On the rationality of Knörr's virtually irreducible lattices

It was established by Reinhard Knörr that the half of Brauer's height zero conjecture which assumes an abelian defect group would follow from the nonexistence of certain integral representations of abelian p-groups. We establish nonexistence of such representations of rank less than 14 for the elementary abelian group of order 8 and believe these to be the first results of this kind. The result is made possible by imposing a simple rationality condition.