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Cusp forms and special values of certain Dirichlet series. (English) Zbl 0744.11026

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
\[ W_g(f)(z):=\sum_{n\geq 1}n^{k-l- 1} L_{f,g;n}(k-1)e^{2\pi inz}. \]

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)
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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
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