Irreducible vector-valued modular forms of dimension less than six

The Theory of Vector-Valued Modular Forms for the Modular Group

“…But outside of our work, the most influential for the theory of vvmf (see e.g. [1,26,24,21]) has been the differential equation coming from the Wronskian (see e.g. [24] for the straightforward proof): Lemma 2.2(a) Let (ρ,w ) be admissible.…”

“…the monodromy corresponding to a small circle about 0 and 1, respectively, equals the value of ρ ′ at the corresponding loops in the fundamental group. There will also be a simple pole at ∞, with residue (21) corresponds to an apparent singularity; it can be dropped if the monodromies about 0 or 1 or ∞ have finite order [7,8] (which will happen for the ρ ′ of interest to us).…”

“…(recall (21)). Moreover, 2 , 3 are diagonalisable, with eigenvalues in {0, 1 2 } and {0, 1 3 , 2 3 }, respectively.…”

“…Several papers (e.g. [24,21]) consider a 'cyclic' class of vvmf where the components of (τ) span the solution space to a monic modular differential equation…”

“…This means in dimension < 6 we get all kinds of things for free (see Proposition 3.3), including identifying the Hilbert-Poincaré series H λ (x ;ρ). These series were first computed in [21], for the special case of exponent λ = λ hol , when T is unitary, and the representation ρ is what Marks calls T-determined, which means that any indecomposable ρ ′ with the same T -matrix is isomorphic to ρ. It turns out that most ρ are T -determined.…”