On the zeros of Hecke-type Faber polynomials

For any McKay-Thompson series which appear in Moonshine, the Hecke-type Faber polynomial⁺ Pn(X) of degree n is defined. The Hecke-type Faber polynomials are of course special cases of the Faber polynomials introduced by Faber a century ago. We first study the locations of the zeros of the Hecke-type Faber polynomials of the 171 monstrous types, as well as those of the 157 non-monstrous types. We have calculated, using a computer, the zeros for all n ≤ 50. These results suggest that in many (about 13%) of the cases, we can expect that all of the zeros of Pn(x) are real numbers. In particular, we prove rigorously that the zeros of the Hecke-type Faber polynomials (of any degree) for the McKaγ-Thompson series of type 2A are real numbers. We also discuss the effect of the existence of harmonics, and the effect of a so-called dash operator. We remark that by the dash operators, we obtain many replicable functions (with rational integer coefficients) which are not necessarily completely replicable functions. Finally, we study more closely the curves on which the zeros of the Hecke-type Faber polynomials for type 5B lie in particular in connection with the fundamental domain (on the upper half plane) of the group Γ ₀(5), which was studied by Shigezumi and Tsutsumi. At the end, we conclude this paper by stating several observations and speculations.