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Numbers as Functions: The Development of an Idea in the Moscow School of Algebraic Geometry

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Mathematical Events of the Twentieth Century

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Bibliography

  1. Algebra, Mathematical Logic, Number Theory, Topology (survey articles on the occasion of the fiftieth anniversary of the Steklov Mathematical Institute of the USSR Academy of Sciences). Proc. Steklov Inst. Math., 1984, 168.

    Google Scholar 

  2. Algebraic Surfaces. Proc. Steklov Inst. Math., 1965, 75, 281 pp.

    Google Scholar 

  3. S. Yu. Arakelov. Intersection theory of divisors on an arithmetic surface. Math. USSR, Izv., 1974, 8, 1167–1180.

    Article  Google Scholar 

  4. S. Yu. Arakelov. Theory of intersections on an arithmetic surface. In: Proceedings of the International Congress of Mathematicians (Vancouver, 1974), V. 1. Montreal: Canad. Math. Congress, 1975, 405–408.

    Google Scholar 

  5. Z. I. Borevich, I. R. Shafarevich. Number Theory (transl. N. Greenleaf). New York-London: Academic Press, 1966 (Pure Appl. Math., 20).

    MATH  Google Scholar 

  6. J. B. Bost. Fibrés déterminants, déterminants régularisés et mesures sur les espaces de modules des courbes complexes. In: Séminaire Bourbaki, février 1987, exposé 676. Astérisque, 152–153. Paris: Soc. Math. France, 1987, 113–149.

    Google Scholar 

  7. H. Brückner. Eine explizite Formel zum Reziprozitätsgesetz für Primzahlexponenten p. In: Algebraische Zahlentheorie (Oberwolfach, 1964). Mannheim: Bibliographisches Institut, 1967, 31–39.

    Google Scholar 

  8. H. Brückner. Explizites Reziprozitätsgesetz und Anwendungen. Vorles. Fachber. Math. Univ. Essen, 1979, Heft 2.

    Google Scholar 

  9. H. Cartan. Séminaire École Normale Supérieure, 1950/51, 1951/52, 1953/54, 1957/58. Secrétariat mathématiques, Paris.

    Google Scholar 

  10. J. W. S. Cassels, A. Frölich. Algebraic Number Theory. London: Academic Press, 1967.

    MATH  Google Scholar 

  11. N. G. Chebotarev. Theory of Algebraic Functions. Moscow-Leningrad: Gostekhizdat, 1948 (Russian).

    Google Scholar 

  12. P. Deligne. Intersections sur les surfaces régulières. In: Groupes de monodromie en géométrie algébrique. II (SGA 7 II, 1967–1969). Berlin-New York: Springer, 1973, exposé X, 1–38 (Lecture Notes in Math., 340).

    Chapter  Google Scholar 

  13. P. Deligne. Le déterminant de la cohomologie. In: Current Trends in Arithmetical Algebraic Geometry (Arcata, CA, 1985). Providence, RI: Amer. Math. Soc., 1987, 93–177 (Contemp. Math, 67).

    Google Scholar 

  14. P. Deligne. Quelques idées maîtresses de l’œuvre de A. Grothendieck. Matériaux pour l’histoire des matématiques au XXième siècle. Actes de colloque à la mémoire de Jean Dieudonné (Nice, 1996). Paris: Soc. Math. France, 1998, 11–20 (Séminaires et Congrès, 3).

    Google Scholar 

  15. S. P. Demushkin et al. Igor’ Rostislavovich Shafarevich (on his sixtieth birthday). Russ. Math. Surveys, 1984, 39(1), 189–200.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Enriques. Le superficie algebriche. Bologna: Nicola Zanichelli, 1949.

    MATH  Google Scholar 

  17. D. K. Faddeev. On Hilbert’s ninth problem. In: The Hilbert Problems. Moscow: Nauka, 1969, 131–140 (Russian).

    Google Scholar 

  18. G. Faltings. Calculus on arithmetic surfaces. Ann. Math., 1984, 119, 387–424.

    Article  MathSciNet  Google Scholar 

  19. G. Faltings. Lectures on the Arithmetic Riemann-Roch Theorem. Princeton, NJ, 1992 (Ann. Math. Studies, 127).

    Google Scholar 

  20. A. Grothendieck. The cohomology theory of abstract algebraic varieties. In: Proceedings of the International Congress of Mathematicians (Edinburgh, 1958). Cambridge University Press, 1960, 103–118.

    Google Scholar 

  21. K. Hensel. Theorie der algebraischen Zahlen. Leipzig: B. G. Teubner, 1908.

    MATH  Google Scholar 

  22. D. Hilbert. Über die Theorie der relativquadratischer Zahlkörper. Jahresber. Deutsch. Math.-Verein., 1899, 6; Gesammelte Abhandlungen [23], Bd. 1, 364–369.

    Google Scholar 

  23. D. Hilbert. Gesammelte Abhandlungen, Bd. 1–3. Berlin: Springer, 1932–1935; New York: Chelsea, 1965 (reprint); Russian translation: D.Hilbert. Selected Works, V. 1–2. Moscow: Factorial, 1998.

    Google Scholar 

  24. D. Hilbert. Mathematische Probleme. In: Gesammelte Abhandlungen [23], Bd. 2, 401–436.

    Google Scholar 

  25. W. V. D. Hodge. The Theory and Applications of Harmonic Integrals. Cambridge University Press, 1941.

    Google Scholar 

  26. A. I. Kostrikin et al. Vasilii Alekseevich Iskovskikh (on his 60th birthday). Russ. Math. Surveys, 1999, 54(4), 863–868.

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Lang. Fundamentals of Diophantine Geometry. New York: Springer, 1983.

