Jacobi forms of degree two

After reviewing a theory of Taylor expansion of scalar valued Jacobi forms of higher degree and
several structure theorems obtained before, we talk new results on vector valued Jacobi forms
of degree two. By using the Taylor expansion and a result on the image of Witt operator, we
determine the module structure of vector valued Jacobi forms of degree two of index one over
the ring of scalar valued Siegel modular forms. We also obtain an explicit dimension formula
for $k > 8$ and confirm Tushima's conjecture.