Binary theta series and modular forms with complex multiplication

Abstract

The main purpose of this paper is to give an intrinsic interpretation of the space Θ(D) generated by the binary theta series ϑf attached to the positive binary quadratic forms f whose discriminant has the form D(f) = D/t2, for some integer t. It turns out that [Formula: see text], the space of modular forms of weight 1 and of level |D| which have complex multiplication (CM) by their Nebentypus character [Formula: see text]. As an application, we obtain a structure theorem of the space [Formula: see text]. The proof of this theorem rests on the results of [The space of binary theta series, Ann. Sci. Math. Québec36 (2012) 501–534] together with a characterization of the newforms f which have CM by their Nebentypus character in terms of properties of the associated Deligne–Serre Galois representationρf.