Structures and Dimensions of Vector Valued Jacobi Forms of Degree Two

Abstract

We give a complete characterization of vector valued holomorphic Jacobi forms of degree two of index one in the sense of Ziegler by the Taylor expansion and vector valued Siegel modular forms of various weights. By this characterization, we also give explicit dimension formulas for spaces of vector valued holomorphic Jacobi forms of index one of degree two, using those for vector valued Siegel modular forms and a certain surjectivity theorem on the Witt operator (the restriction operator to the diagonals). Our characterization also gives a concrete way to give the plus subspace of the space of Siegel modular forms of half-integral weight.