Fourier coefficients of three-dimensional vector-valued modular forms

TL;DR

This paper investigates the Fourier coefficients of three-dimensional vector-valued modular forms, confirming that rational coefficients typically have unbounded denominators, supporting a conjecture about noncongruence modular forms.

Contribution

It verifies that for most three-dimensional irreducible representations, rational Fourier coefficients of associated modular forms have unbounded denominators, extending conjectures to higher dimensions.

Findings

Most such forms have Fourier coefficients with unbounded denominators.

Confirmed the conjecture for all but finitely many cases.

Supports the generalization of the noncongruence modular forms conjecture.

Abstract

A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a finite number of equivalence classes: if a vector-valued modular form associated to such a representation has rational Fourier coefficients, then these coefficients have "unbounded denominators", i.e. there is a prime number p, depending on the representation, which occurs to an arbitrarily high power in the denominators of the coefficients. This provides a verification in the three-dimensional setting of a generalization of a long-standing conjecture about noncongruence modular forms.