Subgroups of some Fuchsian groups defined by two linear congruences
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Abstract:
In this article we define a new family of subgroups of Fuchsian groups $\mathcal {H}(\sqrt {m})$, for a squarefree positive integer $m$, and calculate their index in $\mathcal {H}(\sqrt {m})$ and their parabolic class number. Moreover, we will show that the index of these subgroups is closely related to the solvability of a quadratic congruence $x^2\equiv m(\textrm {mod }n)$ and the number of inequivalent solutions of a quadratic congruence $x^2\equiv 1(\textrm {mod }n)$. Finally, we will show that the results obtained by Yilmaz and Keskin [Acta Math. Sin 25 (2005), 215–222] are immediate corollaries of one of the main theorems of this article.References
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
- İ. N. Cangül and D. Singerman, Normal subgroups of Hecke groups and regular maps, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 1, 59–74. MR 1474865, DOI 10.1017/S0305004197002004
- David S. Dummit and Richard M. Foote, Abstract algebra, Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. MR 1138725
- J. I. Hutchinson, On a class of automorphic functions, Trans. Amer. Math. Soc. 3 (1902), no. 1, 1–11. MR 1500582, DOI 10.1090/S0002-9947-1902-1500582-5
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
- Refik Keskin, On the parabolic class numbers of some Fuchsian groups, Note Mat. 19 (1999), no. 2, 275–283 (2001). MR 1816881
- Refik Keskin, On the parabolic class number of some subgroups of Hecke groups, Turkish J. Math. 22 (1998), no. 2, 199–205. MR 1651026
- M. I. Knopp and M. Newman, Congruence subgroups of positive genus of the modular group, Illinois J. Math. 9 (1965), 577–583. MR 181675
- Kurt Ludwick, Congruence restricted modular forms, Ramanujan J. 9 (2005), no. 3, 341–356. MR 2173493, DOI 10.1007/s11139-005-1872-7
- Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004, DOI 10.1007/3-540-29593-3
- I. Niven, H. S. Zuckerman, and H. L. Montogomery, An Introduction To The Theory Of Numbers, John Wiley & Sons, New York (1991).
- Robert A. Rankin, The modular group and its subgroups, Ramanujan Institute, Madras, 1969. MR 0265289
- David Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J. 21 (1954), 549–563. MR 65632
- David Rosen, Research Problems: Continued Fractions in Algebraic Number Fields, Amer. Math. Monthly 84 (1977), no. 1, 37–39. MR 1538246, DOI 10.2307/2318305
- 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
- David Rosen and Thomas A. Schmidt, Hecke groups and continued fractions, Bull. Austral. Math. Soc. 46 (1992), no. 3, 459–474. MR 1190349, DOI 10.1017/S0004972700012120
- Nihal Yılmaz Özgür and İ. Naci Cangül, On the group structure and parabolic points of the Hecke group $H(\lambda )$, Proc. Estonian Acad. Sci. Phys. Math. 51 (2002), no. 1, 35–46 (English, with English and Estonian summaries). MR 1906718
- Yilmaz Özgür Nihal and Keskhin Refik, Proof of a conjecture related to the parabolic class numbers of some Fuchsian groups, Acta Math. Sci. Ser. B (Engl. Ed.) 25 (2005), no. 2, 215–222. MR 2133061, DOI 10.1016/S0252-9602(17)30278-3
- John Wesley Young, On the group of sign $(0,3;2,4,\infty )$ and the functions belonging to it, Trans. Amer. Math. Soc. 5 (1904), no. 1, 81–104. MR 1500662, DOI 10.1090/S0002-9947-1904-1500662-6
- Jürgen Wolfart, Eine Bemerkung über Heckes Modulgruppen, Arch. Math. (Basel) 29 (1977), no. 1, 72–77 (German). MR 453643, DOI 10.1007/BF01220377
Additional Information
- Omer Yayenie
- Affiliation: Department of Mathematics and Statistics, Murray State University, Murray, Kentucky 42071
- Email: omer.yayenie@murraystate.edu
- Received by editor(s): March 26, 2007
- Published electronically: December 18, 2007
- © Copyright 2007 American Mathematical Society
- Journal: Conform. Geom. Dyn. 11 (2007), 271-287
- MSC (2000): Primary 11F06, 19B37; Secondary 20H05, 20H10
- DOI: https://doi.org/10.1090/S1088-4173-07-00172-5
- MathSciNet review: 2365641