Notes on the Poisson formula

Parshin, A.

doi:10.1090/S1061-0022-2012-01218-5

Notes on the Poisson formula
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by A. N. Parshin
Translated by: The Author

St. Petersburg Math. J. 23 (2012), 781-818

DOI: https://doi.org/10.1090/S1061-0022-2012-01218-5

Published electronically: July 10, 2012

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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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