×

zbMATH — the first resource for mathematics

Notes on the Poisson formula. With an appendix by Irina S. Rezvyakova. (English. Russian original) Zbl 1323.11065
St. Petersbg. Math. J. 23, No. 5, 781-818 (2012); translation from Algebra Anal. 23, No. 5, 1-54 (2011).
Summary: 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.

MSC:
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967.
[2] François Bruhat, Distributions sur un groupe localement compact et applications à l’étude des représentations des groupes ℘-adiques, Bull. Soc. Math. France 89 (1961), 43 – 75 (French). · Zbl 0128.35701
[3] A. Connes, Trace formulas in noncommutative geometry and the zeros of the Riemann zeta function, Preprint IHES/M/98/72, 1998, 88 pp.
[4] Christopher Deninger, Lefschetz trace formulas and explicit formulas in analytic number theory, J. Reine Angew. Math. 441 (1993), 1 – 15. · Zbl 0782.11034 · doi:10.1515/crll.1993.441.1 · doi.org
[5] Max Deuring, Lectures on the theory of algebraic functions of one variable, Lecture Notes in Mathematics, Vol. 314, Springer-Verlag, Berlin-New York, 1973. · Zbl 0249.14008
[6] Jacques Dixmier, Les \?*-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). · Zbl 0288.46055
[7] I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 2, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1968 [1977]. Spaces of fundamental and generalized functions; Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. I. M. Gel\(^{\prime}\)fand and G. E. Shilov, Generalized functions. Vol. 3, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1967 [1977]. Theory of differential equations; Translated from the Russian by Meinhard E. Mayer. I. M. Gel\(^{\prime}\)fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. I. M. Gel\(^{\prime}\)fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1966 [1977]. Integral geometry and representation theory; Translated from the Russian by Eugene Saletan.
[8] Теория представлений и автоморфные функции, Генерализед фунцтионс, Но. 6, Издат. ”Наука”, Мосцощ, 1966 (Руссиан). И. М. Гел\(^{\приме}\)фанд, М. И. Граев, анд И. И. Пятецкии-Шапиро, Репресентатион тхеоры анд аутоморпхиц фунцтионс, Транслатед фром тхе Руссиан бы К. А. Хирсч, Щ. Б. Саундерс Цо., Пхиладелпхиа, Па.-Лондон-Торонто, Онт., 1969.
[9] Jay Jorgenson, Serge Lang, and Dorian Goldfeld, Explicit formulas, Lecture Notes in Mathematics, vol. 1593, Springer-Verlag, Berlin, 1994. · Zbl 0804.00008
[10] E. Hecke, Über die Zetafunktion beliebiger algebraischer Zahlkorper, Gött. Nachr. (1917), 77-89. · JFM 46.0256.02
[11] -, Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verlagsges, Leipzig, 1923. · JFM 49.0106.10
[12] K. Iwasawa, A note on functions, Proc. of the Internat. Congress of Mathematicians (Cambridge, Mass., 1950). Vol. 2, Amer. Math. Soc., Providence, RI, 1952, p. 322.
[13] Kenkichi Iwasawa, Letter to J. Dieudonné, Zeta functions in geometry (Tokyo, 1990) Adv. Stud. Pure Math., vol. 21, Kinokuniya, Tokyo, 1992, pp. 445 – 450. · Zbl 0835.11002
[14] Serge Lang, Algebraic numbers, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto-London, 1964. · Zbl 0211.38501
[15] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. · Zbl 0211.38404
[16] Ralf Meyer, On a representation of the idele class group related to primes and zeros of \?-functions, Duke Math. J. 127 (2005), no. 3, 519 – 595. · Zbl 1079.11044 · doi:10.1215/S0012-7094-04-12734-4 · doi.org
[17] D. V. Osipov and A. N. Parshin, Harmonic analysis on local fields and adelic spaces. I, Izv. Ross. Akad. Nauk Ser. Mat. 72 (2008), no. 5, 77 – 140 (Russian, with Russian summary); English transl., Izv. Math. 72 (2008), no. 5, 915 – 976. · Zbl 1222.11137 · doi:10.1070/IM2008v072n05ABEH002424 · doi.org
