[Paper Review] On F-crystalline representations

Bryden Cais, Tong Liu|arXiv (Cornell University)|Feb 5, 2015

Advanced Algebra and Geometry4 citations

TL;DR

This paper extends Kisin's theory of crystalline representations and Kisin modules to arbitrary finite extensions $F/\mathbb{Q}_p$ and general power series $f(u)$, constructing a new class of infinite, totally wildly ramified extensions $K_\infty/K$ such that the restriction functor $V \mapsto V|_{G_{K_\infty}}$ is fully faithful on $F$-crystalline representations. It establishes a classification of $F$-Barsotti-Tate groups via Kisin modules of height 1 with generalized Frobenius lifts.

ABSTRACT

We extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, for a finite and totally ramified extension $F/\mathbb Q_p$, and an arbitrary finite extension $K/F$, we construct a general class of infinite and totally wildly ramified extensions $K_\infty/K$ so that the functor $V\mapsto V|_{G_{K_\infty}}$ is fully-faithfull on the category of $F$-crystalline representations $V$. We also establish a new classification of $F$-Barsotti-Tate groups via Kisin modules of height 1 which allows more general lifts of Frobenius.

Motivation & Objective

• To generalize Kisin's theory of $\mathbb{Z}_p$-lattices in semistable representations to $F$-crystalline representations for arbitrary finite extensions $F/\mathbb{Q}_p$.
• To construct a new class of infinite, totally wildly ramified extensions $K_\infty/K$ via Frobenius-iterate systems $\{\pi_n\}$ defined by a power series $f(u)$, replacing the classical $K_\infty$ from $p^n$-th roots of unity.
• To establish a fully faithful restriction functor $V \mapsto V|_{G_{K_\infty}}$ on the category of $F$-crystalline representations with Hodge-Tate weights in $\{0,\dots,r\}$.
• To classify $F$-Barsotti-Tate groups over $\mathcal{O}_K$ via Kisin modules of height 1 with generalized Frobenius structure.
• To lay foundational tools for extending the theory of $(\varphi, \hat{G})$-modules and Breuil's theory to more general coefficient fields and Frobenius lifts.

Proposed method

• Define a generalized Frobenius endomorphism $\varphi$ on $\mathfrak{S}_F = \mathcal{O}_F[[u]]$ by acting as the $q$-power Frobenius on $W(k)$, trivially on $\mathcal{O}_F$, and sending $u$ to $f(u)$, where $f(u) \equiv u^q \mod \mathfrak{m}_F$.
• Construct the extension $K_{\underline{\pi}} = \bigcup_n K(\pi_n)$ via a compatible system $\{\pi_n\}$ satisfying $f(\pi_n) = \pi_{n-1}$, forming a non-Galois, totally wildly ramified extension.
• Define Kisin modules of $E(u)$-height $r$ as finite free $\mathfrak{S}_F$-modules $\mathfrak{M}$ with a $\varphi$-semilinear endomorphism $\varphi_\mathfrak{M}$ whose linearization has cokernel killed by $E(u)^r$
• Use the functor $T_{\mathfrak{S}}(\mathfrak{M})$ to associate $\mathcal{O}_F$-lattices in $F$-crystalline representations to Kisin modules, generalizing Kisin's construction.
• Prove that the assignment $V \mapsto \mathfrak{M}$ is fully faithful on $F$-crystalline representations via the construction of a $\mathfrak{S}_F$-linear isomorphism $T_{\mathfrak{S}}(\mathfrak{M}) \simeq T$ for $G_{K_\infty}$-stable lattices $T \subset V$
• Conjecture that Kisin modules associated to different $f(u)$ become isomorphic after base change to $W(R)_F$, generalizing comparison results in the classical case.

Experimental results

Research questions

• RQ1 Can Kisin's theory of $\mathbb{Z}_p$-lattices in semistable representations be extended to $F$-crystalline representations for arbitrary finite extensions $F/\mathbb{Q}_p$?
• RQ2 Does the restriction functor $V \mapsto V|_{G_{K_\infty}}$ remain fully faithful for $F$-crystalline representations when $K_\infty$ is constructed via a general power series $f(u)$ instead of $u^p$?
• RQ3 Can $F$-Barsotti-Tate groups be classified via Kisin modules of height 1 when the Frobenius structure is generalized beyond the classical case?
• RQ4 How do Kisin modules associated to different choices of $f(u)$ relate when $F$ and $\pi$ are fixed?
• RQ5 Can the torsion theory of Kisin modules be extended to this generalized setting, and can it recover known results on reductions of potentially crystalline representations?

Key findings

• The restriction functor $V \mapsto V|_{G_{K_\infty}}$ is fully faithful on the category of $F$-crystalline representations with Hodge-Tate weights in $\{0,\dots,r\}$, generalizing Kisin's classical result.
• For any $F$-crystalline representation $V$ with a $G_{K_\infty}$-stable $\mathcal{O}_F$-lattice $T$, there exists a Kisin module $\mathfrak{M}$ of $E(u)$-height $r$ such that $T_{\mathfrak{S}}(\mathfrak{M}) \simeq T$, extending Kisin's construction to general $F$ and $f(u)$.
• The category of $F$-Barsotti-Tate groups over $\mathcal{O}_K$ is anti-equivalent to the category of Kisin modules of height 1 over $\mathfrak{S}_F$ with the generalized Frobenius structure.
• The construction recovers known cases: when $f(u) = u^p$, it reduces to Kisin's theory; when $v_F(a_1) = 1$, it includes the Lubin-Tate case of Kisin-Ren.
• The paper conjectures that Kisin modules associated to different $f(u)$ become isomorphic after base change to $W(R)_F$, suggesting a universal comparison isomorphism.
• The theory provides a framework for extending Breuil's classification of semistable representations and torsion Kisin modules, though the monodromy operator remains an open challenge for general $f(u)$.