We consider special geometry of the vector multiplet moduli space in
compactifications of the heterotic string on $K3 \times T^2$ or the type IIA
string on $K3$-fibered Calabi-Yau threefolds. In particular, we construct a
modified dilaton that is invariant under $SO(2, n; Z)$ T-duality
transformations at the non-perturbative level and regular everywhere on the
moduli space. The invariant dilaton, together with a set of other coordinates
that transform covariantly under $SO(2, n; Z)$, parameterize the moduli space.
The construction involves a meromorphic automorphic function of $SO(2, n; Z)$,
that also depends on the invariant dilaton. In the weak coupling limit, the
divisor of this automorphic form is an integer linear combination of the
rational quadratic divisors where the gauge symmetry is enhanced classically.
We also show how the non-perturbative prepotential can be expressed in terms of
meromorphic automorphic forms, by expanding a T-duality invariant quantity both
in terms of the standard special coordinates and in terms of the invariant
dilaton and the covariant coordinates.
