  In the paper we study two types of relations: a one is between the elliptic
genus of Calabi-Yau manifolds and Jacobi modular forms, another one is between
the second quantized elliptic genus, Siegel modular forms and Lorentzian
Kac-Moody Lie algebras. We also determine the structure of the graded ring of
the weak Jacobi forms with integral Fourier coefficients. It gives us a number
of applications to the theory of elliptic genus and of the second quantized
elliptic genus.
