Theta series for quadratic forms of signature (n − 1,1) with (spherical) polynomials

Abstract

We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature [Formula: see text]. We include homogeneous and spherical polynomials in the definition of the theta series (generalizing a construction of the second author) to obtain holomorphic, almost holomorphic and modular theta series. We give a criterion for these series to coincide, enabling us to construct almost holomorphic and holomorphic cusp forms on congruence subgroups of the modular group. Further, we provide numerous explicit examples.