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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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  • [A] Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • [Br] 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). MR 0140941
  • [C] A. Connes, Trace formulas in noncommutative geometry and the zeros of the Riemann zeta function, Preprint IHES/M/98/72, 1998, 88 pp.
  • [Den] Christopher Deninger, Lefschetz trace formulas and explicit formulas in analytic number theory, J. Reine Angew. Math. 441 (1993), 1–15. MR 1228608, 10.1515/crll.1993.441.1
  • [D] Max Deuring, Lectures on the theory of algebraic functions of one variable, Lecture Notes in Mathematics, Vol. 314, Springer-Verlag, Berlin-New York, 1973. MR 0344231
  • [Dix] Jacques Dixmier, Les 𝐶*-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). MR 0246136
  • [GSh] I. M. Gel′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. MR 0435831
  • [GGPSh] I. M. Gel′fand, M. I. Graev, and I. I. Pjateckiĭ-Šapiro, Teoriya predstavlenii i avtomorfnye funktsii, Generalized functions, No. 6, Izdat. “Nauka”, Moscow, 1966 (Russian). MR 0220673
    I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Translated from the Russian by K. A. Hirsch, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR 0233772
  • [G] Jay Jorgenson, Serge Lang, and Dorian Goldfeld, Explicit formulas, Lecture Notes in Mathematics, vol. 1593, Springer-Verlag, Berlin, 1994. MR 1329730
  • [H1] E. Hecke, Über die Zetafunktion beliebiger algebraischer Zahlkorper, Gött. Nachr. (1917), 77-89.
  • [H2] -, Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verlagsges, Leipzig, 1923.
  • [I1] 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.
  • [I2] Kenkichi Iwasawa, Letter to J. Dieudonné, Zeta functions in geometry (Tokyo, 1990) Adv. Stud. Pure Math., vol. 21, Kinokuniya, Tokyo, 1992, pp. 445–450. MR 1210798
  • [L1] Serge Lang, Algebraic numbers, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto-London, 1964. MR 0160763
  • [L2] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [M] Ralf Meyer, On a representation of the idele class group related to primes and zeros of 𝐿-functions, Duke Math. J. 127 (2005), no. 3, 519–595. MR 2132868, 10.1215/S0012-7094-04-12734-4
  • [OP1] 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. MR 2473773, 10.1070/IM2008v072n05ABEH002424
  • [OP2] 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. MR 2866188, 10.1070/IM2011v075n04ABEH002552
  • [OP3] -, Harmonic analysis and the Riemann-Roch theorem, e-print arXov:1107.0408.
  • [P1] 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.
  • [P2] -, Numbers as functions: the development of an idea in the Moscow school of algebraic geometry, Mathematical Events of XX Century (A. Bolibrukh et al., eds.), Fazis, Moscow, 2003, pp. 363-397; e-print arXiv: 0912.3785. (Russian)
  • [P3] 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. MR 2681962, 10.1007/s10688-010-0020-3
  • [P4] 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. MR 2827898
  • [P5] -, Lectures on representations of discrete Heisenberg groups, Humboldt Univ., Berlin, October 2010 (Notes by Aaron Greicius and Hartwig Mayer).
  • [R] 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).
  • [Sch] S. N. Bernšteĭn, On the relation of quasi-analytic functions with weight functions, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 773–776 (Russian). MR 0041945
  • [Sh] I. R. Shafarevich, The zeta-function. Notes of lectures 1966-1967, Moskov. Gos. Univ., Moscow, 1969. (Russian)
  • [S] G. E. Šilov, Matematicheskii analiz: Spetsialnyi kurs, 2nd ed, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). MR 0131328
  • [S1] 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). MR 0103191
  • [S2] Jean-Pierre Serre, Sur la rationalité des représentations d’Artin, Ann. of Math. (2) 72 (1960), 405–420 (French). MR 0171775
  • [S3] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968 (French). Deuxième édition; Publications de l’Université de Nancago, No. VIII. MR 0354618
  • [T1] 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. MR 2612222
  • [T2] John Tate, Genus change in inseparable extensions of function fields, Proc. Amer. Math. Soc. 3 (1952), 400–406. MR 0047631, 10.1090/S0002-9939-1952-0047631-9
  • [VK] S. M. Voronin and A. A. Karatsuba, Dzeta-funktsiya Rimana, Fiziko-Matematicheskaya Literatura, Moscow, 1994 (Russian, with Russian summary). MR 1918212
    A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function, de Gruyter Expositions in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1992. Translated from the Russian by Neal Koblitz. MR 1183467
  • [W1] André Weil, Fonction zêta et distributions, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 312, 523–531 (French). MR 1610983
  • [W2] André Weil, Basic number theory, 3rd ed., Springer-Verlag, New York-Berlin, 1974. Die Grundlehren der Mathematischen Wissenschaften, Band 144. MR 0427267
  • [W3] 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). MR 0379440

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Additional Information

A. N. Parshin
Affiliation: Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina St. 8, Moscow 119991, Russia
Email: parshin@mi.ras.ru

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