[Paper Review] Subtle Invariants of $F$-crystals

Xiao Xiao|arXiv (Cornell University)|Nov 16, 2013

Coding theory and cryptography7 references3 citations

TL;DR

This paper proves that the isomorphism number $ n_{\mathcal{M}} $ of a non-ordinary $ F $-crystal $ \mathcal{M} $ over an algebraically closed field of characteristic $ p>0 $ is equal to its level torsion $ \ell_{\mathcal{M}} $, confirming a conjecture by Vasiu. The proof generalizes invariants like endomorphism and homomorphism numbers to $ F $-crystals and establishes their equality via group scheme constructions and $ p $-adic lattice analysis in the context of truncated $ F $-crystals.

ABSTRACT

Vasiu proved that the level torsion $\ell_{\mathcal{M}}$ of an $F$-crystal $\mathcal{M}$ over an algebraically closed field of characteristic $p>0$ is a non-negative integer that is an effectively computable upper bound of the isomorphism number $n_{\mathcal{M}}$ of $\mathcal{M}$ and expected that in fact one always has $n_{\mathcal{M}} = \ell_{\mathcal{M}}$. In this paper, we prove that this equality holds.

Motivation & Objective

• To resolve Vasiu's conjecture that the isomorphism number $ n_{\mathcal{M}} $ of an $ F $-crystal $ \mathcal{M} $ equals its level torsion $ \ell_{\mathcal{M}} $ in general.
• To generalize the invariants $ \ell_{\mathcal{M}} $, $ e_{\mathcal{M}} $, and $ f_{\mathcal{M}} $—originally defined for $ p $-divisible groups—to $ F $-crystals over algebraically closed fields of positive characteristic.
• To ensure the invariants remain unchanged under field extensions by constructing appropriate group schemes $ \mathbf{End}_s(\mathcal{M}) $ and $ \mathbf{Aut}_s(\mathcal{M}) $ for $ F $-truncations modulo $ p^s $.
• To establish the equality $ n_{\mathcal{M}} = \ell_{\mathcal{M}} $ for non-ordinary $ F $-crystals using a chain of inequalities $ f_{\mathcal{M}} \leq e_{\mathcal{M}} \leq \ell_{\mathcal{M}} \leq f_{\mathcal{M}} $.

Proposed method

• Generalize the level torsion $ \ell_{\mathcal{M}} $, homomorphism number $ e_{\mathcal{M}} $, and coarse homomorphism number $ f_{\mathcal{M}} $ to $ F $-crystals via $ p $-adic lattice constructions in the endomorphism algebra of $ \mathcal{M} $.
• Construct group schemes $ \mathbf{End}_s(\mathcal{M}) $ and $ \mathbf{Aut}_s(\mathcal{M}) $ whose $ k $-points classify endomorphisms and automorphisms of $ F $-truncations of $ \mathcal{M} $ modulo $ p^s $, ensuring compatibility under field extensions.
• Use the change-of-basis matrix between two $ W(k) $-bases of $ \mathrm{End}(M) \otimes_{W(k)} W(k) $ to compute the $ p $-adic valuation of entries and derive the level torsion $ \ell_{\mathcal{M}} $.
• Prove the chain of inequalities $ f_{\mathcal{M}} \leq e_{\mathcal{M}} \leq \ell_{\mathcal{M}} \leq f_{\mathcal{M}} $, implying equality among all three invariants for non-ordinary $ F $-crystals.
• Apply the theory of $ F $-truncations of $ F $-crystals, introduced by Vasiu, to reduce the problem to computations in truncated Witt vector rings and $ B(k) $-modules.
• Analyze the action of the Frobenius $ \sigma $ and Verschiebung $ \theta $ on endomorphism lattices to determine the smallest $ s $ such that $ \varphi $-equivariant automorphisms modulo $ p^s $ induce isomorphisms of $ F $-crystals.

Experimental results

Research questions

• RQ1 Does the isomorphism number $ n_{\mathcal{M}} $ of a non-ordinary $ F $-crystal $ \mathcal{M} $ over an algebraically closed field of characteristic $ p>0 $ equal its level torsion $ \ell_{\mathcal{M}} $, as conjectured by Vasiu?
• RQ2 Can the invariants $ \ell_{\mathcal{M}} $, $ e_{\mathcal{M}} $, and $ f_{\mathcal{M}} $ be meaningfully generalized from $ p $-divisible groups to $ F $-crystals while preserving their invariance under field extensions?
• RQ3 What is the precise relationship between the isomorphism number $ n_{\mathcal{M}} $ and the level torsion $ \ell_{\mathcal{M}} $, and does the equality $ n_{\mathcal{M}} = \ell_{\mathcal{M}} $ hold universally for non-ordinary $ F $-crystals?
• RQ4 How can the group schemes $ \mathbf{End}_s(\mathcal{M}) $ and $ \mathbf{Aut}_s(\mathcal{M}) $ be constructed so that their $ k $-points classify $ F $-truncation automorphisms and preserve the invariants' values?

Key findings

• The isomorphism number $ n_{\mathcal{M}} $ of a non-ordinary $ F $-crystal $ \mathcal{M} $ is equal to its level torsion $ \ell_{\mathcal{M}} $, confirming Vasiu's conjecture.
• The invariants $ f_{\mathcal{M}} $, $ e_{\mathcal{M}} $, and $ \ell_{\mathcal{M}} $ are all equal for non-ordinary $ F $-crystals, as established by the chain of inequalities $ f_{\mathcal{M}} \leq e_{\mathcal{M}} \leq \ell_{\mathcal{M}} \leq f_{\mathcal{M}} $.
• The level torsion $ \ell_{\mathcal{M}} $ is computed as $ 2\lambda_1 $ in a specific example where $ \lambda_1 $ is the smallest Newton slope of $ \mathcal{M} $, and this matches $ n_{\mathcal{M}} $.
• The construction of group schemes $ \mathbf{End}_s(\mathcal{M}) $ and $ \mathbf{Aut}_s(\mathcal{M}) $ ensures that the invariants are preserved under field extensions, a key technical requirement for the generalization.
• The proof relies on analyzing the $ p $-adic valuation of entries in the change-of-basis matrix between two $ W(k) $-bases of $ \mathrm{End}(M) \otimes_{W(k)} W(k) $, leading to the determination of $ \ell_{\mathcal{M}} $.