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Jacobi forms and a certain space of modular forms. (English) Zbl 0651.10020
Let \(M_{2k-2}(m)\) be the space of holomorphic modular forms of weight \(2k-2\) on \(\Gamma_0(m)\) and let \(J_{k,m}\) be the space of Jacobi forms of weight \(k\) and index \(m\) in the sense of Eichler-Zagier [M. Eichler and D. Zagier, The theory of Jacobi forms. Prog. Math. 55. Basel et al.: Birkhäuser (1985; Zbl 0554.10018)]. The main point in the proof of the Saito-Kurokawa conjecture was the isomorphism between \(J_{k,1}\) and \(M_{2k-2}(1)\) as modules over the Hecke algebra.
In the impressive paper under review the authors deal with the general case for the index \(m\). There exists a canonical subspace \({\mathfrak M}^{-}_{2k-2}(m)\) of \(M^{-}_{2k-2}(m)\), which can be described by properties of the Euler factors of the \(L\)-series attached to a modular form and which contains the space of newforms. Here “-” means that the \(L\)-series satisfies a functional equation under \(s\mapsto 2k-2-s\) with root number \(-1\). The Main Theorem says that \(J_{k,m}\) and \({\mathfrak M}^{-}_{2k-2}(m)\) are isomorphic as modules over the Hecke algebra.
In §1 the trace of the Hecke operator \(T(\ell)\) on \(J_{k,m}\) with \(\ell\) relatively prime to \(m\) is computed as an application of the general trace formula for Jacobi forms. Then the Eichler-Selberg trace formula is used in order to express \(\operatorname{tr}(T(\ell),J_{k,m})\) as linear combinations of \(\operatorname{tr}(T(\ell),M_{2k-2}^{\text{new},-}(m'))\), \(m'\mid m\). In §3 the isomorphy is proved, where the proof moreover gives a collection of explicit lifting maps. In the Appendix, the authors derive a formula for a certain class number involving Gauss sums associated to binary quadratic forms.

MSC:
11F50 Jacobi forms
11F11 Holomorphic modular forms of integral weight
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11E41 Class numbers of quadratic and Hermitian forms
11L03 Trigonometric and exponential sums (general theory)
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References:
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