Item

Abstract

Let M\sb2k-2(m) be the space of holomorphic modular forms of weight 2k-2 on Γ\sb 0(m) and let J\sbk,m be the space of Jacobi forms of weight k and index m in the sense of Eichler-Zagier [\it M. Eichler and \it D. Zagier, The theory of Jacobi forms (Prog. Math. 55)(Birkhäuser 1985; Zbl 0554.10018)]. The main point in the proof of the Saito-Kurokawa conjecture was the isomorphism between J\sbk,1 and M\sb2k-2(1) as modules over the Hecke algebra. \par In the impressive paper under review the authors deal with the general case for the index m. There exists a canonical subspace \frak M\sp- \sb2k-2(m) of M\sp-\sb2k-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\sbk,m and \frak M\sp- \sb2k-2(m) are isomorphic as modules over the Hecke algebra. \par In \S 1 the trace of the Hecke operator T(ℓ) on J\sbk,m with ℓ 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 \texttr(T(ℓ),J\sbk,m) as linear combinations of \texttr(T(ℓ),M\sb2k-2\spnew,-(m')), m'\vert m. In \S 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.