Kohnen, Winfried Cusp forms and special values of certain Dirichlet series. (English) Zbl 0744.11026 Math. Z. 207, No. 4, 657-660 (1991). Let \(S_k\) denote the space of elliptic cusp forms on \(\mathrm{SL}_2(\mathbb{Z})\) of even weight \(k\). Given \(f\in S_k\), \(g\in S_l\), \(k>l+2\), \(n\geq 1\), the author constructs a cusp form from special values of the Dirichlet series \[ L_{f,g;n}(s):=\sum_{m\geq 1}\alpha_ f(m+n)\cdot\overline{\alpha_ g(m)}\cdot(m+n)^{-s} \] attached to the Fourier expansions of \(f\) and \(g\). More precisely the adjoint with respect to the Petersson scalar product of the map \(S_{k-l}\to S_ k\), \(h\mapsto gh\), is up to a constant given by \(W_g: S_k\to S_{k- l}\), \(f\mapsto W_ g(f)\), where \[ W_g(f)(z):=\sum_{n\geq 1}n^{k-l- 1} L_{f,g;n}(k-1)e^{2\pi inz}. \] Reviewer: S.J.Patterson (Göttingen) Cited in 8 ReviewsCited in 16 Documents MSC: 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations 11F30 Fourier coefficients of automorphic forms 11F11 Holomorphic modular forms of integral weight 11F25 Hecke-Petersson operators, differential operators (one variable) Keywords:Poincaré series; Petersson scalar product; adjoint operator; special values of Dirichlet series; space of elliptic cusp forms; construction of cusp forms PDFBibTeX XMLCite \textit{W. Kohnen}, Math. Z. 207, No. 4, 657--660 (1991; Zbl 0744.11026) Full Text: DOI EuDML References: [1] Good, A.: On various means involving the Fourier coefficients of cusp forms. Math. Z.183, 95–129 (1983) · Zbl 0503.10015 · doi:10.1007/BF01187218 [2] Satoh, T., Jacobi forms and certain special values of Dirichlet series associated to modular forms. Math. Ann.285, 463–480 (1989) · Zbl 0662.10019 · doi:10.1007/BF01455068 [3] Eichler, M., Zagier, D.: The theory of Jacobi forms. (Prog. Math, vol. 55) Boston Basel Stuttgart: Birkhäuser 1985 · Zbl 0554.10018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.