## 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)

Zbl 0554.10018
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### References:

  [E-Z] Eichler, M., Zagier, D.: The theory of Jacobi forms. Boston: Birkhäuser 1985 · Zbl 0554.10018  [G-K-Z] Gross, B., Kohnen, W., Zagier, D.: Heegner points and derivatives ofL-series, II. Math. Ann.278, 497-562 (1987) · Zbl 0641.14013  [H] Hecke, E.: Mathematische Werke. Göttingen: Vandenhoeck & Ruprecht 1959  [M] Maaß, H.: Über eine Spezialschar von Modulformen zweiten Grades. Invent. Math.52, 95-104 (1979) · Zbl 0395.10036  [O] Oesterlé, J.: Sur la trace des opérateurs de Hecke. Thèse, Université de Paris-Sud, 1977  [S-S] Serre, J-P., Stark, H.: Modular forms of weight 1/2. In: Modular functions of one variable, VI (Lecture Notes Math., Vol. 627). Berlin-Heidelberg-New York: Springer 1977  [S] Skoruppa, N.-P.: Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts. Bonn. Math. Schr., Vol. 159. Univ. Bonn 1985  [S-Z] Skoruppa, N.-P., Zagier, D.: A trace formula for Jacobi forms. J. Reine Angew. Math. (To appear) · Zbl 0651.10019  [Y] Yamauchi, M.: On the traces of Hecke operators for a normalizer of ?o(N). J. Math. Kyoto Univ.13, 403-411 (1973) · Zbl 0267.10038  [Z] Zagier, D.: The Eichler-Selberg trace formula onSL 2(?). Appendix in S. Lang, Introduction to modular forms. Berlin-Heidelberg-New York: Springer 1976
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