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 $p\ne 2$. 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 $S{L}_2$. 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 $SL(2,F)$.

arXiv commentIncorporated the comments of the referee, accepted in Communications in Algebra