Vector Valued and Scalar Valued Modular Forms

Vector valued modular forms associated to Weil representations were employed by Borcherds to naturally formulate his theory of automorphic products. Unlike that of scalar valued modular forms, the spaces of these modular forms do not possess as nice structures. In this talk, we prove an isomorphism between spaces of vector valued and scaler valued modular forms, generalizing a result of Bruinier and Bundschuh. We then find canonical bases and discover a beautiful duality, Zagier duality, for their Fourier coefficients. If time permits, we will raise a question on the integrality of such Fourier coefficients.