Finiteness of logarithmic crystalline representations

TL;DR

This paper proves that, over certain $p$-adic fields, there are finitely many logarithmic crystalline local systems of a fixed rank with irreducible residual representations, up to character twists, using advanced $p$-adic Hodge theory.

Contribution

It establishes a finiteness result for log crystalline local systems over $p$-adic fields, extending understanding of their classification and structure.

Findings

Finiteness of log crystalline local systems with fixed rank.

Finiteness holds up to twisting by characters.

Uses $p$-adic nonabelian Hodge theory and Abe/Lafforgue 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 normal crossings divisor. We show that there are only finitely many log crystalline $\mathbb Z_{p^f}$-local systems over $X_K\setminus S_K$ of given rank and with geometrically absolutely irreducible residual representation, up to twisting by a character. The proof uses $p$-adic nonabelian Hodge theory and a finiteness result due Abe/Lafforgue.