2012/12/03 by Dohoon Choi, Choi, Dohoon, Subong Lim +1
Mathematics · #11F11 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1212.0370
openalex publication_date 2012/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \SL2(\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 Poincaré 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 Dk+1 and ξ-k and quadric relations of the critical values of their L-functions.