[Paper Review] Representations attached to vector bundles on curves over finite and p-adic fields, a comparison

Christopher Deninger|ArXiv.org|Mar 17, 2009

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes a precise link between Galois representations attached to vector bundles on curves over p-adic fields and the fundamental group of their special fibers over finite fields. It proves that the mod 𝔪 reduction of a p-adic Galois representation associated to a vector bundle with strongly semistable reduction factors through the fundamental group of the special fiber, using Nori’s fundamental group scheme to classify the resulting mod 𝔪 representation as essentially finite.

ABSTRACT

For a vector bundle E on a model of a smooth projective curve over a p-adic number field a p-adic representation of the geometric fundamental group of X has been defined in work with Annette Werner if the reduction of E is strongly semistable of degree zero. In the present note we calculate the reduction of this representation using the theory of Nori's fundamental group scheme.

Motivation & Objective

• To understand the reduction modulo 𝔪 of Galois representations attached to vector bundles on curves over p-adic fields.
• To clarify the relationship between p-adic representations and the fundamental group of the special fiber over a finite field.
• To show that the mod 𝔪 reduction of a representation associated to a vector bundle with strongly semistable reduction factors through the fundamental group of the reduced special fiber.
• To apply Nori’s theory of fundamental group schemes to classify the resulting mod 𝔪 representations as essentially finite.

Proposed method

• Uses a model 𝔛 of a curve X over 𝔬, with a vector bundle ℰ on 𝔛ₒ extending a bundle E on X_{ℂ_p}.
• Imposes the condition that ℰ has strongly semistable reduction of degree zero on the special fiber 𝔛_k.
• Applies the theory of [DW2] to construct a continuous p-adic Galois representation ρ_{ℰ,xₒ} on the lattice ℰ_{xₒ} in E_x.
• Reduces this representation modulo 𝔪 to obtain a representation ρ_{ℰ,xₒ} ⊗ k with values in GL(ℰ_{x_k}).
• Applies Nori’s fundamental group scheme π(Z,z) to classify essentially finite vector bundles on the special fiber Z = 𝔛^red_k.
• Uses Tannakian duality and the compatibility of pullbacks under finite étale covers to show commutativity of the diagram involving π₁(X,x) and π₁(𝔛^red_k, x_k).

Experimental results

Research questions

• RQ1 How does the mod 𝔪 reduction of a p-adic Galois representation attached to a vector bundle on a curve over ℂ_p relate to the fundamental group of the special fiber over 𝔽_p?
• RQ2 Under what conditions does the mod 𝔪 reduction of such a representation factor through the fundamental group of the special fiber?
• RQ3 Can Nori’s fundamental group scheme be used to classify the mod 𝔪 reduction of these representations?
• RQ4 Is the mod 𝔪 reduction of the representation ρ_{ℰ,xₒ} essentially finite, and if so, how is it related to the geometry of the special fiber?
• RQ5 Does the representation ρ_{ℰ,xₒ} itself factor through the specialization map π₁(X,x) → π₁(𝔛^red_k, x_k), or only its mod 𝔪 reduction?

Key findings

• The mod 𝔪 reduction of the p-adic Galois representation ρ_{ℰ,xₒ} factors through the specialization map π₁(X,x) → π₁(𝔛^red_k, x_k).
• The reduced representation ρ_{ℰ,xₒ} ⊗ k is isomorphic to the representation associated via Nori’s fundamental group scheme to the restriction ℰ^red_k of ℰ to the reduced special fiber.
• The bundle ℰ^red_k on 𝔛^red_k is essentially finite, meaning it arises from a finite group scheme torsor.
• The commutativity of the diagram involving π₁(X,x), π₁(𝔛^red_k, x_k), and GL(ℰ_{x_k}) is established via compatibility with finite étale covers trivializing the bundle.
• The result holds under the assumption that ℰ has strongly semistable reduction of degree zero on the special fiber.
• An example shows that ρ_{ℰ,xₒ} itself does not factor through π₁(𝔛^red_k, x_k), but only its mod 𝔪 reduction does, highlighting the necessity of reduction.