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

TL;DR

This paper extends the construction of theta series for quadratic forms of signature (n-1,1) to include boundary parameters, enabling the study of Eisenstein series, modular forms, and mock theta functions related to quadratic polynomials and class numbers.

Contribution

It generalizes previous theta series constructions by allowing boundary parameters, facilitating the analysis of new examples like Eisenstein series and mock theta functions.

Findings

Constructed theta series with boundary parameters on the cone $C_Q$

Connected theta series to Eisenstein series and modular forms

Linked mock theta functions to Hurwitz class numbers

Abstract

We generalize the construction from arXiv:2102.09329 of theta series for quadratic forms of signature $(n-1,1)$ with homogeneous and spherical polynomials. Namely, we allow that the parameters $c_1,c_2$, which define the theta series and ensure the convergence of the defining series, are located on the boundary of the cone $C_Q$. This enables us to study several interesting examples such as Eisenstein series, modular forms on $\Gamma_0(4)$ which appear during the investigation of quadratic polynomials of a fixed discriminant, and a mock theta function of order 2 that is connected to the generating function of the Hurwitz class numbers $H(8n+7)$.