Let

In this article, the author gives some of his results on Jacobi forms of higher degree without proof. The proof can be found in the references [Y1] and [Y2]. 1 Jacobi Forms First of all, we introduce the notations. We denote by $Z, $ $R $ and $C$ the ring of integers, the field of real numbers and the field of complex numbers respectively. We denote by $Z^{+} $ the set of all positive integers. $F^{\langle k,l)} $ denotes the set of all $k\cross l $ matrices with entries in a commuatative ring $F $. For any $M\in F^{(k,l)},{}^{t}M $ denotes the transpose matrix of $M $. For $A\in F^{(k,l)}, $ $\sigma(A) $ denotes the trace of $A $. For $A\in F^{(k,l)} $ and $B\in F^{(k,k)} $, we set $B[A]={}^{t}ABA $. $E_{n} $ denotes the identity matrix of degree $n $. For any positive integer $g\in Z^{+} $ , we let $H_{g} $ $:=\{Z\in C^{(g,g)}|Z={}^{t}Z, ImZ>0\}$ the Siegel upper half plane of degree $g $. Let $Sp(g, R) $ and $Sp(g, Z) $ be the real symplectic group of degree $g $ and the Siegel modular group of degree $g $ respectively