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GENERATION OF CLASS FIELDS BY SIEGEL-RAMACHANDRA INVARIANTS

신동화
대한수학회 (The Korean Mathematical Society)|2015.09|대한수학회지 (Journal of the KMS)|제52권 제5호|pp. 907-928

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참고문헌

키워드

complex multiplicationclass field theoryclass numberselliptic and modular unitsmodular and automorphic functions
주제분류수학

초록

We show in many cases that the Siegel-Ramachandra invari- ants generate the ray class fields over imaginary quadratic fields. As its application we revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of inequality argument together with Shimura’s reciprocity law.

Abstract

We show in many cases that the Siegel-Ramachandra invari- ants generate the ray class fields over imaginary quadratic fields. As its application we revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of inequality argument together with Shimura’s reciprocity law.

참고문헌 (27)

  1. A. Baker, "Linear forms in the logarithms of algebraic numbers I", Mathematika, 13, pp.204~216 (1968)
  2. B. Baran, "A modular curve of level 9 and the class number one problem", J. Number Theory, 129(3), pp.715~728 (2009)
  3. B. Baran, "Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem", J. Number Theory, 130(12), pp.2753~2772 (2010)
  4. D. A. Cox, "Primes of the form x2 + ny2 : Fermat, Class Field, and Complex Multiplication" (1989), A Wiley-Interscience Publication, John Wiley & Sons, Inc
  5. M. Deuring, "Imagin¨are quadratische Zahlk¨orper mit der Klassenzahl Eins", Invent. Math, 5, pp.169~179 (1968)
  6. in Q(j()), "Math. Ann", 1989, 177~191
  7. A. Gee, "Class invariants by Shimura’s reciprocity law", J. Th´eor. Nombres Bordeaux, 11(1), pp.45~72 (1999)
  8. B. H. Gross, "On singular moduli", J. Reine Angew. Math, 355, pp.191~220 (1985)
  9. K. Heegner, "Diophantische Analysis und Modulfunktionen", Math. Z, 56, pp.227~253 (1952)
  10. G. J. Janusz, "Algebraic Number Fields", Grad. Studies in Math, 7 (1996), Amer. Math. Soc
  11. J. K. Koo, "On some arithmetic properties of Siegel functions", Math. Z, 264(1), pp.137~177 (2010)
  12. D. Kubert, "Modular Units", Grundlehren der mathematischen Wis-senschaften, 244 (1981), Spinger-Verlag
  13. E. Landau, "Uber die Klassenzahl der binaren quadratischen Formen von negativer Discriminante", Math. Ann, 56(4), pp.671~676 (1903)
  14. S. Lang, "Introduction to Modular Forms", Grundlehren der mathematischen Wis-senschaften, (222) (1976), Springer-Verlag
  15. S. Lang, "Elliptic Functions", Grad. Texts in Math, 112 (1987), Spinger-Verlag
  16. S. Lang, "Algebraic Number Theory" (1994), Spinger-Verlag
  17. K. Ramachandra, "Some applications of Kronecker’s limit formula", Ann. of Math. (2), 80, pp.104~148 (1964)
  18. P. Ribenboim, "Classical Theory of Algebraic Numbers, Universitext" (2001), Springer-Verlag
  19. R. Schertz, "Die singul¨aren Werte der Weberschen Funktionen f, f1, f2, 2, 2", J. Reine Angew. Math, 286, pp.287 (1976), 46~74
  20. R. Schertz, "Construction of ray class fields by elliptic units", J. Th´eor. Nombres Bordeaux, 9(2), pp.383~394 (1997)
  21. R. Schertz, "Weber’s class invariants revisited", J. Th´eor. Nombres Bordeaux, 14(1), pp.325~343 (2002)
  22. J.-P. Serre, "Lectures on the Mordell-Weil Theorem", Aspects of Mathematics, E15 (1989), Vieweg & Sohn
  23. G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions" (1971), Iwanami Shoten and Princeton University Pr
  24. H. M. Stark, "A complete determination of the complex quadratic fields of class number one", Michigan Math. J, 14, pp.1~27 (1967)
  25. H. M. Stark, "On the “gap” in a theorem of Heegner", J. Number Theory, 1, pp.16~27 (1969)
  26. P. Stevenhagen, "Hilbert’s 12th problem, complex multiplication and Shimura reciprocity, Class Field Theory-Its Centenary and Prospect", Adv. Stud. Pure Math., 30, pp.161~176 (2001), Math. Soc. Japan
  27. H. Weber, "Lehrbuch der Algebra. Vol. III" (1961), Chelsea

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