Knopp, M. I.; Yayenie, Omer Congruence subgroups of the modular group related to congruence restrictions of modular forms. (English) Zbl 1202.11027 JP J. Algebra Number Theory Appl. 15, No. 1, 1-17 (2009). Let \(\Gamma(1)=\text{PSL}_2(\mathbb Z)\) be the classical (inhomogeneous) modular group and let \[ M=\left(\begin{matrix} a&b\\c&d\end{matrix}\right) \] be a typical element. For each positive integer \(N\) and each divisor \(n\) of \(N\) let \[ \Gamma_{0,n}(N)=\{ M \in \Gamma(1): c \equiv 0\pmod N,\;a \equiv d\pmod n\}. \] Then \(\Gamma_{0,n}(N)\) is a congruence subgroup of \(\Gamma(1)\), contained in \(\Gamma_0(N)\). K. Ludwick [Ramanujan J. 9, No. 3, 341–356 (2005; Zbl 1099.11023)] has used congruence relations to restrict the Fourier series expansion for the standard modular form on \(\Gamma_0(N)\) to a modular form on the smaller subgroup \(\Gamma_{0,n}(N)\). Here the authors determine the index of \(\Gamma_{0,n}(N)\) in \(\Gamma(1)\), together with its specification (as a Fuchsian group). The latter includes its genus and its parabolic class number. Reviewer: A. W. Mason (Glasgow) MSC: 11F06 Structure of modular groups and generalizations; arithmetic groups 19B37 Congruence subgroup problems 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects) 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) Keywords:modular forms; modular group; congruence subgroups; index; signature Citations:Zbl 1099.11023 PDFBibTeX XMLCite \textit{M. I. Knopp} and \textit{O. Yayenie}, JP J. Algebra Number Theory Appl. 15, No. 1, 1--17 (2009; Zbl 1202.11027) Full Text: Link