Congruences of parahoric group schemes

TL;DR

This paper extends the understanding of how certain group schemes over local fields are determined by their truncated data, focusing on parahoric group schemes in the context of Bruhat-Tits theory.

Contribution

It proves an analogous congruence result for parahoric group schemes, generalizing previous results known for tori to the setting of reductive groups.

Findings

Established a canonical determination of parahoric group schemes from truncated data.

Extended Chai and Yu's results from tori to reductive groups.

Provided a new framework for understanding congruences of group schemes.

Abstract

Let $F$ be a non-archimedean local field and let $T$ be a torus over $F$. With $\cT^{NR}$ denoting the N\'eron-Raynaud model of $T$, a result of Chai and Yu asserts that the model $\cT^{NR} \times_{\fO_F} \fO_F/\fp_F^m$ is canonically determined by $(\Tr_l(F), \Lambda)$ for $l>>m$, where $\Tr_l(F) = (\fO_F/\fp_F^l, \fp_F/\fp_F^{l+1}, \epsilon)$ with $\epsilon$ denoting the natural projection of $\fp_F/\fp_F^{l+1}$ on $\fp_F/\fp_F^l$, and $\Lambda:=X_*(T)$. In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat-Tits building of a connected reductive group over $F$.