Chabauty limits of groups of involutions in $SL(2,F)$ for local fields

Abstract

We classify Chabauty limits of groups fixed by various (abstract) involutions over $SL(2,F)$, where $F$ is a finite field-extension of $\mathbb{Q}_p$, with $p\neq 2$. To do so, we first classify abstract involutions over $SL(2,F)$ with $F$ a quadratic extension of $\mathbb{Q}_p$, and prove $p$-adic polar decompositions with respect to various subgroups of $p$-adic $SL_2$. Then we classify Chabauty limits of: $SL(2, F) \subset SL(2,E)$ where $E$ is a quadratic extension of $F$, of $SL(2,\mathbb{R}) \subset SL(2,\mathbb{C})$, and of $H_\theta \subset SL(2,F)$, where $H_\theta$ is the fixed point group of an $F$-involution $\theta$ over $SL(2,F)$.