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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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