Fourier coefficients of modular forms of half integral weight, periods of modular forms and the special values of zeta functions
The author investigates relations between Fourier coefficients of modular forms of half integral weight and special values of zeta functions associated with the corresponding modular forms \(F\) of integral weight. To that end, periods of modular forms are first expressed as finite sums of special values of modified zeta functions associated with modular forms. The values of these modified zeta functions are further expressed in terms of special values of the zeta functions \(L(s,F,\psi) =\sum^\infty_{n=1} A(n)\psi (n)n^{-s}\) with Dirichlet characters \(\psi\). This leads to the result that the Fourier coefficients of the Shintani lift of \(F\) can be explicitly expressed in terms of special values of the zeta functions \(L(s,F, \psi)\). The author's results also yield a result on the arithmeticity of the fundamental periods of modular forms. The author's results moreover give an expression for the Fourier coefficients of Jacobi forms in terms of special values of zeta functions associated with modular forms of integral weight.
Fourier coefficients of modular forms of half integral weight
scientific article; zbMATH DE number 1079838