Cartan subgroups and generosity in $SL_2(\mathbb{Q}_p)$

TL;DR

This paper classifies all Cartan subgroups of SL_2 over the p-adic numbers and identifies the diagonal subgroup as the unique generous Cartan subgroup up to conjugacy.

Contribution

It provides a complete description of Cartan subgroups in SL_2(Q_p) and establishes the uniqueness and generosity of the diagonal Cartan subgroup.

Findings

The diagonal Cartan subgroup is generous in SL_2(Q_p).

All Cartan subgroups are classified and up to conjugacy.

The diagonal subgroup is the only generous Cartan subgroup.

Abstract

We describe all Cartan subgroups of $SL_2(\mathbb{Q}_p)$. We show that the Cartan subgroup consisting of all diagonal matrices is generous and it is the only one up to conjugacy.