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

TL;DR

This paper proves that the number of free subgroups of certain lifts of the modular group exhibits periodic behavior modulo prime powers, using Riccati differential equations and Padé approximants.

Contribution

It introduces a method to analyze free subgroup numbers via Riccati equations and extends results to lifts of the modular and Hecke groups.

Findings

Free subgroup numbers are ultimately periodic modulo prime powers.

Explicit solutions involve Padé approximants to Riccati equations.

Results extend previous work on complex modular subgroup behaviors.

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.