p-adic Hodge parameters in the crystalline representations of GSp4

TL;DR

This paper generalizes Ding's work to GSp4, associating crystalline Galois representations with explicit locally analytic representations, advancing understanding of p-adic Hodge parameters and local-global compatibility.

Contribution

It introduces a new explicit construction of locally analytic representations for GSp4 that encodes crystalline Galois representations, extending prior work to a higher-dimensional setting.

Findings

Constructs explicit locally analytic GSp4 representations from crystalline Galois representations.

Establishes a link between these representations and p-adic Hodge parameters.

Demonstrates local-global compatibility in certain cases.

Abstract

This article gives a generalization of the work of Y.Ding in the context of $\mathrm{GSp}_4(\mathbb{Q}_p)$, where $p$ is an odd prime number. Let $\rho$ be a 4-dimensional generic non-critical crystalline representations of the absolute Galois group of $\mathbb{Q}_p$ of regular Hodge-Tate weights which is valued in $\mathrm{GSp}_4(E)$, where $E$ is a finite extension of $\mathbb{Q}_p$, we associate to $\rho$ an explicit locally analytic $E$-representation $\pi_\mathrm{min}(\rho)$ of $\mathrm{GSp}_4(\mathbb{Q}_p)$, which encodes enough information to determines $\rho$. Moreover, under certain settings, this construction follows the local-global compatibility.