[18] D. V. Osipov and A. N. Parshin, Harmonic analysis on local fields and adelic spaces. II, Izv. Ross. Akad. Nauk Ser. Mat. 75 (2011), no. 4, 91 – 164 (Russian, with Russian summary); English transl., Izv. Math. 75 (2011), no. 4, 749 – 814. · Zbl 1232.11122 · doi:10.1070/IM2011v075n04ABEH002552 · doi.org
[19] -, Harmonic analysis and the Riemann-Roch theorem, e-print arXov:1107.0408. · Zbl 1350.14015
[20] A. N. Parshin, Harmonic analysis on adelic spaces and local fields, Mathematisches Forschungsinstitut Oberwolfach, Report No. 43/2005, Arakelov Geometry, September 2005, pp. 2471-2474.
[21] -, Numbers as functions: the development of an idea in the Moscow school of algebraic geometry, Mathematical Events of XX Century , Fazis, Moscow, 2003, pp. 363-397; e-print arXiv: 0912.3785. (Russian)
[22] A. N. Parshin, On holomorphic representations of discrete Heisenberg groups, Funktsional. Anal. i Prilozhen. 44 (2010), no. 2, 92 – 96 (Russian); English transl., Funct. Anal. Appl. 44 (2010), no. 2, 156 – 159. · Zbl 1232.43005 · doi:10.1007/s10688-010-0020-3 · doi.org
[23] A. N. Parshin, Representations of higher adelic groups and arithmetic, Proceedings of the International Congress of Mathematicians. Volume I, Hindustan Book Agency, New Delhi, 2010, pp. 362 – 392. · Zbl 1266.11118
[24] -, Lectures on representations of discrete Heisenberg groups, Humboldt Univ., Berlin, October 2010 (Notes by Aaron Greicius and Hartwig Mayer).
[25] B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsb. der Berliner Akad. 1858/1860, 671-680 (Gesamm. Math. Werke, Teubner, Leipzig, 1982, No. VII, S. 145-155).
[26] S. N. Bernšteĭn, On the relation of quasi-analytic functions with weight functions, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 773 – 776 (Russian).
[27] I. R. Shafarevich, The zeta-function. Notes of lectures 1966-1967, Moskov. Gos. Univ., Moscow, 1969. (Russian)
[28] Математический анализ: Специал\(^{\приме}\)ный курс, 2нд ед, Государств. Издат. Физ.-Мат. Лит., Мосцощ, 1961 (Руссиан).
[29] Jean-Pierre Serre, Groupes algébriques et corps de classes, Publications de l’institut de mathématique de l’université de Nancago, VII. Hermann, Paris, 1959 (French). · Zbl 0097.35604
[30] Jean-Pierre Serre, Sur la rationalité des représentations d’Artin, Ann. of Math. (2) 72 (1960), 405 – 420 (French). · Zbl 0202.32803 · doi:10.2307/1970142 · doi.org
[31] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968 (French). Deuxième édition; Publications de l’Université de Nancago, No. VIII. · Zbl 1095.11504
[32] John Torrence Tate Jr, FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS, ProQuest LLC, Ann Arbor, MI, 1950. Thesis (Ph.D.) – Princeton University.
[33] John Tate, Genus change in inseparable extensions of function fields, Proc. Amer. Math. Soc. 3 (1952), 400 – 406. · Zbl 0047.03901
[34] Дзета-функция Римана, Физико-Математическая Литература, Мосцощ, 1994 (Руссиан, щитх Руссиан суммары). А. А. Карацуба анд С. М. Воронин, Тхе Риеманн зета-фунцтион, Де Груытер Ешпоситионс ин Матхематицс, вол. 5, Щалтер де Груытер & Цо., Берлин, 1992. Транслатед фром тхе Руссиан бы Неал Коблитз.
[35] André Weil, Fonction zêta et distributions, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 312, 523 – 531 (French).
[36] André Weil, Basic number theory, 3rd ed., Springer-Verlag, New York-Berlin, 1974. Die Grundlehren der Mathematischen Wissenschaften, Band 144. · Zbl 0326.12001
[37] André Weil, Sur les formules explicites de la théorie des nombres, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3 – 18 (French, with Russian summary). · Zbl 0245.12010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.