On the normalizer of the congruence subgroup $H^5_0 (I)$ of the Hecke group $H^5$
On the normalizer of the congruence subgroup $H^5_0 (I)$ of the Hecke group $H^5$
Let $lambda = 2cos frac{pi}{5}$ and let $H^5$ be the Hecke group associated to $lambda$. In this paper, the normalizersof the congruence subgroups $H^5_0 (I)$ in $PSL(2,Bbb{Z}[lambda])$ are studied in the case where $I = (2)^{alpha}I', (2, I')$ = 1 and I' is a prime ideal.
___
- [1] Akbaş, M., Singerman, D.: The Normalizer Γ(N) in PSL(2,R), Glasgow Math. J. 32,317-327 (1990).
- [2] Atkin, A. O. L., Lehner, J.: Hecke operatorson Γ0(m), Math. Ann. 185,134-160 (1970).
- [3] Chan, S.-P., Lang, M.-L., Lim, C.-H., Tan, S.-P.: The invariants of the congruence subgroups $G_5(I)$ of the Hecke group $G_5$, Illinois J. of Math. 38,636-652 (1994).
- [4] Conway, J.H., Norton, S.P.: Montrous Moonshine, Bull. London Math. Soc. 11,308-339 (1979).
- [5] Conway, J.H.: Understanding Groups like $Gamma_0(N)$, Groups, difference sets and the monester (Columbus, Ohio, 1993), Ohio State Univ. Math. Res. Inst. Publ., 4, de Gruyter, Berlin, 327-343 (1996).
- [6] Hecke, E.: Uber die Bestimmung Dirichletscher Reichen durch ihre Funktionalgleichungen, Math. Ann. 112,664-699(1936).
- [7] Hungerfort, T.W.: Algebra, Springer Verlag, Springer Verlag, New York, Heidelberg, Berlin, 1980.
- [8] Ivrissimtzis, I.P.: Congruence subgroups of Hecke groups and regular dessins, Ph.D. Thesis, University of Southampton, 1998.
- [9] Lang, M.-L.: Normalizersof the congruence subgroups of the Hecke groups $G_4$ and $G_6$, Journal of Number Theory, 90,31-43 (2001).
- [10] Lang, M.-L., Lim, C.-H., Tan, S.-P.: Principal Congruence Subgroupsof the Hecke Groups, Journal of Number Theory 85, 220-130 (2000).
- [11] Lang, M.-L., Tan, S.-P.: Normalizersof the congruence subgroups of the Hecke group $G_5$ II, Proc.Amer.Math.Soc. 127, 2271-2280 (2000).
- [12] Lang, M.-L., Tan, S.-P.: Normalizersof the congruence subgroups of the Hecke group $G_5$, Proc.Amer.Math. Soc. 127, 3131-3140 (1999).
- [13] Lang, M.-L., Lim, C.-H., Tan, S.-P.: Independent generatorsf or congruence subgroups of Hecke groups, Math. Z. 220, 569-594 (1995).
- [14] Leutbecher, A.: Uber die Heckeschen Gruppen $G(lambda)$, Abh. Math. Sem. Hambg. 31, 199-205 (1967).
- [15] Leutbecher, A.: Uber die Heckeschen Gruppen $G(lambda)$, II, Math. Ann. 211, 63-68 (1974).
- [16] Newman, M.: The normalizer of certain Modular Subgroups, Canada J. Math. 8, 29-31 (1956).
- [17] Parson, L.A.: Generalized Kloosterman sums and the Fourier coefficients of cusp forms, Trans. Amer. Math. Soc. 217, 329-350 (1976).
- [18] Rosen, D.: The substitutions of the Hecke group $Gamma (2cos frac{pi}{5})$, Arch. Math.46, 533-538 (1986).