[Paper Review] Crystalline representations of G_Qp^a with coefficients

Hui June Zhu|ArXiv.org|Jul 7, 2008

Advanced Algebra and Geometry17 references3 citations

TL;DR

This paper establishes a Fontaine-Laffaille-type correspondence for crystalline representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_{p^a}/\mathbb{Q}_{p^a})$ with coefficients in a $p$-adic field $E$, proving a bijection between Galois-stable lattices in crystalline representations and strongly divisible $\varphi$-lattices when the $\sigma$-invariant Hodge-Tate weights are less than $p-1$. It classifies 2-dimensional crystalline representations and their mod $p$ reductions, generalizing results of Deligne, Fontaine-Serre, and Edixhoven to Hilbert modular forms under this condition.

ABSTRACT

This paper studies crystalline representations of G_K with coefficients of any dimension, where K is the unramified extension of Q_p of degree a. We prove a theorem of Fontaine-Laffaille type when σ-invariant Hodge-Tate weight less than p-1, which establishes the bijection between Galois stable lattices in crystalline representations and strongly divisible ϕ-lattice. In generalizing Breuil's work, we classify all reducible and irreducible crystalline representations of G_K of dimensional 2, then describe their mod p reductions. We generalize some results (of Deligne, Fontaine-Serre, and Edixhoven) to representations arising from Hilbert modular forms when σ-invariant Hodge-Tate weight less than p-1.

Motivation & Objective

• To extend Fontaine-Laffaille theory to crystalline representations with coefficients in a $p$-adic field $E$ containing $\mathbb{Q}_{p^a}$, generalizing the classical correspondence to higher-dimensional and coefficient-adapted settings.
• To classify all irreducible and reducible 2-dimensional crystalline representations of $G_{\mathbb{Q}_{p^a}}$ and describe their mod $p$ reductions.
• To generalize results of Deligne, Fontaine-Serre, and Edixhoven on mod $p$ reductions of Galois representations arising from Hilbert modular forms under the condition that $\sigma$-invariant Hodge-Tate weights are less than $p-1$.
• To construct crystalline representations via a structural bijection between $\mathrm{GL}_d(\mathcal{O}_K)/\sim_{\sigma,\mathbf{k}}$ and isomorphism classes of crystalline representations with prescribed Hodge polygon $\mathbf{k}$.

Proposed method

• Constructs crystalline representations using a bijection $\Theta: \mathrm{GL}_d(\mathcal{O}_K)/\sim_{\sigma,\mathbf{k}} \to \mathrm{Rep}_{\mathrm{cris}/E}^{\mathbf{k}}(G_K)$, where each matrix $A$ in $\mathrm{GL}_d(\mathcal{O}_K)$ defines a representation via its action on a filtered $\varphi$-module with prescribed Hodge-Tate weights.
• Defines embedded strongly divisible $\varphi$-lattices and uses Newton and Hodge polygons with $\sigma$-invariant variation to analyze the structure of filtered $\varphi$-modules.
• Applies embedded Wach modules and their continuity properties to lift mod $p$ reductions and study $\varphi$-action on reductions, particularly in the 2-dimensional case.
• Uses the $\varphi$-action on mod $p$ reductions of Wach modules to prove irreducibility by showing no nontrivial $\varphi$-invariant submodules exist in $\mathbb{F}_p((\pi))$.
• Employs the determinant of the mod $p$ reduction to identify the associated character $\omega_{2a}^{\sum k_j p^j}$, linking the Galois representation to a twist of the mod $p$ cyclotomic character.
• Compares the mod $p$ reduction of the representation to a direct sum of two characters via a change of basis, showing $\overline{V}|_{I_K} \cong \omega_{2a}^{\sum k_j p^j} \oplus \omega_{2a}^{p^a \sum k_j p^j}$.

Experimental results

Research questions

• RQ1 Under what conditions does a Fontaine-Laffaille-type correspondence hold for crystalline representations with coefficients in a $p$-adic field $E$?
• RQ2 How can all 2-dimensional crystalline representations of $G_{\mathbb{Q}_{p^a}}$ be classified, and what are their mod $p$ reductions?
• RQ3 When do mod $p$ reductions of crystalline representations depend only on the Hodge-Tate weights and not on the choice of lattice?
• RQ4 How do the results of Deligne, Fontaine-Serre, and Edixhoven extend to Galois representations from Hilbert modular forms under the $\sigma$-invariant Hodge-Tate weight condition $< p-1$?

Key findings

• A bijection $\Theta: \mathrm{GL}_d(\mathcal{O}_K)/\sim_{\sigma,\mathbf{k}} \to \mathrm{Rep}_{\mathrm{cris}/E}^{\mathbf{k}}(G_K)$ is constructed, explicitly linking matrices in $\mathrm{GL}_d(\mathcal{O}_K)$ to crystalline representations with prescribed Hodge polygon $\mathbf{k}$.
• When $\sigma$-invariant Hodge-Tate weights are less than $p-1$, the functor $\mathbf{D}_{\mathrm{cris}}^*$ induces a bijection between Galois-stable lattices in crystalline representations and strongly divisible $\varphi$-lattices in the associated filtered $\varphi$-module.
• For 2-dimensional crystalline representations $V_{{\vec{k}},\vec{v},\vec{1}}$ with $\mathrm{ord}_p(\vec{v}) > 0$, the mod $p$ reduction satisfies $\overline{V}_{{\vec{k}},\vec{v},\vec{1}} \cong \overline{V}_{{\vec{k}},{\vec{0}},\vec{1}}$, showing independence from $\vec{v}$ under this condition.
• The mod $p$ reduction $\overline{V}_{{\vec{k}},{\vec{0}},\vec{1}}$ is irreducible as an étale $(\varphi,\Gamma)$-module over $\mathbb{F}_p((\pi))^{\mathrm{sep}}$, proven by showing no solution exists in $\mathbb{F}_p((\pi))$ to the characteristic equation of $\varphi^a$.
• The restriction of $\overline{V}_{{\vec{k}},{\vec{0}},\vec{1}}$ to the inertia group $I_K$ is isomorphic to $\omega_{2a}^{\sum k_j p^j} \oplus \omega_{2a}^{p^a \sum k_j p^j}$, explicitly identifying the mod $p$ Galois representation as a sum of two characters.