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 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.
We define a class of monomial not necessary unitary representations of discrete Heisenberg groups. Their moduli space is a complex-analytic manifold. There exist characters of the representations which are automorphic forms on the moduli space.
This paper is a continuation of papers: arXiv:0707.1766 [math.AG] and
arXiv:0912.1577 [math.AG]. Using the two-dimensional Poisson formulas from
these papers and two-dimensional adelic theory we obtain the Riemann-Roch
formula on a projective smooth algebraic surface over a finite field.Comment: 7 pages; to appear in Doklady Mathematic
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