Binary Theta Series and Modular Forms with Complex Multiplication

Let D be a negative discriminant, and let Θ(D) be the complex vector space generated by the binary theta series θf attached to the positive definite binary quadratic forms f(x, y) = ax + bxy + cy whose discriminant D(f) = b − 4ac equals D/t, for some integer t. It is a well-known classical fact that Θ(D) is a subspace of the space M1(|D|, ψD) of modular forms of weight 1, level |D| and Nebentypus ψD, where ψD = (D· ) is the Kronecker-Legendre character. The purpose of this paper is to give an intrinsic interpretation of Θ(D) as a subspace of M1(|D|, ψD). More precisely, it turns out that Θ(D) is precisely the subspace M 1 (|D|, ψD) of modular forms which have complex multiplication (CM) by their Nebentypus character ψD (in the sense of Ribet[10]):