[論文レビュー] Bruhat-Tits group schemes over higher dimensional base-II

V. Balaji, Yashonidhi Pandey|arXiv (Cornell University)|Mar 5, 2026

Algebraic Geometry and Number Theory被引用数 0

ひとこと要約

The paper proves that split reductive Bruhat–Tits (BT) group schemes over higher dimensional bases are affine, and it provides a new construction of higher BT-group schemes beyond parahoric ones using Yu’s recursive approach, dilatations, and structure theory.

ABSTRACT

We prove that split reductive BT group schemes over a higher dimensional base are {\em affine}. Our method also gives a new construction of higher BT-group schemes more general than parahoric ones. The new ingredients are an extension of J.-K.Yu's construction in \cite{yu} to higher dimensional bases, Néron-Raynaud dilatations of subgroup schemes on divisors, combined with techniques from \cite{bt2} and the structure theory developed in \cite{bp}.

研究の動機と目的

• Motivate the construction of higher BT-group-schemes on higher dimensional bases under mild residue field characteristics.
• Generalize the base case from parahoric to broader higher BT-group schemes.
• Establish affineness and smoothness of the resulting group schemes with connected fibers.
• Extend Yu’s recursive construction to higher dimensional bases and relate to dilatations.

提案手法

• Extend J.-K. Yu’s recursive step to higher dimensional bases within the BT-framework.
• Use Néron-Raynaud dilatations of subgroup schemes on divisors to glue local data into a global BT-group scheme.
• Leverage affine apartment data and concave functions to interpolate group schemes across a base with a normal crossing divisor.
• Prove reductive Levi-decompositions in non-perfect residue settings to ensure affineness and big-cell structure.
• Handle cases where G is split reductive, including non-simply-connected scenarios, by lifting to simply-connected covers and factoring centers.
• Apply base-case results from Bruhat–Tits theory and extend with dilatations to obtain the global higher BT-group schemes.

実験結果

リサーチクエスチョン

• RQ1 Can split reductive BT-group schemes be realized as affine group schemes over higher dimensional bases?
• RQ2 How can one extend Yu’s recursive construction to multi-divisor bases using dilatations while preserving smoothness and connected fibers?
• RQ3 What are the structural properties (e.g., Levi decompositions) of the closed fibers in non-perfect residue settings?
• RQ4 How do concave functions and parahoric data assemble into higher BT-group schemes beyond type I (parahoric)?

主な発見

• Split reductive BT-group schemes over higher dimensional bases are affine and smooth with connected fibers.
• A new construction of higher BT-group schemes (types II and III) is achieved via an extension of Yu’s recursion plus dilatations.
• The method provides a big-cell structure for the interpolated group schemes and clarifies their reductive quotients and Levi decompositions.
• The paper handles non-perfect residue fields and non-simply-connected G by passing to simply-connected covers and controlling centers.
• For dimension two bases, the recursive step and affineness are established; higher dimensions are handled via inductive dilatations along divisors.
• A unified framework connects the parahoric (type I) case with the broader higher BT-group schemes through dilatation techniques.