On essentially cuspidal noncongruence subgroups of PSL(2,\({\mathbb{Z}})\)
Let \(\Gamma\) be a cofinite subgroup of \(PSL_ 2({\mathbb{R}})\) acting on the upper half-plane H, and let \(N_{\Gamma}(\lambda)\) be the number of eigenvalues of -\(\Delta\) acting on \(L^ 2(\Gamma \setminus H)\) which are less than or equal to \(\lambda\). The group \(\Gamma\) is called essentially cuspidal if \(N_{\Gamma}(\lambda)\) satisfies Weyl's law, that is, if \[ N_{\Gamma}(\lambda)\sim \frac{1}{4\pi}vol(\Gamma \setminus H) \lambda \quad as\quad \lambda \to \infty. \] A subgroup \(\Gamma '<\Gamma\) of finite index in \(\Gamma\) is called a generalized cycloidal subgroup if every cusp orbit \(\Gamma\) \(\zeta\) (\(\zeta\in {\mathbb{R}}\cup \{\infty \})\) of \(\Gamma\) consists of a single orbit with respect to \(\Gamma '\). Comparing the Fourier expansions of the Eisenstein series for \(\Gamma\) and \(\Gamma '\) the author proves Theorem 1: If \(\Gamma '\) is a generalized cycloidal subgroup of \(\Gamma\) of index d then \[ N_{\Gamma '}(\lambda)\sim N_{\Gamma}(\lambda)+\frac{1}{4\pi}\frac{d-1}{d}vol(\Gamma \quad \setminus H)\lambda \quad as\quad \lambda \to \infty. \] Hence, if \(\Gamma\) is an arbitrary congruence subgroup of \(PSL_ 2({\mathbb{Z}})\), then any cycloidal subgroup of \(\Gamma\) is essentially cuspidal. This allows the author to give infinitely many examples of noncongruence subgroups of \(PSL_ 2({\mathbb{Z}})\) which are essentially cuspidal. This adds further evidence for Sarnak's conjecture which claims that \(\Gamma\) is essentially cuspidal iff \(\Gamma\) is arithmetic. (Reviewer's remark: The following papers should be added to the list of references: \textit{H. Petersson}: Über die Konstruktion zykloider Kongruenzgruppen in der rationalen Modulgruppe, J. Reine Angew. Math. 250, 182-212 (1971; Zbl 0232.20103); Konstruktionsprinzipien für Untergruppen der Modulgruppe mit einer oder zwei Spitzenklassen, J. Reine Angew. Math. 268/269, 94-109 (1974; Zbl 0316.20035)].)
scientific article; zbMATH DE number 4146008
scientific article