Normalizers of the congruence subgroups of the Hecke group 𝐺₅

Lang, Mong-Lung; Tan, Ser-Peow

doi:10.1090/S0002-9939-99-05154-0

Normalizers of the congruence subgroups of the Hecke group $G_5$
HTML articles powered by AMS MathViewer

by Mong-Lung Lang and Ser-Peow Tan

Proc. Amer. Math. Soc. 127 (1999), 3131-3140

DOI: https://doi.org/10.1090/S0002-9939-99-05154-0

Published electronically: May 4, 1999

PDF | Request permission

Abstract:

Let $\lambda = 2$cos$(\pi /5)$ and let $G$ be the Hecke group associated to $\lambda$. In this article, we show that for $\tau$ a prime ideal in $\mathbb {Z}[\lambda ]$, the congruence subgroups $G_{0}(\tau )$ of $G$ are self-normalized in $PSL_{2}(\mathbb {R})$.

References

A. O. L. Atkin and J. Lehner, Hecke operators on $\Gamma _{0}(m)$, Math. Ann. 185 (1970), 134–160. MR 268123, DOI 10.1007/BF01359701
J. H. Conway, Understanding groups like $\Gamma _0(N)$, Groups, difference sets, and the Monster (Columbus, OH, 1993) Ohio State Univ. Math. Res. Inst. Publ., vol. 4, de Gruyter, Berlin, 1996, pp. 327–343. MR 1400424
S. P. Chan, M. L. Lang, C. H. Lim, S. P. Tan, The invariants of the congruence subgroups $G_{0}(P)$ of the Hecke group, Illinois J. of Math. $38$ ($1994$), $636-652$.
Armin Leutbecher, Über die Heckeschen Gruppen ${\mathfrak {G}}(\lambda )$, Abh. Math. Sem. Univ. Hamburg 31 (1967), 199–205 (German). MR 228438, DOI 10.1007/BF02992399
Armin Leutbecher, Über die Heckeschen Gruppen $G(\lambda )$. II, Math. Ann. 211 (1974), 63–86 (German). MR 347736, DOI 10.1007/BF01344143
M. L. Lang, S. P. Tan, Normalizer of the congruence subgroups of the Hecke groups $G_{4}$ and $G_{6}$., (in preparation).
Mong-Lung Lang, Chong-Hai Lim, and Ser Peow Tan, Independent generators for congruence subgroups of Hecke groups, Math. Z. 220 (1995), no. 4, 569–594. MR 1363856, DOI 10.1007/BF02572632
M. L. Lang, C. H. Lim, S. P. Tan, Principal congruence subgroups of the Hecke groups, (submitted for publication).
J. Lehner and M. Newman, Weierstrass points of $\Gamma _{0}\,(n)$, Ann. of Math. (2) 79 (1964), 360–368. MR 161841, DOI 10.2307/1970550
L. Alayne Parson, Generalized Kloosterman sums and the Fourier coefficients of cusp forms, Trans. Amer. Math. Soc. 217 (1976), 329–350. MR 412112, DOI 10.1090/S0002-9947-1976-0412112-8
David Rosen, The substitutions of Hecke group $\Gamma (2\,\textrm {cos}(\pi /5))$, Arch. Math. (Basel) 46 (1986), no. 6, 533–538. MR 849858, DOI 10.1007/BF01195021