Harmonic analysis on local fields and adelic spaces. II

Abstract

We develop harmonic analysis in some categories of filtered abelian groups and vector spaces. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.where the group G 0 is the connected component of the group G which contains the identity element e , the group K is a maximal compact subgroup of the group G 0 , the group G tor /G 0 is the maximal torsion subgroup of the discrete group G/G 0 . Then we have the following isomorphisms

Harmonic analysis and the Riemann-Roch theorem

“…We consider the fol lowing admissible triple of complete C 2 spaces over k: (4) where we use that ‫ށ‬ 0 = ‫ށ‬ 01 ∩ ‫ށ‬ 02 . The space ‫ށ‬ 02 ‫ށ/‬ 0 is a cfC 2 space, and the space ‫ށ(/ށ‬ 01 + ‫ށ‬ 02 ) is a dfC 2 space.…”

Harmonic analysis on the rank- value group of a two-dimensional local field

“…Earlier in papers of authors [6] and [7], harmonic analysis was constructed on adele groups of an algebraic surface (function and distribution spaces, Fourier transforms, Poisson formulas). These groups appear from adele ring A of the algebraic surface.…”

“…In paper [11] the discrete adele groups, in particular various groups of divisors, were constructed in this situation. The question on construction of harmonic analysis on these groups (such that this analysis is based on already constructed harmonic analysis from papers [6] and [7]) by analogy with the case of curves in [12] was not considered at that time. After the appearance of paper [13] and the above mentioned reformulation of properties of the zeta-function of a surface in the form of analysis on the group of 0 -cycles the following program appears.…”

“…First, one needs to transfer harmonic analysis of papers [6] and [7] to discrete groups of divisors and divisor classes. Then, a transition from the groups associated with divisors to the groups of 0 -cycles, which are directly related to the study of zeta-and L -functions, is necessary.…”

“…We define also Fourier transforms on these spaces. If the group Γ is the group Γ K , and the last residue field of a two-dimensional local field K is a finite field F q , then for r = q the constructed function and distribution spaces and Fourier transforms on them are connected with the function and distribution spaces on K and Fourier transforms there (they were constructed and investigated in [6] and [7]).…”