## Tohoku Mathematical Journal 2012 September SECOND SERIES VOL. 64, NO. 3

 Tohoku Math. J. 64 (2012), 361-385

Title ON THE FOURIER COEFFICIENTS OF JACOBI FORMS OF INDEX $N$ OVER TOTALLY REAL NUMBER FIELDS

Author Hisashi Kojima

(Received March 31, 2011, revised November 24, 2011)
Abstract. Skoruppa and Zagier established a bijective correspondence from the space of Jacobi forms $\phi$ of index $m$ to that of elliptic modular forms $f$ of level $m$. Gross, Kohnen and Zagier formulated this correspondence by means of kernel functions. Moreover, they proved that the squares of Fourier coefficients of $\phi$ are essentially equal to the critical values of the zeta functions $L(s,f,\chi)$ of $f$ twisted by a quadratic character $\chi$.
The purpose of this paper is to prove a generalization of such results concerning liftings and Fourier coefficients of Jacobi forms to the case of Jacobi forms of index $N$ over totally real number fields $F$. Using kernel functions associated with the space of quadratic forms, we shall establish the existence of a lifting from the space of Jacobi forms $\phi$ of index $N$ over $F$ to that of Hilbert modular forms $f$ of level $N$ over $F$. Moreover, we determine explicitly the Fourier coefficients of $f$ from those of $\phi$. We prove that an analogue of Waldspurger's theorem in the case of Jacobi forms of index $N$ over $F$ holds.

2000 Mathematics Subject Classification. Primary 11F50; Secondary 11F30, 11F67.

Key words and phrases. Jacobi forms, Jacobi forms over totally real number fields, lifting of modular forms, Fourier-Jacobi forms, special values of zeta functions.