2025/09/14 by Corina Ciobotaru, Ciobotaru, Corina
Mathematics · #advanced mathematical theories #Advanced Topology and Set Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2509.11202
We study Chabauty limits of the fixed-point group of k-points Hk associated with an involutive k-automorphism θ of a connected linear reductive group G defined over a non-Archimedean local field k of characteristic zero. Leveraging the geometry of the Bruhat--Tits building, the structure of (θ,k)-split tori, and the KBkHk decomposition of Gk, we establish that any nontrivial Chabauty limit L of Hk is Gk-conjugate to a subgroup of Uσ+(k) \rtimes (Ker(α)0 ⋅ (Hk ∩ M_σ±)) ≤ Pσ+(k), where α is a projection map arising from a Levi factor M_σ± of a parabolic subgroup Pσ+ ⊂ G, and Ker(α)0 denotes the subgroup of elliptic elements in the kernel of α. Our analysis distinguishes between elliptic and hyperbolic elements and constructs explicit unipotent elements in the limit group L using the Moufang property of Gk. Furthermore, we show that L acts transitively on the set of ideal simplices opposite to σ+. These results yield a detailed description of the Chabauty compactification of Hk, and provide new insights into its interaction with the non-Archimedean geometry of Gk.