Automorphic Forms and Holomorphic Functions on the Upper Half-plane

TL;DR

This paper introduces a class of holomorphic functions derived from Hauptmodul of quotient Riemann surfaces, demonstrating their automorphic properties of weight k on the upper half-plane.

Contribution

It establishes that these functions are holomorphic and automorphic forms of weight k, linking Hauptmodul to automorphic form theory.

Findings

Functions are holomorphic on the upper half-plane

Functions are automorphic forms of weight k

Connection between Hauptmodul and automorphic forms

Abstract

We define a set of holomorphic functions in terms of the Hauptmodul of a quotient Riemann surface and prove that these functions are holomorphic on the upper half-plane. It is also shown that these functions are automorphic forms of weight k with respect to a Fuchsian group.