ON A CORRESPONDENCE BETWEEN JACOBI CUSP FORMS AND ELLIPTIC CUSP FORMS

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z. 253 (2006) 735-752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler-Zagier correspondences, Trans. Amer. Math. Soc. 352 (2000) 2601-2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168-185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math. 411 (1990) 66-95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].