Zagier duality for real weights
Tom 207 / 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.