First cohomology groups of Chevalley groups in cross characteristic

Let G be a simple Chevalley group defined over F_q. We show that if r does not divide q and k is an algebraically closed field of characteristic r, then very few irreducible kG-modules have nonzero H¹(G, V). We also give an explicit upper bound for dim H¹(G, V) for V an irreducible kG-module that does not depend on q, but only on the rank of the group. Cline, Parshall and Scott showed that such a bound exists when r|q. We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on V.