Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products

TL;DR

This paper develops algebraic formulas for mock theta function coefficients, constructs harmonic Maass forms with rational coefficients, and explores their applications to Petersson inner products and Weyl vectors of Borcherds products.

Contribution

It provides finite algebraic formulas for mock theta coefficients and constructs harmonic Maass forms with rational coefficients, advancing understanding of their arithmetic properties.

Findings

Finite algebraic formulas for Ramanujan's mock theta coefficients.

Construction of harmonic Maass forms with rational coefficients.

Formulas for Petersson inner products and Weyl vectors of Borcherds products.

Abstract

We present some applications of the Kudla-Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight $3/2$ and $1/2$, respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions $f(q)$ and $\omega(q)$ in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficients of mock theta functions, i.e., harmonic Maass forms whose shadows lie in the space of unary theta functions. Moreover, the harmonic Maass forms we construct can be used to evaluate the Petersson inner products of unary theta functions with harmonic Maass forms, giving formulas and rationality results for the Weyl vectors of Borcherds products.