On wonderful compactifications of SL(2,𝔽) for non-Archimedean local fields 𝔽

Abstract

We compute the wonderful compactification of symmetric varieties of $\text{SL}(2,\mathbb{F})$, where $\mathbb{F}$ is a finite field-extension of ${\mathbb{Q}}_p$ with $p\ne 2$, that comes from either an abstract or an $\mathbb{F}$-involution of $\text{SL}(2,\mathbb{F})$. For each of those wonderful compactifications we find the $\text{SL}(2,\mathbb{F})$-stabilizers of the accumulation points of the corresponding symmetric varieties and show they agree with the Chabauty limits found in an earlier paper by the author and Arielle Leitner from 2023.

KEYWORDS:

• Bruhat–Tits trees
• groups over local fields
• wonderful compactification

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20
• 22
• 51

Acknowledgments

The author would like to thank Aarhus University and Oberwolfach Research Institute for Mathematics for the perfect working conditions they provide, and to Linus Kramer and Maneesh Thakur for very helpful discussions regarding reference [19]. As well, author thanks Arielle Leitner for the wonderful collaboration on our joint paper [7] that gave rise to this article, for reading it and providing useful comments. We also thank the referee for an incredibly job of making many suggestions which greatly improved the clarity and readability of the paper.

Additional information

Funding

The author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 754513 and The Aarhus University Research Foundation, and a research grant (VIL53023) from VILLUM FONDEN. The author was partially supported by The Mathematisches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics).