Indecomposable vector-valued modular forms and periods of modular curves

TL;DR

This paper classifies certain indecomposable three-dimensional modular group representations and uses them to compute modular curve periods without Hecke operators, aiding the study of both congruence and noncongruence curves.

Contribution

It introduces a classification of reducible but indecomposable three-dimensional representations and applies them to compute modular curve periods without Hecke operators.

Findings

Classified all such indecomposable representations.

Developed a new method for computing periods of modular curves.

Enabled analysis of noncongruence modular curves.

Abstract

We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.