On the problem of the 10th discriminant

Shafarevich, I.

doi:10.1090/S1061-0022-2014-01312-X

On the problem of the 10th discriminant
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by I. R. Shafarevich
Translated by: B. M. Bekker

St. Petersburg Math. J. 25 (2014), 699-711

DOI: https://doi.org/10.1090/S1061-0022-2014-01312-X

Published electronically: June 5, 2014

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Abstract:

An elementary proof is given for Heegner’s theorem describing imaginary quadratic fields with class number one.

References

Carl Friedrich Gauss, Disquisitiones arithmeticae, Springer-Verlag, New York, 1986. Translated and with a preface by Arthur A. Clarke; Revised by William C. Waterhouse, Cornelius Greither and A. W. Grootendorst and with a preface by Waterhouse. MR 837656
Carl Ludwig Siegel, Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin-New York, 1966 (German). Herausgegeben von K. Chandrasekharan und H. Maass. MR 0197270
Serge Lang, Algebraic numbers, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto-London, 1964. MR 0160763
H. Heilbronn, On the class numbers of imaginary quadratic fields, Quart. J. Math. 5 (1935), 150–160.
Max Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl 1, Math. Z. 37 (1933), no. 1, 405–415 (German). MR 1545403, DOI 10.1007/BF01474583
H. Heilbronn and E. H. Linfoot, On the imaginary quadratic corpora of class number one, Quart. J. Math. 5 (1934), 293–301.
Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Z. 56 (1952), 227–253 (German). MR 53135, DOI 10.1007/BF01174749
H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050
A. Baker, Linear forms in the logarithms of algebraic numbers. IV, Mathematika 15 (1968), 204–216. MR 258756, DOI 10.1112/S0025579300002588
A. Baker, A remark on the class number of quadratic fields, Bull. London Math. Soc. 1 (1969), 98–102. MR 241383, DOI 10.1112/blms/1.1.98
Max Deuring, Imaginäre quadratische Zahlkörper mit der Klassenzahl Eins, Invent. Math. 5 (1968), 169–179 (German). MR 228464, DOI 10.1007/BF01425548
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
Z. I. Borevich and I. R. Shafarevich, Teoriya chisel, 3rd ed., “Nauka”, Moscow, 1985 (Russian). MR 816135
R. Füeter, Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen, Bd. 1, Teubner, Berlin, 1924.
A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, and J.-P. Serre, Seminar on complex multiplication, Lecture Notes in Mathematics, No. 21, Springer-Verlag, Berlin-New York, 1966. Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. MR 0201394
A. G. Kurosh, The theory of groups. Vol. II, Chelsea Publishing Co., New York, N.Y., 1956. Translated from the Russian and edited by K. A. Hirsch. MR 0080089
Carl Ludwig Siegel, Gesammelte Abhandlungen. Band IV, Springer-Verlag, Berlin-New York, 1979 (German). Edited by K. Chandrasekharan and H. Maass; With corrections to the first three volumes. MR 543842
David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
V. A. Abraškin, Determination of the imaginary quadratic fields of class number two with even discriminant by the method of Heegner, Mat. Zametki 15 (1974), 241–246 (Russian). MR 354608
H. M. Stark, On complex quadratic fields wth class-number two, Math. Comp. 29 (1975), 289–302. MR 369313, DOI 10.1090/S0025-5718-1975-0369313-X