  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.
