Mock Modular Forms with Integral Fourier Coefficients

TL;DR

This paper constructs mock modular forms with integral Fourier coefficients using regularized inner products and improves bounds on denominators for forms with specific shadows, advancing understanding of their arithmetic properties.

Contribution

It provides explicit constructions of mock modular forms with integral Fourier coefficients and refines bounds on their denominators, enhancing the theoretical framework.

Findings

Explicit construction of mock modular forms with integral Fourier coefficients.

Improved bounds for denominators of coefficients in certain mock modular forms.

Evaluation of regularized Petersson inner products involving unary theta functions.

Abstract

In this note, we explicitly construct mock modular forms with integral Fourier coefficients by evaluating regularized Petersson inner products involving their shadows, which are unary theta functions of weights 1/2 and 3/2 . In addition, we also improve the known bounds for the denominators of the coefficients of mock modular forms whose shadows are holomorphic weight one cusp forms constructed by Hecke.