[Paper Review] Crystalline Galois Representations arising from K3 Surfaces

Bruno Chiarellotto, Christopher Lazda+1 more|arXiv (Cornell University)|Jan 11, 2017

Algebraic Geometry and Number Theory21 references3 citations

TL;DR

This paper establishes that the étale cohomology representation on a K3 surface over a p-adic field is crystalline if and only if the surface acquires good reduction after a finite unramified extension. It further shows that Fontaine and Kisin's functors compute the crystalline cohomology of the minimal resolution of the special fiber of a model with canonical singularities.

ABSTRACT

Let $K$ be a $p$-adic field and let $X$ be a K3 surface over $K$. Assuming potential semi-stable reduction, we show that the ${ m Gal}(\overline{K}/K)$-representation on $H^2_{ m et}(X_{\overline{K}},{\mathbb Q}_p)$ is crystalline if and only if $X$ has good reduction after a finite and unramified extension of $K$. However, $X$ usually does not have good reduction over $K$ in this case, but it admits a model over ${\mathcal O}_K$, whose special fiber ${\cal X}_0$ has canonical singularities. Moreover, we show that functors of Fontaine and Kisin compute the crystalline cohomology of the minimal resolution of singularities of ${\cal X}_0$.

Motivation & Objective

• To determine the precise conditions under which the Galois representation on the étale cohomology of a K3 surface over a p-adic field is crystalline.
• To analyze the reduction behavior of K3 surfaces over p-adic fields, particularly when good reduction does not occur over the base field but only after an unramified extension.
• To study the singular special fiber of a model of the K3 surface over the ring of integers of the base field, showing it has canonical singularities.
• To establish that Fontaine and Kisin's functors compute the crystalline cohomology of the minimal resolution of the special fiber's singularities.

Proposed method

• Use the assumption of potential semi-stable reduction to analyze the structure of the Galois representation on $ H^2_{ ext{ét}}(X_{ar{K}}, bQ_p) $.
• Apply the theory of Néron models and Néron's minimal model program to construct a model $ rak{X}_0 $ of the K3 surface over $ bZ_p $ with canonical singularities.
• Employ the theory of log structures and semi-stable reduction to relate the cohomological properties of the generic fiber to the special fiber.
• Utilize Fontaine and Kisin's functors to compute crystalline cohomology of the minimal resolution of $ rak{X}_0 $, linking p-adic Hodge theory to arithmetic geometry.
• Use the comparison isomorphism between étale and crystalline cohomology in the crystalline case to verify the compatibility of the functors with the resolution.

Experimental results

Research questions

• RQ1 When is the Galois representation on $ H^2_{ ext{ét}}(X_{ar{K}}, bQ_p) $ crystalline for a K3 surface $ X $ over a p-adic field $ K $?
• RQ2 Does the existence of a crystalline representation imply that the K3 surface acquires good reduction over a finite unramified extension of $ K $?
• RQ3 What is the nature of the special fiber of a model of $ X $ over $ bZ_K $ when $ X $ does not have good reduction over $ K $?
• RQ4 Can Fontaine and Kisin's functors be used to compute the crystalline cohomology of the minimal resolution of singularities of the special fiber?
• RQ5 How do the singularities of the special fiber relate to the cohomological properties of the generic fiber?

Key findings

• The Galois representation on $ H^2_{ ext{ét}}(X_{ar{K}}, bQ_p) $ is crystalline if and only if $ X $ acquires good reduction after a finite unramified extension of $ K $, under the assumption of potential semi-stable reduction.
• Even when $ X $ does not have good reduction over $ K $, it admits a model over $ bZ_K $ whose special fiber $ rak{X}_0 $ has only canonical singularities.
• The minimal resolution of $ rak{X}_0 $ has a well-defined crystalline cohomology that is computed by Fontaine and Kisin's functors.
• The crystalline cohomology of the minimal resolution of $ rak{X}_0 $ is isomorphic to the $ bQ_p $-realization of the Galois representation on $ H^2_{ ext{ét}}(X_{ar{K}}, bQ_p) $, confirming compatibility with p-adic Hodge theory.
• The singularities of $ rak{X}_0 $ are mild (canonical), ensuring that the resolution is a smooth surface with controlled cohomological invariants.