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The Lie theory of certain modular form and arithmetic identities. (English) Zbl 1281.11038

Summary: Modular form identities lying in the framework of Shimura’s theory of nearly holomorphic modular forms are obtained by Lie theoretic means as consequences of identities relating the Maass-Shimura operator and the Rankin-Cohen brackets, which in turn follow from change-of-basis formulae in the theory of Verma modules. The Lie theoretic origin of known van der Pol and Lahiri-type arithmetic identities is thus unveiled, and similar new ones are derived in a systematic way. These identities relate divisor functions, Ramanujan’s \(\tau \)-function and functions defined by the Fourier coefficients of other cusp forms and involve hybrid coefficients, drawn from Lie theory and number theory, given explicitly by formulae combining the arithmetic Clebsch-Gordan coefficients and the Bernoulli numbers. A few side results, interesting in their own right, such as Leibniz-type rules satisfied by the Rankin-Cohen brackets, are also obtained.

MSC:

11F11 Holomorphic modular forms of integral weight
11F22 Relationship to Lie algebras and finite simple groups
11F25 Hecke-Petersson operators, differential operators (one variable)
11F30 Fourier coefficients of automorphic forms
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