#### Vol. 13, No. 6, 2019

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Congruences of parahoric group schemes

Vol. 13 (2019), No. 6, 1475–1499
DOI: 10.2140/ant.2019.13.1475
##### Abstract

Let $F$ be a nonarchimedean local field and let $T$ be a torus over $F$. With ${\mathsc{T}}^{N\phantom{\rule{0.3em}{0ex}}R}$ denoting the Néron–Raynaud model of $T$, a result of Chai and Yu asserts that the model ${\mathsc{T}}^{N\phantom{\rule{0.3em}{0ex}}R}{×}_{{\mathfrak{O}}_{F}}{\mathfrak{O}}_{F}∕{\mathfrak{p}}_{F}^{m}$ is canonically determined by $\left({Tr}_{l}\left(F\right),\Lambda \right)$ for $l\gg m$, where ${Tr}_{l}\left(F\right)=\left({\mathfrak{O}}_{F}∕{\mathfrak{p}}_{F}^{l},{\mathfrak{p}}_{F}∕{\mathfrak{p}}_{F}^{l+1},ϵ\right)$ with $ϵ$ denoting the natural projection of ${\mathfrak{p}}_{F}∕{\mathfrak{p}}_{F}^{l+1}$ on ${\mathfrak{p}}_{F}∕{\mathfrak{p}}_{F}^{l}$, and $\Lambda :={X}_{\ast }\left(T\right)$. 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 $F$.

##### Keywords
parahoric, close local fields
Primary: 22E50
Secondary: 11F70