    MATH  Google Scholar 

  28. S. Lang. Introduction to Arakelov Theory. New York: Springer, 1988.

    MATH  Google Scholar 

  29. A. I. Lapin. The theory of the Shafarevich symbol. Izv. Akad. Nauk SSSR, Ser. Mat., 1953, 17, 31–50 (Russian).

    MathSciNet  MATH  Google Scholar 

  30. A. I. Lapin. On the theory of the Shafarevich symbol. Izv. Akad. Nauk SSSR, Ser. Mat., 1954, 18, 145–158 (Russian).

    MathSciNet  MATH  Google Scholar 

  31. A. I. Lapin. The general reciprocity law and a new foundation of class-field theory. Izv. Akad. Nauk SSSR, Ser. Mat., 1954, 18, 335–378 (Russian).

    MathSciNet  MATH  Google Scholar 

  32. S. Lichtenbaum. Curves over discrete valuation rings. Amer. J. Math., 1968, 15, 380–405.

    Article  MathSciNet  Google Scholar 

  33. A. Néron. Quasi-fonctions et hauteurs sur les variétés abéliennes. Ann. Math., 1965, 82, 249–331.

    Article  Google Scholar 

  34. I. Newton. The Mathematical Works, Vol. I, II. Assembled with an introduction by D. T. Whiteside. New York-London: Johnson Reprint Corp., 1964; 1967.

    Google Scholar 

  35. O. A. Oleinik. On Hilbert’s sixteenth problem. In: The Hilbert Problems. Moscow: Nauka, 1969, 182–195.

    Google Scholar 

  36. A. N. Parshin. Modular correspondences, heights, and isogeny of Abelian varieties. Trudy Matem. Inst. im. Steklova, 1973, 132, 211–236 (Russian).

    Google Scholar 

  37. A. N. Parshin. On the application of ramified coverings in the theory of Diophantine equations. Math. USSR, Sb., 1990, 66(1), 249–264.

    Article  MathSciNet  MATH  Google Scholar 

  38. Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), V. 1. Providence, RI: Amer. Math. Soc., 1952, p. 122.

    Google Scholar 

  39. J.-P. Serre. Faisceaux algébriques cohérents. Ann. Math., 1955, 61, 197–278.

    Article  MathSciNet  Google Scholar 

  40. I. R. Shafarevich. The general reciprocity law. Matem. Sb., 1950, 26, 113–146 (Russian).

    Google Scholar 

  41. I. R. Shafarevich. Algebraic number fields. In: Proceedings of the International Congress of Mathematicians (Stockholm, 1962). Djursholm: Institut Mittag-Leffler, 1963, 163–176 (Russian).

    Google Scholar 

  42. I. R. Shafarevich. Lectures on Minimal Models and Birational Transformations of Two-Dimensional Schemes. Bombay: Tata Institute of Fundamental Research, 1966.

    MATH  Google Scholar 

  43. C. L. Siegel. Analytic Functions of Several Complex Variables. Lectures delivered at the Institute for Advanced Study, 1948–1949.

    Google Scholar 

  44. C. L. Siegel. Automorphic Functions of Several Complex Variables. Moscow: Inostrannaya Literatura, 1954 (Russian translation).

    Google Scholar 

  45. C. Soulé (with D. Abramovich, J.-F. Burnol, and J. Kramer). Lectures on Arakelov Geometry. Cambridge University Press, 1992.

    Google Scholar 

  46. C. Soulé. Hermitian vector bundles on arithmetic varieties. Proc. Symp. Pure Math., 1997, 62, Part 1, 383–419.

    Google Scholar 

  47. C. Soulé. On the field with one element. Talk given at the Arbeitstagung, Bonn, June 1999. Preprint IHES/M/99/55.

    Google Scholar 

  48. C. Soulé, S. Bloch, H. Gillet. Non-Archimedean Arakelov theory. J. Alg. Geom., 1995, 4, 427–485.

    MATH  Google Scholar 

  49. L. Szpiro, ed. Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell. Astérisque, 127. Paris: Soc. Math. France, 1985.

    Google Scholar 

  50. V. S. Vladimirov, I. V. Volovich, E. I. Zelenov. p-Adic Analysis and Mathematical Physics. River Edge, NJ: World Scientific, 1994.

    Google Scholar 

  51. S. V. Vostokov. Explicit form of the law of reciprocity. Math. USSR, Izv., 1978, 13, 557–588.

    Article  Google Scholar 

  52. A. Weil. Number-theory and algebraic geometry. In: Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. 2. Providence, RI: Amer. Math. Soc., 1952, 90–110; André Weil. Œuvres Scientifiques/Collected Papers, Vol. 1 (1926–1951). Heidelberg-Berlin: Springer, 1980 (corrected second printing), 442–452.

    Google Scholar 

  53. A. Weil. Introduction à l’étude des variétés kählériennes. Paris: Hermann, 1958.

    MATH  Google Scholar 

  54. Yu. G. Zarkhin, A. N. Parshin. Finiteness problems in Diophantine geometry. Appendix to the Russian translation of: S. Lang. Fundamentals of Diophantine Geometry. Moscow: Mir, 1986, 369–448 (Russian).

    Google Scholar 

  55. S. Zdravkovska, P. Duren, eds. The Golden Years of Moscow Mathematics. Providence, RI: Amer. Math. Soc., 1993.

    Google Scholar 

  56. S. Zhang. Admissible pairing on a curve. Invent. Math., 1993, 112(1), 171–193.

    Article  MathSciNet  MATH  Google Scholar 

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Parshin, A.N. (2006). Numbers as Functions: The Development of an Idea in the Moscow School of Algebraic Geometry. In: Bolibruch, †.A.A., et al. Mathematical Events of the Twentieth Century. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-29462-7_15

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