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# Communications in Number Theory and Physics

## Volume 1 (2007)

### Number 4

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

Pages: 651 – 680

DOI: http://dx.doi.org/10.4310/CNTP.2007.v1.n4.a2

#### Authors

#### Abstract

A general theory of vector-valued modular functions, holomorphicin the upper half-plane, is presented for finite-dimensionalrepresentations of the modular group. This also provides adescription of vector-valued modular forms of arbitraryhalf-integer weight. It is shown that the space of these modularfunctions is spanned, as a module over the polynomials in $J$,by the columns of a matrix that satisfies an abstracthypergeometric equation, providing a simple solution of theRiemann–Hilbert problem for representations of the modulargroup. Restrictions on the coefficients of this differentialequation implied by analyticity are discussed, and an inversionformula is presented that allows the determination of anarbitrary vector-valued modular function from its singularbehavior. Questions of rationality and positivity of expansioncoefficients are addressed. Closed expressions for the number ofvector-valued modular forms of half-integer weight are given,and the general theory is illustrated on simple examples.