Abstract
For a fixed modular form we consider a family of linear maps constructed using Rankin–Cohen brackets. We explicitly compute the adjoint of these maps with respect to the Petersson scalar product. The Fourier coefficients of the image of a cusp form under the adjoint maps are, up to a constant, a special value of a certain shifted Rankin–Selberg convolution attached to them. This is a generalization of the work due to Kohnen (Math. Z. 207 (1991), 657–660) and Herrero (Ramanujan J. 36 (2014), no. 3, 529–536) in the case of integral weight modular forms to half-integral weight modular forms. As a consequence we get non-vanishing and asymptotic bound for the special values of a certain shifted Rankin–Selberg convolution of modular forms.
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