Finiteness of logarithmic crystalline representations II

TL;DR

This paper proves the finiteness of certain irreducible Galois representations arising from log crystalline local systems over smooth proper schemes with normal crossings divisors, using p-adic nonabelian Hodge theory.

Contribution

It establishes a finiteness result for absolutely irreducible log crystalline representations of the fundamental group over unramified p-adic fields, extending previous work.

Findings

Finiteness of irreducible representations with specified properties.

Application of p-adic nonabelian Hodge theory.

Utilization of Abe/Lafforgue's finiteness results.

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 $\pi_1(X_{\bar K})\rightarrow \mathrm{GL}_r(\mathbb{Z}_{p^f})$ 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.