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A general theory of vector-valued modular functions, holomorphic in the upperhalf-plane, is presented for finite dimensional representations of the modulargroup. This also provides a description of vector-valued modular forms ofarbitrary half-integer weight. It is shown that the space of these modularfunctions is spanned, as a module over the polynomials in J, by the columns ofa matrix that satisfies an abstract hypergeometric equation, providing a simplesolution of the Riemann-Hilbert problem for representations of the modulargroup. Restrictions on the coefficients of this differential equation impliedby analyticity are discussed, and an inversion formula is presented that allowsthe determination of an arbitrary vector-valued modular function from itssingular behavior. Questions of rationality and positivity of expansioncoefficients are addressed. Closed expressions for the number of vector-valuedmodular forms of half-integer weight are given, and the general theory isillustrated on simple examples.

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