$p$-adic Hodge parameters in the crystabelline representations of $\mathrm{GL}_n$

TL;DR

This paper constructs an explicit locally $Q_p$-analytic representation of $ ext{GL}_n(K)$ that encodes key $p$-adic Hodge parameters of certain crystabelline Galois representations, providing a new link between Galois and automorphic representations.

Contribution

It introduces a novel explicit construction of a locally $Q_p$-analytic representation associated to crystabelline Galois representations, capturing their $p$-adic Hodge parameters.

Findings

The representation $ ho$ determines the associated $ ext{GL}_n(K)$-representation $ ext{pi}_1( ho)$.

When $K= Q_p$, $ ext{pi}_1( ho)$ encodes all information about $ ho$.

Under mild conditions, $ ext{pi}_1( ho)$ appears as a subrepresentation of the automorphic representation linked to $ ho$.

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $\rho$ be an $n$-dimensional (non-critical generic) crystabelline representation of the absolute Galois group of $K$ of regular Hodge-Tate weights. We associate to $\rho$ an explicit locally $\mathbb{Q}_p$-analytic representation $\pi_1(\rho)$ of $\mathrm{GL}_n(K)$, which encodes some $p$-adic Hodge parameters of $\rho$. When $K=\mathbb{Q}_p$, it encodes the full information hence reciprocally determines $\rho$. When $\rho$ is associated to $p$-adic automorphic representations, we show under mild hypotheses that $\pi_1(\rho)$ is a subrepresentation of the $\mathrm{GL}_n(K)$-representation globally associated to $\rho$.