Über die Spitzenbreiten von Kongruenzuntergruppen der rationalen Modulgruppe. (On the cusp amplitudes of congruence subgroups of the rational modular group) (Q919386)

In a previous paper it was shown by the author that the level of a congruence subgroup of the classical modular group always equals the least common multiple (lcm) of all the cusp amplitudes. Stemming on a finer result of \textit{H. Larcher} [Ill. J. Math. 26, 164-172 (1982; Zbl 0467.20039)] the author proves by elementary number-theoretic arguments that for a congruence subgroup with cusps \(\infty,0,1\) the level is already given by the lcm of the amplitudes of the cusps. Furthermore he characterizes those numbers e,f,g to be amplitudes of cusps \(\infty,0,1\) respectively and determines all the amplitudes h of the group \(\Gamma(e,f,g)\), more precisely: \(\gcd(e,f,g) | h | lcm(e,f,g)\).