  In this paper we derive topological and number theoretical consequences of
the rigidity of elliptic genera, which are special modular forms associated to
each compact almost complex manifold. In particular, on the geometry side, we
prove that rigidity implies relations between the Betti numbers and the index
of a compact symplectic manifold of dimension $2n$ admitting a Hamiltonian
action of a circle with isolated fixed points. We investigate the case of
maximal index and toric actions. On the number theoretical side we prove that
from each compact almost complex manifold of index greater than one, that can
be endowed with the action of a circle with isolated fixed points, one can
derive non-trivial relations among Eisenstein series. We give explicit formulas
coming from the standard action on $\mathbb{C} P^n$.
