1,314 research outputs found
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 onto a smooth
algebraic curve , both given over a perfect field , we construct the
direct image morphism in two cases: from to
and when from to
. (If i=2, then the last map is the Gysin map from
to .)
To do this in the first case we use the known adelic resolution for sheafs
and . In the second case we construct a -adelic
resolution of a sheaf . 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 ( is a closed
point on an irredicuble curve ) to 1-dimensional local fields
associated with closed points on the curve .
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
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