Nonclassical harmonic analysis and hyper zeta function ofarithmetic
schemes
(Un exposé de Ivan Fesenko au séminaire de théorie
des nombres de Chevaleret le 3 décembre 2001)
Résumé :
The talk presents parts of a generalization
of Tate's thesis to arithmetic schemes.
A generalization of the Haar measure and Fourier transform to
higher local fields
(which are not locally compact groups) will be explained.
A new measure on higher dimensional local fields
takes values in hyperreal numbers.
Local hyper zeta function is introduced as a certain integral
against a measure on topological K-groups of the fields;
it satisfies a functional equation.
Global hyper zeta function is then defined as a certain product of
local zeta functions.