Mock modular forms whose shadows are Eisenstein series of integral weight

TL;DR

This paper provides explicit constructions of mock modular forms with shadows as Eisenstein series of any integral weight, including special cases related to imaginary quadratic fields and powers of theta functions.

Contribution

It introduces a simple method to explicitly construct mock modular forms with prescribed Eisenstein series shadows, extending previous results and covering various weights and characters.

Findings

Constructed mock modular forms with Eisenstein series shadows of arbitrary weight

Recovered known results for Eisenstein series associated to imaginary quadratic fields

Produced forms with shadows equal to powers of Jacobi's theta function

Abstract

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are Hecke's Eisenstein series of weight one associated to imaginary quadratic fields, recovering some results by Kudla, Rapoport and Yang (1999), and Schofer (2009), and forms whose shadows equal $\Theta^{2k}(z)$ for $k\in \{1,2,3,4\}$, where $\Theta(z)$ denotes Jacobi's theta function.