The rationality of vector valued modular forms associated with the Weil representation

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Generalized moonshine, II: Borcherds products (2009, background): “i,k ∈ Z/NZ, n < 0, and c0,k(0) ∈ Q for all k, then all coeﬃcients of F are rational. The analogous statement holds if we replace Q with any subring of C containing Q. Proof. By the main theorem of [M03], there exists a basis {fi} for the space of vector-valued functions of type ρM with integer co”
Heegner divisors, $L$-functions and harmonic weak Maass forms (2007, background): “usp form f′ ∈ S 1/2,ρ˜ L such that f+f ′ has rational coeﬃcients. Proof. This follows from the fact that the spaces M! 1/2,˜ρ L and S1/2,ρ˜ L have bases of modular forms with rational coeﬃcients (see [McG]).28 JANH.BRUINIERANDKENONO Remark 17. Observe that M1/2,ρ¯ L is always trivial, by a result of”
Gromov-Witten theory and Noether-Lefschetz theory (2007, background): “For the representations ρ arising in [5], MacGraw proves in [39] that Mod(Mp2(Z), k, ρ) admits a basis with rational coefficients.”