Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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)).
For L-functions: Wikipedia Rankin–Selberg method – Modern adelic theory
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.
Analogies with the Feynman path integral are pointed out in (Fesenko 06, section 18)…
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
Last revised on November 8, 2019 at 13:17:35. See the history of this page for a list of all contributions to it.