Let $X$ be a smooth projective variety over a field $k$. Then if $\ell\neq \text{char} k$, $k$ is finite, and $X$ is an abelian variety it was shown by Weil that the $\ell$-adic cohomology of $X\_{k^{sep}}$ is a semisimple representation of $G\_k$. The conjectural reason for this, if I've understood this correctly, is that the $\ell$-adic representation gives an equivalence of the category of "geometric" Galois representations to the category of pure motives ([see for instance](https://mathoverflow.net/q/120608)), and we furthermore conjecture that the later category is semisimple.


What I would be curious about is the "bad coefficient" case, that is to say do we expect $H^i(X\_{k^{sep}},\mathbb{F}\_p)$ to be semisimple as $G\_k$ representation? By [this argument](https://mathoverflow.net/q/275379), we know that the $p$-adic Tate module gives a semisimple representation, but, unless I'm mistaken, we would need an additional argument, as the minimal polynomial might become divisible by a square mod $p$. I suspect this might be a way of providing a counter-example for the semisimplicity of $H^1(A\_{k^{sep}},\mathbb{F}\_p)$, but I'm stuck at that point.



Semisimplicity of the étale cohomology mod $p$
Take an elliptic curve $E$ with a (single) rational $2$-torsion point, say $y^2 = x(x^2+1)$ over $\mathbb{Q}$ (or even $\mathbb{R}$). Then the Galois action on $E[2]$ factors through a group of order $2$ and so is not semisimple.


An easier example is the variety $x^2+1=0$ and $H^0(X,\mathbb{F}\_2)$.


If you insist that the characteristic of the field equals $p$ you could take $x^2+x+1=0$ over $\mathbb{F}\_2$ and look at $H^0(X,\mathbb{F}\_2)$.


