  Let $G({\mathbb F}_{q})$ be a finite Chevalley group defined over the field
of $q=p^{r}$ elements, and $k$ be an algebraically closed field of
characteristic $p>0$. A fundamental open and elusive problem has been the
computation of the cohomology ring $\opH^{\bullet}(G({\mathbb F}_{q}),k)$. In
this paper we determine initial vanishing ranges which improves upon known
results. For root systems of type $A_n$ and $C_n$, the first non-trivial
cohomology classes are determined when $p$ is larger than the Coxeter number
(larger than twice the Coxeter number for type $A_n$ with $n>1$ and $r >1$). In
the process we make use of techniques involving line bundle cohomology for the
flag variety $G/B$ and its relation to combinatorial data from Kostant
Partition Functions.
