First cohomology groups for finite groups of Lie type in defining characteristic

Let G be a finite group of Lie type, defined over a field k of characteristic p > 0 . We find explicit bounds for the dimensions of the first cohomology groups for with coefficients in simple kG-modules. We proceed by bounding the number of composition factors of Weyl modules for simple algebraic groups independently of P and using this to deduce bounds for the 1-cohomology of simple algebraic groups. If γ_l denotes the (finite) maximum of the dimensions of the 1-cohomology groups over all Lie groups of rank l we find bounds for the growth rate of the sequence. {γ_l}We show that log γ_l is O(l³log l).