Harmonic analysis on local fields and adelic spaces. I

Abstract

Harmonic analysis is developed on objects of some category C 2 of infinitedimensional filtered vector spaces over a finite field. This category includes twodimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and obtaining of two-dimensional Poisson formulas.

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“…Concerning the general formalism of harmonic analysis over n-dimensional local fields and adelic groups (n = 0, 1, 2), we refer to [P1,OP1,OP2]. The basic problem solved there was to extend the classical analysis known for locally compact (and first of all for finite) groups to the case of fields and groups arising from two-dimensional schemes.…”

Geometric and analytic structures on the higher adèles

“…There is also a variant for C := Ab or including some abelian real Lie groups, the categories C fin n or C ar n of [41].…”

Harmonic analysis and the Riemann-Roch theorem

“…Then A is a cC 2 space and B is a dC 2 space (see [3,Section 5.1]). Let o ∈ I and μ ∈ μ(W/F(o) ∩ W), ν ∈ μ(F(o)/F(o) ∩ W)*.…”

“…Let E = (I, F, V) be a C 2 space over the field k (see [2]). Recall that for any i, j ∈ I we have constructed in [3,Section 5.2] a one dimensional ‫ރ‬ vector space of virtual measures μ(F(i) | F(j)) = μ(F(i)/F(l))* ⊗ ‫ރ‬ μ(F(l)/F(j)), where l ∈ I such that l ≤ i, l ≤ j, and μ(H) is the space of ‫ރ‬ valued Haar measures on a C 1 space H. The space μ(F(i) | F(j)) does not depend on the choice of l ∈ I up to a canonical isomorphism.…”

“…In this note we show, how to solve the analogous problem for the case of dimension two. Namely, we will show how the Rie mann-Roch theorem for invertible sheaves on a pro jective smooth algebraic surface X over k (in a variant without the Noether formula, see, e.g., [7]) is obtained from the two dimensional Poisson formulas (see [3,Section 5.9] and [4, Section 13]).…”