Vector-valued modular functions for the modular group and the hypergeometric equation

Abstract

A general theory of vector-valued modular functions, holomorphic in the upper half-plane, is presented for finite-dimensional representations of the modular group. This also provides a description of vector-valued modular forms of arbitrary half-integer weight. It is shown that the space of these modular functions is spanned, as a module over the polynomials in J, by the columns of a matrix that satisfies an abstract hypergeometric equation, providing a simple solution of the Riemann-Hilbert problem for representations of the modular group. Restrictions on the coefficients of this differential equation implied by analyticity are discussed, and an inversion formula is presented that allows the determination of an arbitrary vector-valued modular function from its singular behavior. Questions of rationality and positivity of expansion coefficients are addressed. Closed expressions for the number of vector-valued modular forms of half-integer weight are given, and the general theory is illustrated on simple examples.

On the Existence of Vector-Valued Automorphic Forms

“…study of vector-valued forms, and several important results have been obtained by various authors [1,11,19,10,[22][23][24]. However, as far as we know, most of what is found in the literature deals only with the typical case of being the modular group SL 2 (Z) or its subgroups and the convenient condition that R is trivial at the parabolic elements of and sometimes diagonalizable.…”

Partition Functions for Supersymmetric Black Holes

“…[143]. Dimension formulas for vector-valued modular forms are also mentioned in [139], [144] and [145]. Ref.…”

Vector-valued modular forms and the modular orbifold of elliptic curves

“…It is worth noting that many of the results presented below have been obtained by Gannon [10] using a different approach, and in a slightly more general context of admissible multiplier systems of weight w ∈ C. His approach builds on work of Borcherds [4] and joint work between Bantay and Gannon [3], and it makes essential use of the solution to the Riemann-Hilbert problem. In this paper, we restrict to the case when w ∈ Z.…”

“…A systematic treatment of their theory has only been initiated in recent years by Bantay, Gannon [2], [3], [10], Knopp, Mason [15], [18], [19] and others [16], [17], [22]. Most of these approaches are based upon the Riemann-Hilbert correspondence, or vector valued Poincaré series.…”

“…In [24] this restriction arises due to an application of a trace formula, while in [4] and [9] this finiteness condition allows the authors to work on a finite cover of the modular orbifold that is in fact a scheme. In [3], the authors assume ρ(T ) is of finite order, while Gannon ([10], Lemma 3.2) avoids imposing any finiteness condition via an application of the solution to the Riemann-Hilbert problem, but assumes ρ(T ) diagonal. The present paper avoids imposing any finiteness or diagonalizability conditions via a technique modeled after the construction of extensions of a regular connection on a punctured sphere -see [5], [21] and Proposition 3.2 of the present paper.…”