Transcendence of Zeros of Automorphic Forms For Cuspidal Triangle Groups

TL;DR

This paper investigates the zeros of automorphic forms for cuspidal triangle groups, showing they are either transcendental or have complex multiplication, extending prior results to non-arithmetic groups.

Contribution

It generalizes the transcendence results of automorphic form zeros to all cuspidal triangle groups, including non-arithmetic cases.

Findings

Zeros are either transcendental or CM.

First such result for non-arithmetic groups.

Extends previous work on elliptic modular forms.

Abstract

We extend some results of Gun, Murty, and Rath on elliptic modular forms. We take ANY Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in the natural parameter have algebraic coefficients. We show that the zeros of such forms are either transcendental or are "CM." By "CM," we mean they correspond to abelian varieties with complex multiplication. This result is the first of its kind in the case of non-arithmetic groups.