Maximality of hyperspecial compact subgroups avoiding Bruhat-Tits theory

Geometric monodromy -- semisimplicity and maximality

Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $π_1(X,x)$ acting on the étale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of those of $π_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $F_\ell$-coefficients of the Grothendiec…