Let $p>5$ be a prime. For any integer $n\geq 2$, let $M_{n}^{0}(\mathbb{F}_p)$ denote the $n\times n$ matrices with trace $0$ over a finite field $\mathbb{F}_p$ of order $p$. Then we have an action of ${\rm SL}_{n}(\mathbb{F}_p)$ on $M_{n}^{0}(\mathbb{F}_p)$ by conjugation and hence $M_{n}^{0}(\mathbb{F}_p)$ is a $\mathbb{F}_p[{\rm SL}_{n}(\mathbb{F}_p)]$-module. Suppose that $p$ does not divide $n$.
In Theorem 3.2 of A structure theorem for subgroups of GLn over complete local Noetherian rings with large residual image, it's proved that the cohomology group $H^{i}({\rm SL}_{n}(\mathbb{F}_p),M_{n}^{0}(\mathbb{F}))$ is always $0$ for $i=1$. Also, it's proved in 2-Cohomologies of the groups SL(n, q) that the second cohomology group $H^{2}({\rm SL}_{n}(\mathbb{F}_p),M_{n}^{0}(\mathbb{F}_p))$ is non-zero if and only if $n=2$.
So the question is that: What do we know about $H^{i}({\rm SL}_{n}(\mathbb{F}_p),M_{n}^{0}(\mathbb{F}_p))$ for $i\geq 3$?
It would be great if someone could point to relevant literature.
