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Zagier duality for real weights

Volume 207 / 2023

Youngmin Lee, Subong Lim Acta Arithmetica 207 (2023), 235-249 MSC: Primary 11F37; Secondary 11F30. DOI: 10.4064/aa220614-13-1 Published online: 6 March 2023


Zagier proved a duality, known as Zagier duality, between the Fourier coefficients of two sequences of weakly holomorphic modular forms of half-integral weights. Guerzhoy formulated the Zagier duality by saying that Fourier coefficients constitute a grid and proved that a grid exists and is unique for every positive even integral weight and level $1$. We prove that the grid exists and is unique for any level and for any real weight $k$ with $k \gt 2$. We also prove some relations between Fourier coefficients appearing in a grid by using Hecke operators. Moreover, we compute the number of ratios of the Fourier coefficients of two weakly holomorphic modular forms in a grid.


  • Youngmin LeeSchool of Mathematics
    Korea Institute for Advanced Study
    Seoul 02455, Republic of Korea
  • Subong LimDepartment of Mathematics Education
    Sungkyunkwan University
    Seoul 110-745, Republic of Korea

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