Vector valued modular forms and the modular orbifold of elliptic curves

TL;DR

This paper develops a geometric framework for holomorphic vector valued modular forms using vector bundles on the modular orbifold of elliptic curves, simplifying the theory and enabling algebraic geometry techniques.

Contribution

It introduces a geometric perspective on vector valued modular forms via vector bundles on the modular orbifold, clarifying the role of representation exponents and deriving key theorems.

Findings

Simplifies the theory of vector valued modular forms

Establishes free-module theorems and dimension formulas

Identifies the modular orbifold with weighted projective line P(4,6)

Abstract

This paper presents the theory of holomorphic vector valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of SL_2(Z) play in the holomorphic theory of vector valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line P(4, 6).