  The existence and construction of vector-valued modular forms (vvmf) for any
arbitrary Fuchsian group $\mathrm{G}$, for any representation $\rho:\mathrm{G}
\longrightarrow \mathrm{GL}_{d}(\mathbb{C})$ of finite image can be established
by lifting scalar-valued modular forms of the finite index subgroup $Ker(\rho)$
of $\mathrm{G}$. In this article vvmf are explicitly constructed for any
admissible multiplier (representation) $\rho$, see Section 3 for the definition
of admissible multiplier. In other words, the following question has been
partially answered: For which representations $\rho$ of a given $\mathrm{G}$,
is there a vvmf with at least one nonzero component ?

Number Theory