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Conformal Geometry and Dynamics

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Subgroups of some Fuchsian groups defined by two linear congruences


Author: Omer Yayenie
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
Published electronically: December 18, 2007
MathSciNet review: 2365641
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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.


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Additional Information

Omer Yayenie
Affiliation: Department of Mathematics and Statistics, Murray State University, Murray, Kentucky 42071
Email: omer.yayenie@murraystate.edu

DOI: https://doi.org/10.1090/S1088-4173-07-00172-5
Keywords: Fuchsian groups, Hecke groups, modular group, congruence subgroups, and modular forms
Received by editor(s): March 26, 2007
Published electronically: December 18, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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