Notes on the Poisson formula

A. N. Parshin

Algebra i analiz, Tome 23 (2011) no. 5, pp. 1-54.

Bibliographie

[1] J. W. S. Cassels, A. Fröhlich (eds.), Algebraic Number Theory, Acad. Press, London–New York, 1967 ; Algebraicheskaya teoriya chisel, Mir, M., 1969 | MR | Zbl | MR

[2] Bruhat F., “Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes $\mathcal W$-adiques”, Bull. Soc. Math. France, 89 (1961), 43–75 | MR | Zbl

[3] Connes A., Trace formulas in noncommutative geometry and the zeros of the Riemann zeta function, Preprint IHES/M/98/72, 1998, 88 pp.

[4] Deninger C., “Lefschetz trace formulas and explicit formulas in analytic number theory”, J. Reine Angew. Math., 441 (1993), 1–15 | DOI | MR | Zbl

[5] Deuring M., Lectures on the theory of algebraic functions of one variable, Lecture Notes in Math., 314, Springer-Verlag, Berlin–New York, 1973 | MR | Zbl

[6] Dixmier J., Les $C^*$-algèbres et leurs représentations, Cahies Sci., 29, Gauthier-Villar, Paris, 1969 ; Diksme Zh., $C^*$-algebry i ikh predstavleniya, Nauka, M., 1974 | MR | Zbl | MR

[7] Gelfand I. M., Shilov G. E., Obobschennye funktsii i deistviya nad nimi, Obobschennye funktsii, 1, Fizmatgiz, M., 1959

[8] Gelfand I. M., Graev M. I., Pyatetskii-Shapiro I. I., Teoriya predstavlenii i avtomorfnye funktsii, Obobschennye funktsii, 6, Nauka, M., 1966 | MR

[9] Goldfeld D., “A spectral interpretation of Weil's explicit formula”, Explicit formulas, Lecture Notes in Math., 1593, Springer-Verlag, Berlin etc., 1994, 135–152 | MR | Zbl

[10] Hecke E., “Über die Zetafunktion beliebiger algebraischer Zahlkorper”, Gött. Nachr., 1917, 77–89 | Zbl

[11] Hecke E., Vorlesungen über die Theorie der algebraischen Zahlen, Akad. Verlagsges, Leipzig, 1923 ; Gekke E., Lektsii po teorii algebraicheskikh chisel, Gostekhizdat, M.–L., 1940 | Zbl

[12] Iwasawa K., “A note on functions”, Proc. of the Internat. Congress of Mathematicians (Cambridge, Mass., 1950), v. 2, Amer. Math. Soc., Providence, RI, 1952, 322

[13] Iwasawa K., “Letter to J. Dieudonné”, Zeta Functions in Geometry (Tokyo, 1990), Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo, 1992, 445–450 | MR | Zbl

[14] Lang S., Algebraic numbers, Addison-Wesley, Reading, 1964 ; Leng S., Algebraicheskie chisla, Mir, M., 1966 | MR | Zbl | MR

[15] Lang S., Algebraic number theory, Addison-Wesley, Reading, 1970 | MR | Zbl

[16] Meyer R., “On a representation of the idele class group related to primes and zeros of $L$-functions”, Duke Math. J., 127 (2005), 519–595 | DOI | MR | Zbl

[17] Osipov D. V., Parshin A. N., “Garmonicheskii analiz na lokalnykh polyakh i prostranstvakh adelei, I”, Izv. RAN. Ser. mat., 72:5 (2008), 77–140, arXiv: 0707.1766 | MR | Zbl

[18] Osipov D. V., Parshin A. N., “Garmonicheskii analiz na lokalnykh polyakh i prostranstvakh adelei, II”, Izv. RAN. Ser. mat., 75:4 (2011), 91–164, arXiv: 0912.1577

[19] Osipov D. V., Parshin A. N., Harmonic analysis and the Riemann–Roch theorem, arXiv: 1107.0408

[20] Parshin A. N., “Harmonic analysis on adelic spaces and local fields”, Arakelov Geometry, Mathematisches Forschungsinstitut, Oberwolfach, September 2005, Report No. 43/2005, 2471–2474

[21] Parshin A. N., “Chisla kak funktsii: razvitie odnoi idei v moskovskoi shkole algebraicheskoi geometrii”, Matematicheskie sobytiya dvadtsatogo veka, eds. A. Bolibrukh i dr., Fazis, M., 2003, 363–397, arXiv: 0912.3785

[22] Parshin A. N., “O golomorfnykh predstavleniyakh diskretnykh grupp Geizenberga”, Funkts. anal. i ego pril., 44:2 (2010), 92–96 | MR | Zbl

[23] Parshin A. N., “Representations of higher adelic groups and arithmetic”, Proc. ICM, v. 1, Hyderabad, 2010, 362–392, arXiv: 1012.0486

[24] Parshin A. N., Lectures on representations of discrete Heisenberg groups, Notes by Aaron Greicius and Hartwig Mayer, Humboldt Univ., Berlin, October 2010

[25] Riemann B., “Über die Anzahl der Primzahlen unter einer gegebenen Grösse”, Monatsb. der Berliner Akad., 1859, 671–680; Gesamm. Math. Werke, VII, Teubner, Leipzig, 1982, 145–155

[26] Schwartz L., Théorie des distributions, v. II, Actualites Sci. Indust., 1122, Hermann, Paris, 1951 | MR | Zbl

[27] Shafarevich I. R., Dzeta-funktsiya, Zap. lektsii 1966–1967 gg., MGU, M., 1969

[28] Shilov G. E., Matematicheskii analiz (spetsialnyi kurs), Fizmatgiz, M., 1961 | MR

[29] Serre J.-P., Groupes algébriques et corps de classes, Publ. Inst. Math. Univ. Nancago, 7, Hermann, Paris, 1959 ; Serr Zh.-P., Algebraicheskie gruppy i polya klassov, Mir, M., 1968 | MR | Zbl

[30] Serre J.-P., “Sur la rationalité des représentations d'Artin”, Ann. of Math. (2), 72 (1960), 405–420 | DOI | MR | Zbl

[31] Serre J.-P., Corps locaux, Publ. Univ. Nancago, 8, Hermann, Paris, 1968 | MR

[32] Tate J., Fourier analysis in number fields and Hecke's zeta-function, Thesis, Princeton Univ., Ann Arbor, 1950, (izlozheno v [14] i opublikovano v [1])

[33] Tate J., “Genus change in inseparable extensions of function fields”, Proc. Amer. Math. Soc., 3 (1952), 400–406 | DOI | MR | Zbl

[34] Voronin S. M., Karatsuba A. A., Dzeta-funktsiya Rimana, Fizmatlit, M., 1994 | MR | Zbl

[35] Weil A., “Fonctions zêta et distributions”, Séminaire Bourbaki, 9, Soc. Math. France, Paris, 1995, Exp. No 312, 523–531 | MR

[36] Weil A., Basic number theory, Grundlehren Math. Wiss., 144, 3rd ed., Springer-Verlag, Berlin, 1974 ; Veil A., Osnovy teorii chisel, Mir, M., 1972 | MR | Zbl | MR

[37] Weil A., “Sur les formules explicites de la théorie des nombres”, Izv. AN SSSR. Ser. mat., 36:1 (1972), 3–18 | MR | Zbl