Christian Krattenthaler and Thomas W. Müller

A Riccati differential equation and free subgroup numbers for lifts of PSL2(Z) modulo prime powers

(30 pages)

Abstract. It is shown that the sequence of numbers f\lambda of free subgroups of index 6\lambda in the modular group PSL2(Z), when considered modulo a prime power p\alpha with p >= 5, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing PSL2(Z) as a special case), and for a one-parameter family of lifts of the Hecke group H(4) = C2*C4. All this is achieved by explicitly determining Padé approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complement previous work by Kauers and the authors ([Electron. J. Combin. 18(2) (2012), Article P37] and ["A method for determining the mod-3k behaviour of recursive sequences," preprint]), where it is shown, among other things, that the free subgroup numbers of PSL2(Z) and its lifts display rather complex behaviour modulo powers of 2 and 3.

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The paper is accompanied by the Mathematica input files: Both Mathematica files provide a function, FreeGF, which produces automatically the rational function form of the generating function for the numbers of free subgroups in PSL2(Z) (file psl2zmod.m), respectively in H(4) (file h4lift.m) of given index, when reduced modulo a prime power pe, as predicted by Theorem 11. For using it, you have to first load the file:

In[1]:= <<psl2zmod.m

and then call the function:

In[2]:= FreeGF[p,e]

(where, clearly, p has to be a specific prime number and e a specific exponent). For the file h4lift.m this works analogously.

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