Theta Series for Quadratic Forms of Signature $(n-1,1)$ with (Spherical) Polynomials

TL;DR

This paper constructs various types of modular forms using theta series associated with quadratic forms of signature (n-1,1), incorporating spherical polynomials to generate holomorphic and almost holomorphic cusp forms with explicit examples.

Contribution

It introduces a new method of constructing holomorphic and almost holomorphic modular forms via theta series with spherical polynomials for quadratic forms of signature (n-1,1).

Findings

Established a criterion for theta series to coincide.

Constructed explicit examples of cusp forms.

Extended previous constructions to include spherical polynomials.

Abstract

We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature $(n-1,1)$. 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.