Non-holomorphic Eisenstein series for certain Fuchsian groups and class numbers

TL;DR

This paper investigates specific Fuchsian groups, characterizes their elliptic points and cusps, and derives explicit Fourier expansions of associated non-holomorphic Eisenstein series, linking them to class groups and polyharmonic Maass forms.

Contribution

It provides a detailed analysis of Fuchsian groups $R(N)$, including explicit formulas for Eisenstein series and their relation to class groups, advancing understanding of automorphic forms on these groups.

Findings

Elliptic points of $R(p)$ correspond to imaginary quadratic class groups.

Explicit Fourier expansions of non-holomorphic Eisenstein series are derived.

Eisenstein series and cusp forms form a basis for polyharmonic Maass forms.

Abstract

We study certain types of Fuchsian groups of the first kind denoted by $R(N)$, which coincide with the Fricke groups or the arithmetic Hecke triangle groups of low levels. We find all elliptic points and cusps of $R(p)$ for a prime $p$, and prove that there is a one-to-one correspondence between the set of equivalence classes of elliptic points of $R(p)$ and the imaginary quadratic class group. We also find the explicit formula of the Fourier expansion of the non-holomorphic Eisenstein series for $R(N)$ and study their analytic properties. These non-holomorphic Eisenstein series together with cusp forms provide a basis for the space of polyharmonic Maass forms for $R(N)$.