Instanton Numbers and Exchange Symmetries in $N=2$ Dual String Pairs

In this note, we comment on Calabi-Yau spaces with Hodge numbers $h_{1,1}=3$ and $h_{2,1}=243$. We focus on the Calabi-Yau space $WP_{1,1,2,8,12}(24)$ and show how some of its instanton numbers are related to coefficients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain $N=2$ dual models to six dimensional heterotic/heterotic string duality.