TITLE: Mod-p cohomology of $GL(n,p^d)$ 
QUESTION [6 upvotes]: In the classic paper On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field, Quillen proved (Theorem 6): 

$H^i(GL(n,p^d),\mathbb{F}_p)=0$ for $0 < i < d(p-1)$ and all $n$. 

On the other hand, the cohomology of a finite group doesn't completely vanish (this is nicely discussed on MO: Non-vanishing of group cohomology in sufficiently high degree). So there is a minimal integer $i_0=i_0(n,d) \ge d(p-1)$ satisfying $$H^{i_0}(GL(n,p^d),\mathbb{F}_p) \neq 0$$  Is Quillen's lower bound $d(p-1)$ sharp ? (I couldn't find any information about sharpness in Quillen's paper). If not, is the precise value of $i_0$ known ?

REPLY [6 votes]: There are some results known in this direction; see the paper On the vanishing ranges for the cohomology of finite groups of Lie type, by Christopher Bendel, Daniel Nakano, and Cornelius Pillen (Int. Math. Res. Not. (2012), 2817-2866). In particular, they show that if $p \geq n+2$, then the first degree in which nonzero cohomology appears is $d(2p-3)$.
Bendel, Nakano, and Pillen also have a sequel paper posted on the arXiv that treats a number of cases for other Lie types that aren't already handled in the reference I gave above.