Free modules of vector-valued modular forms and applications to Jacobi forms

The space M(ρ;E) of weakly holomorphic modular forms for a dimension-r complex representation $ρ$ of SL₂(Z) with growth prescribed by E at the cusp is a free module over C[E₄,E₆]. We first prove that a C[E₄,E₆]-linearly independent homogeneous family F₁,…,Fᵣ in this module gives a basis if and only if ∑ᵢ₌₁ʳ wt(Fᵢ)=12trE. We then extend this criterion to Jacobi forms. As applications, we determine the generator weights of free modules of O(D8d⁺)-invariant Jacobi forms of index D8d⁺(p), where D8d⁺ is an even unimodular lattice extending D8d, d≥2, and $p=2,3,5$.