Finiteness of logarithmic crystalline representations II

Abstract

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 ${\pi }_1(X_{\bar{K}})\to {\mathrm{GL}}_r({\mathbb{Z}}_{p^f})$ 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.

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References in corpus (1)

• Deformation theory of periodic Higgs-de Rham flows