Equivariant functions for the M\"{o}bius subgroups and applications

TL;DR

This paper generalizes the theory of equivariant functions to arbitrary subgroups of PSL2(R), revealing deep geometric, analytic, and algebraic connections, and applies these insights to classify automorphic forms and analyze their critical points.

Contribution

It extends equivariant function theory to all subgroups of PSL2(R), linking group geometry with function properties and providing new proofs and results in automorphic form classification.

Findings

Deep relation between group geometry and function properties

New proof of automorphic form classification for non-discrete groups

Automorphic forms have infinitely many non-equivalent critical points

Abstract

The aim of this paper is to give a generalization of the theory equivariant functions, initiated in [17, 4], to arbitrary subgroups of PSL2(R). We show that there is a deep relation between the geometry of these groups and some analytic and algebraic properties of these functions. As an application, we give a new proof of the classification of automorphic forms for non discrete groups. Also, we prove the following automorphy condition: If $f$ is an automorphic form for a Fuchsian group of the first kind $\Gamma$, then $f$ has infinitely many non $\Gamma$-equivalent critical points.