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 Q p , with p ≠ 2 . To do so, we first classify abstract involutions over SL ( 2 , F ) with F a quadratic extension of 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 ) ⊂ SL ( 2 , E ) where E is a quadratic extension of F, of SL ( 2 , R ) ⊂ SL ( 2 , C ) , and of H θ ⊂ SL ( 2 , F ) , where H θ is the fixed point group of an F-involution θ over SL ( 2 , F ) .