p-adic Hodge parameters in the crystalline representations of GSp4
Xiaozheng Han
TL;DR
This work extends Y. Ding’s ideas to the group GSp_4, constructing a locally analytic representation π_min(ρ) for generic crystalline GSp_4-valued Galois representations that encodes the full GSp_4-structure via D_cris(ρ) and Hodge parameters. It develops a deformation-theoretic framework for crystalline (φ,Γ)-modules with GSp_4-structure, defines Hodge parameters a_D,b_D through refined triangulations, and constructs universal extensions that yield π_min(D) with a precise socle/cosocle structure. The global theory connects these local objects to automorphic representations via eigenvarieties, establishing local-global compatibility in a setting that includes a pseudo-character approach and density of classical points, thereby advancing a GSp_4-specific p-adic Langlands program. The methods generalize to reductive groups with suitable GL_n embeddings, providing a scalable framework for p-adic Hodge-theoretic parameters in reductive-group contexts.
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 $ρ$ 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 $ρ$ an explicit locally analytic $E$-representation $π_\mathrm{min}(ρ)$ of $\mathrm{GSp}_4(\mathbb{Q}_p)$, which encodes enough information to determines $ρ$. Moreover, under certain settings, this construction follows the local-global compatibility.