A Riccati differential equation and free subgroup numbers for lifts of ${\text{PSL}}_2(\mathbb{Z})$ modulo powers of primes

It is shown that the number $f_{\lambda }$ of free subgroups of index $6\lambda$ in the modular group ${\text{PSL}}_2(\mathbb{Z})$, when considered modulo a prime power $p^{\text{al}}$ with $p\ge 5$, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing ${\text{PSL}}_2(\mathbb{Z})$ as a special case), and for a one-parameter family of lifts of the Hecke group $\mathfrak{H}(4)=C_2\ast C_4$. All this is achieved by explicitly determining Pad'e approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complement previous work by Kauers and the authors (arXiv:1107.2015 and [“A method for determining the mod-$3^k$ behaviour of recursive sequences”}, preprint]), where it is shown, among other things, that the free subgroup numbers of ${\text{PSL}}_2(\mathbb{Z})$ and its lifts display rather complex behaviour modulo powers of 2 and 3.