Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Abstract

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.