[Paper Review] Finite slope subspace without Y-smallness

Ruochuan Liu|arXiv (Cornell University)|Feb 10, 2012

Algebraic Geometry and Number Theory1 citations

TL;DR

This paper proves the global triangulation conjecture for refined p-adic representations under a mild condition, showing that the associated (φ, Γ)-modules admit a global triangulation on a Zariski open and dense subspace containing all regular non-critical points. It establishes that all specializations of a refined family are trianguline and identifies a large class of points in the global triangulation locus, providing a key ingredient for proving the properness of the Coleman-Mazur eigencurve.

ABSTRACT

We prove the global triangulation conjecture for families of refined p-adic representations under a mild condition. That is, for a refined family, the associated family of (phi, Gamma)-modules admits a global triangulation on a Zariski open and dense subspace of the base that contains all regular non-critical points. We also determine a large class of points which belongs to the locus of global triangulation. Furthermore, we prove that all the specializations of a refined family are trianguline. In the case of the Coleman-Mazur eigencurve, our results provide the key ingredient for showing its properness in a subsequent work.

Motivation & Objective

• To prove the global triangulation conjecture for refined families of p-adic representations under a mild condition.
• To identify a large class of points lying in the locus of global triangulation within the base space of a refined family.
• To establish that all specializations of a refined family are trianguline, extending local triangulation properties globally.
• To provide foundational results necessary for proving the properness of the Coleman-Mazur eigencurve in subsequent work.

Proposed method

• Utilizes the theory of (φ, Γ)-modules associated to p-adic representations to analyze the structure of refined families.
• Applies techniques from p-adic Hodge theory and deformation theory to study the geometry of the base space of refined families.
• Employs Zariski openness and density arguments to show that global triangulation holds on a large open subset of the base.
• Identifies regular non-critical points as part of the global triangulation locus using finiteness and non-criticality conditions.
• Leverages the refinedness condition to control the structure of the (φ, Γ)-module and ensure compatibility with global triangulation.
• Uses the fact that triangulinity is preserved under specialization to deduce that all specializations of a refined family are trianguline.

Experimental results

Research questions

• RQ1 Does the global triangulation conjecture hold for refined families of p-adic representations under a mild condition?
• RQ2 Which points in the base space of a refined family belong to the locus of global triangulation?
• RQ3 Are all specializations of a refined family trianguline, even at non-regular or critical points?
• RQ4 How does the global triangulation structure relate to the geometry of the Coleman-Mazur eigencurve?
• RQ5 What conditions ensure that a Zariski open and dense subspace of the base supports a global triangulation?

Key findings

• The global triangulation conjecture is proven for refined families of p-adic representations under a mild condition.
• The associated (φ, Γ)-modules admit a global triangulation on a Zariski open and dense subspace of the base that includes all regular non-critical points.
• A large class of points in the base space is explicitly identified as belonging to the locus of global triangulation.
• All specializations of a refined family are trianguline, confirming that triangulinity is preserved under specialization.
• The results provide a crucial technical ingredient for proving the properness of the Coleman-Mazur eigencurve in subsequent work.
• The global triangulation structure is shown to be compatible with the refinedness condition and the non-criticality of points.