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On Shimura, Shintani and Eichler-Zagier correspondences. (English) Zbl 0985.11020

The authors set up the Shimura and Shintani correspondence between the space \(J^{\text{cusp}}_{k,m}(M,\chi)\) of Jacobi cusp forms of weight \(k\), index \(m\) and level \(M\) with character \(\chi\) and the space \(S_{2k-2} (mM,\chi^2)\) of elliptic cusp forms. This is done using Poincaré series. Then the Eichler-Zagier isomorphism is extended to a linear map between \(J^{\text{cusp}}_{k,m}(M,\chi)\) and \(S^m_{k-1/2} (mM,\chi)\). The restriction to newforms yields an isomorphism. Invoking known results, this leads to a strong multiplicity 1 theorem in certain cases for both Jacobi newforms and half-integral weight newforms. As a consequence, the Waldspurger result is derived for Jacobi newforms.

MSC:

11F50 Jacobi forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F11 Holomorphic modular forms of integral weight
11F32 Modular correspondences, etc.
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