A Riccati differential equation and free subgroup numbers for lifts of $\PSL_2(\Z)$ modulo powers of primes

Abstract

It is shown that the number $f_\lambda$ of free subgroups of index $6\lambda$ in the modular group $\PSL_2(\Z)$, when considered modulo a prime power $p^\al$ with $p\ge5$, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing $\PSL_2(\Z)$ as a special case), and for a one-parameter family of lifts of the Hecke group $\mathfrak{H}(4)=C_2*C_4$. 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 $\PSL_2(\Z)$ and its lifts display rather complex behaviour modulo powers of 2 and 3.