Generation of Hauptmoduln of Γ1(N) by Weierstrass units and application to class fields

Chang Kim; Ja Koo

Open Mathematics (2011)

  • Volume: 9, Issue: 6, page 1389-1402
  • ISSN: 2391-5455

Abstract

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We show that the modular functions j 1,N generate function fields of the modular curve X 1(N), N ∈ {7; 8; 9; 10; 12}, and apply them to construct ray class fields over imaginary quadratic fields.

How to cite

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Chang Kim, and Ja Koo. "Generation of Hauptmoduln of Γ1(N) by Weierstrass units and application to class fields." Open Mathematics 9.6 (2011): 1389-1402. <http://eudml.org/doc/269704>.

@article{ChangKim2011,
abstract = {We show that the modular functions j 1,N generate function fields of the modular curve X 1(N), N ∈ \{7; 8; 9; 10; 12\}, and apply them to construct ray class fields over imaginary quadratic fields.},
author = {Chang Kim, Ja Koo},
journal = {Open Mathematics},
keywords = {Modular curve; Modular function; Class field; Modular function:Class field},
language = {eng},
number = {6},
pages = {1389-1402},
title = {Generation of Hauptmoduln of Γ1(N) by Weierstrass units and application to class fields},
url = {http://eudml.org/doc/269704},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Chang Kim
AU - Ja Koo
TI - Generation of Hauptmoduln of Γ1(N) by Weierstrass units and application to class fields
JO - Open Mathematics
PY - 2011
VL - 9
IS - 6
SP - 1389
EP - 1402
AB - We show that the modular functions j 1,N generate function fields of the modular curve X 1(N), N ∈ {7; 8; 9; 10; 12}, and apply them to construct ray class fields over imaginary quadratic fields.
LA - eng
KW - Modular curve; Modular function; Class field; Modular function:Class field
UR - http://eudml.org/doc/269704
ER -

References

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  1. [1] Chen I., Yui N., Singular values of Thompson series, In: Groups, Difference Sets, and the Monster, Columbus, 1993, Ohio State Univ. Math. Res. Inst. Publ., 4, de Gruyter, Berlin, 1996, 255–326 http://dx.doi.org/10.1515/9783110893106.255 
  2. [2] Conway J.H., Norton S.P., Monstrous moonshine, Bull. London Math. Soc., 1979, 11(3), 308–339 http://dx.doi.org/10.1112/blms/11.3.308 Zbl0424.20010
  3. [3] Darmon H., Note on a polynomial of Emma Lehmer, Math. Comp., 1991, 56(194), 795–800 http://dx.doi.org/10.1090/S0025-5718-1991-1068821-5 
  4. [4] Deuring M., Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ., 1941, 14, 197–272 http://dx.doi.org/10.1007/BF02940746 Zbl0025.02003
  5. [5] Ishida N., Ishii N., Generators and defining equation of the modular function field of the group Γ1(N), Acta Arith., 2002, 101(4), 303–320 http://dx.doi.org/10.4064/aa101-4-1 Zbl1004.11036
  6. [6] Janusz G.J., Algebraic Number Fields, Pure Appl. Math., 55, Academic Press, New York-London, 1973 
  7. [7] Kim C.H., Koo J.K., On the genus of some modular curve of level N, Bull. Austral. Math. Soc., 1996, 54(2), 291–297 http://dx.doi.org/10.1017/S0004972700017755 Zbl0894.11018
  8. [8] Kim C.H., Koo J.K., Arithmetic of the modular function j 1,8, Ramanujan J., 2000, 4(3), 317–338 http://dx.doi.org/10.1023/A:1009857205327 
  9. [9] Kubert D.S., Lang S., Modular Units, Grundlehren Math. Wiss., 244 Springer, New York-Berlin, 1981 
  10. [10] Lang S., Elliptic Functions, 2nd ed., Grad. Texts in Math., 112, Springer, New York, 1987 http://dx.doi.org/10.1007/978-1-4612-4752-4 
  11. [11] Lecacheux O., Unités d’une famille de corps cycliques réeles de degré 6 liés à la courbe modulaire X 1(13), J. Number Theory, 1989, 31(1), 54–63 http://dx.doi.org/10.1016/0022-314X(89)90051-6 Zbl0664.12004
  12. [12] Lecacheux O., Unités d’une famille de corps liés à la courbe X 1(25), Ann. Inst. Fourier (Grenoble), 1990, 40(2), 237–253 http://dx.doi.org/10.5802/aif.1212 Zbl0739.11023
  13. [13] Miyake T., Modular Forms, Springer, Berlin, 1989 
  14. [14] Néron A., Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. Inst. Hautes Études Sci., 1964, 21, 5–125 http://dx.doi.org/10.1007/BF02684271 Zbl0132.41403
  15. [15] Ogg A.P., Rational points on certain elliptic modular curves, In: Analytic Number Theory, St. Louis University, St. Louis, 1972, Proc. Sympos. Pure Math., 24, American Mathematical Society, Providence, 1973, 221–231 
  16. [16] Schoeneberg B., Elliptic Modular Functions, Grundlehren Math. Wiss., 203, Springer, New York-Heidelberg, 1974 
  17. [17] Serre J.-P., Tate J., Good reduction of abelian varieties, Ann. of Math., 1968, 88, 492–517 http://dx.doi.org/10.2307/1970722 Zbl0172.46101
  18. [18] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan, 11, Iwanami Shoten, Tokyo, 1971 Zbl0221.10029
  19. [19] Silverman J.H., Advanced Topics in the Arithmetic of Elliptic Curves, Grad. Texts in Math., 151, Springer, New York, 1994 Zbl0911.14015
  20. [20] Stevens G., Arithmetic on Modular Curves, Progr. Math., 20, Birkhäuser, Boston, 1982 Zbl0529.10028
  21. [21] Washington L.C., A family of cyclic quartic fields arising from modular curves, Math. Comp., 1991, 57(196), 763–775 http://dx.doi.org/10.1090/S0025-5718-1991-1094964-6 Zbl0743.11058

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