Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Gauss’ class number problem for imaginary quadratic fields
HTML articles powered by AMS MathViewer

by Dorian Goldfeld PDF
Bull. Amer. Math. Soc. 13 (1985), 23-37
References
  • A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI 10.1112/S0025579300002588
  • A. Baker, Imaginary quadratic fields with class number $2$, Ann. of Math. (2) 94 (1971), 139–152. MR 299583, DOI 10.2307/1970739
  • 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
  • B. J. Birch and N. M. Stephens, Heegner’s construction of points on the curve $y^{2}=x^{3}-1728e^{3}$, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982) Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 1–19. MR 729156, DOI 10.1007/BF02591848
  • B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108. MR 179168, DOI 10.1515/crll.1965.218.79
  • S. Chowla, The Heegner-Stark-Baker-Deuring-Siegel theorem, J. Reine Angew. Math. 241 (1970), 47–48. MR 258762, DOI 10.1515/crll.1970.241.47
  • 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, DOI 10.1007/978-1-4757-5927-3
  • Max Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl 1, Math. Z. 37 (1933), no. 1, 405–415 (German). MR 1545403, DOI 10.1007/BF01474583
  • Max Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins, Invent. Math. 5 (1968), 169–179 (German). MR 228464, DOI 10.1007/BF01425548
  • 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.
  • A. O. Gel′fond, Transcendental and algebraic numbers, Dover Publications, Inc., New York, 1960. Translated from the first Russian edition by Leo F. Boron. MR 0111736
  • 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 404212
  • Dorian M. Goldfeld, A simple proof of Siegel’s theorem, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 1055. MR 344222, DOI 10.1073/pnas.71.4.1055
  • 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 450233
  • 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) Astérisque No. 41–42, Soc. Math. France, Paris, 1977, pp. 219–227. MR 0447176
  • 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.
  • Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253 (German). MR 53135, DOI 10.1007/BF01174749
  • 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.
  • Jeffrey Hoffstein, On the Siegel-Tatuzawa theorem, Acta Arith. 38 (1980/81), no. 2, 167–174. MR 604232, DOI 10.4064/aa-38-2-167-174
  • 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.
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287, DOI 10.1007/BF02684339
  • 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.
  • H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta Arith. 24 (1973/74), 529–542. MR 357373, DOI 10.4064/aa-24-5-529-542
  • 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.
  • Carl Ludwig Siegel, Zum Beweise des Starkschen Satzes, Invent. Math. 5 (1968), 180–191 (German). MR 228465, DOI 10.1007/BF01425549
  • H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050, DOI 10.1307/mmj/1028999653
  • H. M. Stark, A historical note on complex quadratic fields with class-number one, Proc. Amer. Math. Soc. 21 (1969), 254–255. MR 237461, DOI 10.1090/S0002-9939-1969-0237461-X
  • H. M. Stark, On the “gap” in a theorem of Heegner, J. Number Theory 1 (1969), 16–27. MR 241384, DOI 10.1016/0022-314X(69)90023-7
  • H. M. Stark, A transcendence theorem for class-number problems, Ann. of Math. (2) 94 (1971), 153–173. MR 297715, DOI 10.2307/1970740
  • Tikao Tatuzawa, On a theorem of Siegel, Jpn. J. Math. 21 (1951), 163–178 (1952). MR 51262, DOI 10.4099/jjm1924.21.0_{1}63
  • 42. A. Weil, Sur un théorème de Mordell, Bull. Sci. Math. (2) 54 (1930), 182-191.
  • André Weil, Sur les fonctions algébriques à corps de constantes fini, C. R. Acad. Sci. Paris 210 (1940), 592–594 (French). MR 2863
  • André Weil, Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156 (German). MR 207658, DOI 10.1007/BF01361551
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
  • Journal: Bull. Amer. Math. Soc. 13 (1985), 23-37
  • MSC (1980): Primary 12A50, 12A25; Secondary 12-03
  • DOI: https://doi.org/10.1090/S0273-0979-1985-15352-2
  • MathSciNet review: 788386