The heterotic string compactified on $T^2$ has a large discrete symmetry group $SO(2, 18; \mathbb{Z})$, which acts on the scalars in the theory in a natural way; there have been a number of attempts to construct models in which these scalars are allowed to vary by using $SO(2, 18; \mathbb{Z})$-invariant functions. In our new work (which is joint work with David Morrison), we give  a more complete construction of these models in the special cases in which either there are no Wilson lines - and $SO(2, 2;\mathbb{Z})$ symmetry - or there is a single Wilson line - and $SO(2, 3; \mathbb{Z})$ symmetry. In those cases, the modular forms can be analyzed in detail and there turns out to be a precise theory of K3 surfaces with prescribed singularities which corresponds to the structure of the modular forms. We work out precise relations between the function theory and geometry on these K3 surfaces. This allows us to compute explicitly the periods and period relations for their two-forms and many-valued modular forms, and describe the transcendental lattices and the Riemann matrices for associated Kuga-Satake varieties explicitly in terms of the periods.