  We derive the period structure of several one-modulus Calabi-Yau manifolds.
With this knowledge we then obtain the generators of the duality group and the
mirror map that defines the physical variable $t$ representing the radius of
compactification. We also describe the fundamental region of $t$ and discuss
its relation with automorphic functions. As a byproduct of our analysis we
compute the non-perturbative corrections of Yukawa couplings.
