Logarithmic vector-valued modular forms

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We consider logarithmic vector- and matrix-valued modular forms of integral weight k associated with a p-dimensional representation ρ: SL₂(Z) → GLₚ(C) of the modular group, subject only to the condition that ρ(T) has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincaré series associated to ρ for all large enough weights. The component functions are logarithmic q-series, i.e., finite sums of products of q-series and powers of q. We derive several consequences, in particular we show that the space H(ρ)=⊕ₖ H(k, ρ) of all holomorphic logarithmic vector-valued modular forms associated to ρ is a free module of rank p over the ring of classical holomorphic modular forms on SL₂(Z).