The Space of Binary Theta Series

The purpose of this paper is to study the space ΘD generated by the binary theta series θf attached to the primitive, positive-definite binary quadratic forms f(x, y) = ax + bxy + cy of discriminant D = b − 4ac < 0. It is curious that while much has been written about the space generated by theta series attached to quadratic forms in 2k ≥ 4 variables (cf. [10] and the references therein), the binary case does not seem to have been treated in detail in the literature. By the work of Weber[25], Hecke[8] and Schoeneberg[18], it is known that ΘD 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. Contrary to the case of higher weight, the space of binary theta series is often a proper subspace of M1(|D|, ψD) (cf. Remark 16) and so it is of interest to be able to identify it inside the space of modular forms. As a first step towards this, we explain in this paper how each theta series θf can be expressed as a linear combination of the canonical (extended) Atkin-Lehner basis of M1(|D|, ψD); cf. Remark 30(b). This will be used in the next paper[12] to give an intrinsic description of the space of theta series. To this end we first observe that ΘD has a natural basis {θχ} indexed by the characters χ ∈ Cl(D)∗ of the class group Cl(D) of forms of discriminant D. It turns out that each θχ is a normalized eigenform with respect to Hecke algebra T(D); the latter is the algebra generated by all Hecke operators Tn with (n,D) = 1. Theorem 1 The space ΘD is a T(D)-submodule of M1(|D|, ψD) of multiplicity one, and has a canonical basis {θχ} consisting of normalized T(D)-eigenforms. Furthermore, θχ is a cusp form if and only if χ is not a quadratic character.