  We study BPS saturated one-loop amplitudes in type II string theory
compactified on K3 x T^2. The classes of amplitudes we consider are only
sensitive to the very basic topological data of the internal K3 manifold. As a
consequence, the integrands of the former are related to the elliptic genus of
K3, which can be decomposed into representations of the internal N=4
superconformal algebra. Depending on the precise choice of external states
these amplitudes capture either only the contribution of the short multiplets
or the full series including intermediate multiplets. In the latter case we can
define a generating functional for the whole class, which we show is given by
the weight ten Igusa cusp form chi_{10} of Sp(4,Z). We speculate on possible
algebraic implications of our result on the BPS states of the N=4 type II
compactification.
