Construction of Jacobi cusp forms

Sakata, Hiroshi

doi:10.3792/pjaa.74.117

Cited by 9 publications

(1 citation statement)

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“…Using the properties of Poincaré series and adjoint of linear maps, W. Kohnen in [14] constructed the adjoint map of the product map by a fixed cusp form, with respect to the Petersson scalar product. After Kohnen's work, similar results in various other spaces have been obtained by many mathematicians, since the Fourier coefficients of the image of a form involve special values of certain shifted Dirichlet series attached to these forms, e.g., generalization to Jacobi forms (see [5], [10] and [18]), Siegel modular forms (see [15] and [11]), Hilbert modular forms (see [21]) and half-integral weight modular forms (see [9]).…”

Section: Introductionmentioning

confidence: 58%

The adjoint map of the Serre derivative and special values of shifted Dirichlet series

Kumar

2017

Journal of Number Theory

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Abstract. We compute the adjoint of the Serre derivative map with respect to the Petersson scalar product by using existing tools of nearly holomorphic modular forms. The Fourier coefficients of a cusp form of integer weight k, constructed using this method, involve special values of certain shifted Dirichlet series associated with a given cusp form f of weight k + 2. As application, we get an asymptotic bound for the special values of these shifted Dirichlet series and also relate these special values with the Fourier coefficients of f . We also give a formula for the Ramanujan tau function in terms of the special values of the shifted Dirichlet series associated to the Ramanujan delta function.

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