Taylor Expansions of Jacobi Forms and Applications to Explicit Structures of Degree Two

Natural mappings from the coefficients of the Taylor expansion of Jacobi forms of general degree to products of certain spaces of vector valued Siegel modular forms are constructed. Proving surjectivity of these mappings in some special cases, we also clarify explicit structures of Jacobi forms of degree two of index one and of even weight of index two as modules over the graded ring of Siegel modular forms of even weight. We also prove surjectivity of the diagonal restriction of Jacobi forms of index one of degree two to the symmetric tensors of Jacobi forms of degree one. 2010 Mathematics Subject Classification: Primary 11F50; Secondary 11F46.