New examples of geometrically special varieties: K3 surfaces, Enriques surfaces, and algebraic groups

We verify that elliptic K3 surfaces and algebraic groups have many rational points over function fields, i.e., they are geometrically special in the sense of Javanpeykar-Rousseau. We also show that under additional assumptions, this geometric specialness persists under removal of closed subsets of codimension at least two.