  Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of
integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a simple
normal crossings divisor and fix positive integers $r$ and $f$. We show that
the set of absolutely irreducible representations $\pi_1(X_{\bar K})\rightarrow
\mathrm{GL}_r(\mathbb{Z}_{p^f})$ that come from log crystalline $\mathbb
Z_{p^f}$-local systems over $(X_K,S_K)$ of rank $r$ is finite. The proof uses
$p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.

Algebraic Geometry