integration is a concept of integration in arithmetic geometry over rings of adeles.

This was introduced for arithmetic curves in (Tate 50, Iwasawa 52, Iwasawa 92) and used to establish analytic continuation, functional equations and Euler product-form of L-functions and zeta functions, see below. Here one speaks of Iwasawa-Tate theory. In (Fesenko 08, chapter 3) adelic integration is generalized to higher arithmetic geometry. A textbook account is in (Goldfeld-Hundley 11, section 2,2)).

Examples

Zeta functions and L-functions

• The completed Riemann zeta function (see there) is naturally an adelic integral (reviewed e.g. in Goldfeld-Hundley 11, (2.2.6)).

For L-functions: Wikipedia Rankin–Selberg method – Modern adelic theory

$p$-Adic string scattering amplitudes

In the context of p-adic string theory (see there) one considers adelic integral version of the Veneziano amplitude and of some further string scattering amplitudes for the open string.

Adelic path integral

Analogies with the Feynman path integral are pointed out in (Fesenko 06, section 18)…

Related entries

• Iwasawa-Tate theory

• Mellin transform

References

The original articles are

• John Tate, Fourier analysis in number fields, and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950

• 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

The generalization to higher arithmetic geometry and hence to arithmetic zeta functions is due to

• Ivan Fesenko, Measure, integration and elements of harmonic analysis on generalized loop spaces, Proc. of

  St Petersburg Math. Soc. 12(2005), 179–199; English transl. in AMS Transl. Series 2, 219 (2006), 149–164 (pdf)

• Ivan Fesenko, chaper 3 of Adelic approch to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273–317 (pdf)

• Ivan Fesenko, Analysis on arithmetic schemes I (pdf)

• Ivan Fesenko, Analysis on arithmetic schemes II, Journal of K-theory 5: 437–557, 2010 (pdf)

A textbook account is in

• Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)