Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Gauss' class number problem for imaginary quadratic fields


Author: Dorian Goldfeld
Journal: Bull. Amer. Math. Soc. 13 (1985), 23-37
MSC (1980): Primary 12A50, 12A25; Secondary 12-03
MathSciNet review: 788386
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 0258756
  • 2. A. Baker, Imaginary quadratic fields with class number 2, Ann. of Math. (2) 94 (1971), 139–152. MR 0299583
  • 3. B. J. Birch, Diophantine analysis and modular functions, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 35–42. MR 0258832
  • 4. B. J. Birch and N. M. Stephens, Heegner’s construction of points on the curve 𝑦²=𝑥³-1728𝑒³, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982) Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 1–19. MR 729156, 10.1007/BF02591848
  • 5. B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108. MR 0179168
  • 6. S. Chowla, The Heegner-Stark-Baker-Deuring-Siegel theorem, J. Reine Angew. Math. 241 (1970), 47–48. MR 0258762
  • 7. Harold Davenport, Multiplicative number theory, 2nd ed., Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York-Berlin, 1980. Revised by Hugh L. Montgomery. MR 606931
  • 8. Max Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl 1, Math. Z. 37 (1933), no. 1, 405–415 (German). MR 1545403, 10.1007/BF01474583
  • 9. Max Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins, Invent. Math. 5 (1968), 169–179 (German). MR 0228464
  • 10. L. Dirichlet, Recherches sur diverse applications de l'analyse infinitésimale à la théorie des nombres, J. Reine Angew. Math. 19 (1839); ibid. 21 (1840).
  • 11. L. Euler, Mém. de Berlin, année 1722, 36; Comm. Arith. 1, 584.
  • 12. C. F. Gauss, Disquisitiones arithmeticae, 1801.
  • 13. A. O. Gel′fond, Transcendental and algebraic numbers, Translated from the first Russian edition by Leo F. Boron, Dover Publications, Inc., New York, 1960. MR 0111736
  • 14. Dorian Goldfeld, An asymptotic formula relating the Siegel zero and the class number of quadratic fields, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 611–615. MR 0404212
  • 15. Dorian M. Goldfeld, A simple proof of Siegel’s theorem, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 1055. MR 0344222
  • 16. Dorian M. Goldfeld, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 624–663. MR 0450233
  • 17. Dorian M. Goldfeld, The conjectures of Birch and Swinnerton-Dyer and the class numbers of quadratic fields, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976) Soc. Math. France, Paris, 1977, pp. 219–227. Astérisque No. 41–42. MR 0447176
  • 18. Benedict Gross and Don Zagier, Points de Heegner et dérivées de fonctions 𝐿, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 2, 85–87 (French, with English summary). MR 720914
  • 19. H. Hasse, Beweis analogous der Riemannschen Vermutung für die Artinsche und F. K. Schmidtschen Kongruenz-zetafunktionen in gewisse elliptischen Fallen, Nachr. Akad. Wiss. Göttingen (1933), 253-262.
  • 20. Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253 (German). MR 0053135
  • 21. H. Heilbronn, On the class number in imaginary quadratic fields, Quart. J. Math. Oxford Ser. 2 5 (1934), 150-160.
  • 22. H. Heilbronn and E. H. Linfoot, On the imaginary quadratic corpora of class number one, Quart. J. Math. Oxford Ser 2 5 (1934), 293-301.
  • 23. Jeffrey Hoffstein, On the Siegel-Tatuzawa theorem, Acta Arith. 38 (1980/81), no. 2, 167–174. MR 604232
  • 24. C. G. J. Jacobi, J. Math. 9 (1832), 189-192.
  • 25. E. Landau, Über die Klassenzahl der binären quadratischen Formen von negativer Discriminante, Math. Ann. 56 (1902), 671-676.
  • 26. E. Landau, Über die Klassenzahl imaginär-quadratischer Zahlkörper, Göttinger Nachr. (1918), 285-295.
  • 27. J. L. Lagrange, Recherches d'arithmétique, Nouv. Mém. Acad. Berlin (1773), 265-312; Oeuvres, III, pp. 693-758.
  • 28. A. M. Legendre, Théorie des nombres, Libraire Scientifique A. Hermann, Paris, 1798, pp. 69-76; 2nd éd., 1808, pp. 61-67; 3rd ed., 1830, pp. 72-80.
  • 29. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). MR 488287
  • 30. L. J. Mordell, On the Riemann hypothesis and imaginary quadratic fields with a given class number, J. London Math. Soc. 9 (1934), 289-298.
  • 31. L. J. Mordell, On the rational solutions of the indeterminate equations of the 3rd and 4th degrees, Proc. Camb. Phil. Soc. 21 (1922), 179-192.
  • 32. H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta Arith. 24 (1973/74), 529–542. Collection of articles dedicated to Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, V. MR 0357373
  • 33. Joseph Oesterlé, Nombres de classes des corps quadratiques imaginaires, Astérisque 121-122 (1985), 309–323 (French). Seminar Bourbaki, Vol. 1983/84. MR 768967
  • 34. G. Rabinovitch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern, Proc. Fifth Internat. Congress Math. (Cambridge), vo. I, 1913, pp. 418-421.
  • 35. C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83-86.
  • 36. Carl Ludwig Siegel, Zum Beweise des Starkschen Satzes, Invent. Math. 5 (1968), 180–191 (German). MR 0228465
  • 37. H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 0222050
  • 38. H. M. Stark, A historical note on complex quadratic fields with class-number one., Proc. Amer. Math. Soc. 21 (1969), 254–255. MR 0237461, 10.1090/S0002-9939-1969-0237461-X
  • 39. H. M. Stark, On the “gap” in a theorem of Heegner, J. Number Theory 1 (1969), 16–27. MR 0241384
  • 40. H. M. Stark, A transcendence theorem for class-number problems, Ann. of Math. (2) 94 (1971), 153–173. MR 0297715
  • 41. Tikao Tatuzawa, On a theorem of Siegel, Jap. J. Math. 21 (1951), 163–178 (1952). MR 0051262
  • 42. A. Weil, Sur un théorème de Mordell, Bull. Sci. Math. (2) 54 (1930), 182-191.
  • 43. André Weil, Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci. Paris 210 (1940), 592–594 (French). MR 0002863
  • 44. André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156 (German). MR 0207658

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 12A50, 12A25, 12-03

Retrieve articles in all journals with MSC (1980): 12A50, 12A25, 12-03


Additional Information

DOI: http://dx.doi.org/10.1090/S0273-0979-1985-15352-2