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St. Petersburg Mathematical Journal

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Notes on the Poisson formula

Author: A. N. Parshin
Translated by: The Author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 5.
Journal: St. Petersburg Math. J. 23 (2012), 781-818
MSC (2010): Primary 11M41
Published electronically: July 10, 2012
MathSciNet review: 2918423
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Abstract | References | Similar Articles | Additional Information

Abstract: This is a survey of applications of harmonic analysis to the study of the zeta-functions of one-dimensional schemes. A new version of the Tate-Iwasawa method is suggested that involves holomorphic duality for discrete groups instead of Pontryagin duality. A relationship is found between the Poisson formula and the residue formula on the compactification of the holomorphically dual group. Links to explicit formulas for zeta-functions of algebraic curves are found. A numerical analog of these constructions is considered in the appendix written by I. S. Rezvyakova.

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Additional Information

A. N. Parshin
Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina St. 8, Moscow 119991, Russia

Keywords: Fourier transform, Poisson formula, zeta-function, holomorphic duality, residues, explicit formulas
Received by editor(s): July 7, 2010
Published electronically: July 10, 2012
Additional Notes: Supported by RFBR (grant no. 11-01-00145-a)
Article copyright: © Copyright 2012 American Mathematical Society

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