  This paper develops families of complex Enriques surfaces whose Brauer groups
pull back identically to zero on the covering K3 surfaces. Our methods rely on
isogenies with Kummer surfaces of product type. We offer both lattice theoretic
and geometric constructions. We also sketch how the construction connects to
string theory and Picard-Fuchs equations in the context of Enriques Calabi-Yau
threefolds.
