Structures for pairs of mock modular forms with the Zagier duality

TL;DR

This paper constructs pairs of vector-valued harmonic weak Maass forms exhibiting Zagier duality using Eichler cohomology, holomorphic Poincaré series, and supplementary functions, revealing their structural properties and relations to L-functions.

Contribution

It introduces a new construction of harmonic weak Maass forms satisfying Zagier duality via Eichler cohomology and Poincaré series, expanding understanding of their structure and duality.

Findings

Constructed dual pairs of harmonic weak Maass forms with specific weights.

Analyzed the images under differential operators and their structural properties.

Explored relations of critical L-values and quadric relations.

Abstract

Zagier introduced special bases for weakly holomorphic modular forms to give the new proof of Borcherds' theorem on the infinite product expansions of integer weight modular forms on $\SL_2(\ZZ)$ with a Heegner divisor. These good bases appear in pairs, and they satisfy a striking duality, which is now called the Zagier duality. After the result of Zagier, this type duality was studied broadly in various view points including the theory of a mock modular form. In this paper, we consider this problem with the Eichler cohomology theory, especially the supplementary function theory developed by Knopp. Using holomorphic Poincar\'e series and their supplementary functions, we construct a pair of families of vector-valued harmonic weak Maass forms satisfying the Zagier duality with integer weights $-k$ and $k+2$ respectively, $k>0$, for a $H$-group. We also investigate the structures of them such as the images under the differential operators $D^{k+1}$ and $\xi_{-k}$ and quadric relations of the critical values of their $L$-functions.