Combinatorics of Maass-Shimura operators. (English) Zbl 1213.11098
Summary: Maass-Shimura operators on holomorphic modular forms preserve the modularity of modular forms but not holomorphy, whereas the derivative preserves holomorphy but not modularity. Rankin-Cohen brackets are bilinear operators that preserve both and are expressed in terms of the derivatives of modular forms. We give identities relating Maass-Shimura operators and Rankin-Cohen brackets on modular forms and obtain a natural expression of the Rankin-Cohen brackets in terms of Maass-Shimura operators. We also give applications to values of \(L\)-functions and Fourier coefficients of modular forms.
MSC:
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
Software:
MagmaReferences:
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