Measure, integration and elements of harmonic analysis on generalized loop spaces

Abstract

In this work we extend the first part of the previous paper [F4] to higher dimensional local fields. We introduce a nontrivial translation invariant measure on the additive group of higher dimensional local fields, and then develop elements of integration and harmonic analysis. We also discuss its relation with several other measure theories.For an n-dimensional local field F a translation invariant measure µ is defined on a certain ring A of measurable sets and takes values in R ((X 1 )) . . . ((X n−1 )) (which itself is an n-dimensional local field whose last residue field is the archimedean field R). The algebra A in the case of higher dimensional fields with finite last residue field is the algebra generated by characteristic functions of shifts of fractional ideals, i.e. a + bO with a, b ∈ F and O the ring of integers of F with respect to the n-dimensional structure. The measure is countably additive in a refined sense (see sections 7 and 8). Elements of integration theory are introduced in sections 9-13.The additive group of a higher dimensional field has a certain topology on it with respect to which it is not locally compact for n > 1. This topology plays a key role in higher class field theory [F1-F3]. The additive group of F is self dual, which together with the measure and integration leads to analogs of many classical results in Fourier analysis (section 14). In particular, for functions in certain space we introduce their transform F(f )(α) = ψ(αβ) f (β) dµ(β) and show that F 2 (f )(α) = f (−α).In section 16 we discuss the theory in the case where the last residue field is R or C.In sections 17-21 we extend the previous theory to the case of generalized algebraic loop and path spaces (including in particular the complexified space of smooth loops), and indicate how the measure of this work can be used to extend p-adic zeta integrals associated to schemes over complete discrete valuation fields, and how it is related to nonarchimedean measures on spaces of arcs, which have recently found applications in algebraic geometry.Theory of this paper can be viewed as a part of yet unknown general theory of harmonic analysis on certain classes of non locally compact groups. It is expected to find further applications, in particular towards integration on path spaces, which is important for mathematical explanation of quantum physics (see section 18).

Positivity of certain functions associated with analysis on elliptic surfaces

“…The paper can be read independently of other related works, in particular in two-dimensional adelic analysis [2,3,5,6], and that the paper is organized as self-contained one as possible.…”

“…Both n E (s) ±1 are meromorphic on C, and are holomorphic on (s) > 1. Recently a new approach to ζ E (s) was proposed by I. Fesenko in his series of works [2,3,6] (see also the survey article [5]). His proposal is treating zeta functions ζ E (s) directly without through L-functions by establishing the theory of "two-dimensional" zeta integrals defined for functions on two-dimensional adelic objects attached to arithmetic elliptic surfaces.…”

Analysis on arithmetic schemes. II

“…The key ingredient, which has not been available until recently, is the theory of translation invariant measure and integration on higher dimensional local fields, developed in [11], [12]. To introduce the zeta integral we first construct the theory of measure and integration on new adelic objects.…”

“…Prerequisites for this work are the one-dimensional theory [57], [24] and the higher dimensional, local theory [11], [12]. A closely related text which explains the main ideas, methods, constructions, and directions of applications and which is relatively free from technical details is [14].…”

“…The sections form rather natural divisions of the material of the paper and they are of varying length; such a simple marking of the text aims to simplify its reading. The two-dimensional theory of [11] was extended to higher dimensional local fields in [12], which also contains a discussion of the links between the integration on higher local fields and the Feynman path integral. For an alternative approach to the measure, integration and zeta integrals on valuation fields whose residue field is a local field see [39]; its method uses the lifting from the residue level in a systematic way and is related to section 13 of [11].…”