In 1984, Oesterle asked whether a wound unipotent group admitting infinitely many rational points over a global function field necessarily contains a nontrivial unirational subgroup -- that is, do rational points only arise for a good geometric reason? Though he formulated the question for wound unipotent groups, it makes sense for arbitrary linear algebraic groups -- although one may show that the more general question ultimately boils down to the wound unipotent case. Using some recent work of mine on the structure of wound unipotent groups, I shall outline a positive answer to Oesterle's question, and in fact discuss a stronger result which holds over more general fields of arithmetic interest. Time permitting, I will also discuss how these structural results can be combined with local Tate duality to give a criterion for which linear algebraic groups over local function fields have finite cohomology.