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Congruence subgroups of the modular group related to congruence restrictions of modular forms. (English) Zbl 1202.11027

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.

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)

Citations:

Zbl 1099.11023
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