Mock modular forms whose shadows are Eisenstein series of integral weight

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 $k \in \lbrace 1,2,3,4 \rbrace$, where $\Theta (z)$ denotes Jacobi’s theta function.

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The first-named author was partially funded by the Royal Swedish Academy of Sciences. The second-named author acknowledges support from the LOEWE research unit Uniformized Structures in Arithmetic and Geometry.

Received 31 August 2018

Accepted 28 February 2019

Published 8 June 2020