Harmonic analysis and the Riemann-Roch theorem

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

Adelic constructions of direct images for differentials and symbols

For a projective morphism of an smooth algebraic surface $X$ onto a smooth algebraic curve $S$, both given over a perfect field $k$, we construct the direct image morphism in two cases: from $H^i(X,\Omega^2_X)$ to $H^{i-1}(S,\Omega^1_S)$ and when $char k =0$ from $H^i(X,K_2(X))$ to $H^{i-1}(S,K_1(S))$. (If i=2, then the last map is the Gysin map from $CH^2(X)$ to $CH^1(S)$.) To do this in the first case we use the known adelic resolution for sheafs $\Omega^2_X$ and $\Omega^1_S$. In the second case we construct a $K_2$-adelic resolution of a sheaf $K_2(X)$. And thus we reduce the direct image morphism to the construction of some residues and symbols from differentials and symbols of 2-dimensional local fields associated with pairs $x \in C$ ($x$ is a closed point on an irredicuble curve $C \in X$) to 1-dimensional local fields associated with closed points on the curve $S$. We prove reciprocity laws for these maps.Comment: 29 pages, modified version of the article, appeared in "Matematicheskiy Sbornik" 5(188) (1997

n-dimensional local fields and adeles on n-dimensional schemes

It is a survey paper on n-dimensional local fields and adeles on n-dimensional schemes.Comment: 30 pages, submitted for publication in the LMS Lecture Notes Serie

On some questions related to the Krichever correspondence

We investigate various new properties and examples of one-dimensional and two-dimensional Krichever correspondence developed by Parshin. In particular, we give explicit examples of the Krichever-Parshin map for various plane curves, we introduce analogs of the Schur pairs in a two-dimensional local field and show that they are oft geometrical. At the end we investigate analogs of the KP hierarchy for two-dimensional local skew-fields with arbitrary commutation law instead of the usual law of Weyl algebra. We derive for these hierarchies new partial differential equations, which coincide with the usual KP equation for certain values of parameters.Comment: 13