  Several independent articles have observed that the Hirzebruch $\chi_y$-genus
has an important feature, which the author calls -1-phenomenon and tells us
that the coefficients of the Taylor expansion of the $\chi_y$-genus at $y=-1$
have explicit expressions. Hirzebruch's original $\chi_y$-genus can be extended
towards two directions: the pluri-case and the case of elliptic genus. This
paper contains two parts in which we investigate the -1-phenomena in these two
generalized cases respectively and show that in each case there exists a
-1-phenomenon in a suitable sense. Our main results in the first part have an
application, which states that all characteristic numbers (Chern numbers and
Pontrjagin numbers) on manifolds can be expressed, in a very explicit way, in
terms of some rationally linear combination of indices of some elliptic
operators. This gives an analytic interpretation of characteristic numbers and
affirmatively answers a question posed by the author several years ago. The
second part contains our attempt to generalize this -1-phenomenon to elliptic
genus, a modern version of the $\chi_y$-genus. We first extend the elliptic
genus of an almost-complex manifold to a twisted version where an extra complex
vector bundle is involved, and show that it is a weak Jacobi form under some
assumptions. A suitable manipulation on the theory of Jacobi form will produce
new modular forms from this weak Jacobi form and thus much arithmetic
information related to the underlying manifold can be obtained, in which the
-1-phenomenon of the original $\chi_y$-genus is hidden.
