On zeros of Eisenstein series for genus zero Fuchsian groups

Abstract

Let r < SL 2 (R) be a genus zero Fuchsian group of the first kind with ∞ as a cusp, and let E Γ 2k be the holomorphic Eisenstein series of weight 2k on r that is nonvanishing at ∞ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on Γ, and on a choice of a fundamental domain F, we prove that all but possibly c(Γ, F) of the nontrivial zeros of E Γ 2k lie on a certain subset of {z ∈ h: j Γ (z) ∈ R}. Here c(Γ, F) is a constant that does not depend on the weight, S) is the upper half-plane, and j Γ is the canonical hauptmodul for Γ.