Signatures of the special congruence subgroup of the extended modular group (Q2459030)

The extended modular group \(\overline{\Pi}\) is obtained by extending the classical modular group \(\text{PSL}_2({\mathbb Z})\) of Möbius transformations of the complex upper halfplane \({\mathbb U}\) by the involution \(c_1:\;z\mapsto -\overline{z}\). The article calculates the signature of the congruence subgroups \[ \overline{\Pi}_0(p)=\langle\Gamma_0(p), c_1\rangle, \] i.e essentially the genus of the (in general non-orientable) quotient space \({\mathbb U}/\overline{\Pi}_0(p)\) and the branching orders. The result is already contained in reference [4], the unpublished PhD thesis by \textit{S. J. Harding} [Some arithmetic and geometric problems concerning discrete groups, Southampton University (via British Library) (1985)].