Arithmetic of automorphic forms for certain Fuchsian groups including $\Gamma_0^+(2)$ and $\Gamma_0^+(3)$

Abstract

We study several arithmetic properties of automorphic forms for certain Fuchsian groups including $\Gamma_0^+(2)$ and $\Gamma_0^+(3)$. The result includes the algebraic independence of the values of the Eisenstein series for the arithmetic Hecke group and the investigation of the common zeros and the multiplicities of zeros of quasi-modular forms. More precisely, we prove that for any $\alpha$ in the upper half-plane, at least three of the numbers $e^{2\pi i \alpha}, E_{2,m}(\alpha), E_{4,m}(\alpha), E_{6,m}(\alpha)$ are algebraically independent over $\mathbb Q$, which is an extension of Nesterenko's result. As its application, we study the zeros of quasi-modular forms of depth 1 for $\Gamma_0^+(N)$, for $N=2,3$. In addition, we give the formula of the non-holomorphic Eisenstein series for certain Fuchsian groups whose elliptic points coincide with the quadratic class group and provide an explicit basis for the space of polyharmonic Maass forms for these groups.