# Chabauty Limits of Groups of Involutions In $\operatorname{SL}(2,F)$ for
local fields

Corina Ciobotaru and Arielle Leitner

(Date: August 25, 2022)

###### Abstract.

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

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##  1\. Introduction

Let $k$ be a local field, which is not necessarily algebraically closed. Let
$G$ be a reductive algebraic group defined over $k$. A symmetric $k$-variety
of $G$ is the quotient $G(k)/H(k)$, where $H$ is the fixed point group of an
involution $\theta$ defined over $k$ of $G$, and $G(k),H(k)$ denote the sets
of $k$-rational points of $G$, $H$. Symmetric $k$-varieties appear naturally
and play a central role in the representation theory of algebraic groups, the
Langlands program, or the Plancherel formulas for Riemannian symmetric spaces
(see [Hel_k_invol, Introduction]).

When $k={\mathbb{R}}$, or ${\mathbb{C}}$, a space $G(k)/H(k)$ as above is an
affine symmetric space, generalizing the theory of Riemannian symmetric
spaces. Familiar examples of affine symmetric spaces come from quadratic forms
on ${\mathbb{R}}^{n}$, or ${\mathbb{C}}^{n}$, of signature $(p,q)$, where the
fixed point group $H(k)$ is the corresponding orthogonal group $O(p,q)$
preserving that quadratic form. In this way one obtains spherical geometry,
hyperbolic geometry, de Sitter geometry, or anti de Sitter geometry. Although
these geometries have different curvature, one can find geometric transitions
between them: continuous paths of geometric structures that change the type of
the model geometry in the limit, also known as a limit of geometries.
Geometric transitions arise in physics: deforming general relativity into
special relativity, or quantum mechanics into Newtonian mechanics. Cooper,
Danciger and Weinhard [CDW] classify limits of geometries coming from affine
symmetric spaces over ${\mathbb{R}}$. In particular, they classify the limits
of geometries of all of the groups $\operatorname{SO}(p,q)$ inside of
$\operatorname{GL}(n,{\mathbb{R}})$. Their approach uses a root space
decomposition of real Lie algebras. Trettel [Trettel] provides another
approach in Chapter 6 of his PhD thesis using the wonderful compactification.

Felix Klein’s Erlangen Program encodes geometries and uniquely determines them
by their groups of isometries. Therefore, studying limits of geometries is
equivalent to studying limits of groups of isometries (i.e. Lie groups) in the
_Chabauty topology_ , (see Section 2 for the concrete definitions and
properties).

Our article is the first installment of an analagous classification of
Chabauty limits as in [CDW] for $p$-adic groups fixed by (abstract)
involutions over $\operatorname{SL}(2,k)$, where $k$ is a finite field-
extension of ${\mathbb{Q}}_{p}$ with $p\neq 2$. In future work we will study
the general case of $\operatorname{SL}(n,k)$.

We use the classification of the isomorphism classes of $k$-involutions of a
connected reductive algebraic group defined over $k$ given in [Hel_k_invol]. A
simple characterization of the isomorphism classes of $k$-involutions of
$\operatorname{SL}(n,k)$ is given in [HelWD]. Further, [Beun, HW_class,
Sutherland] study ${\mathbb{Q}}_{p}$-involutions and their fixed point groups
for $\operatorname{SL}(2,{\mathbb{Q}}_{p})$.

Using the results of Borel–Tits [BoTi], and Steinberg [Stein], our first main
result is in Section 4 and gives a classification of all abstract and
$E$-involutions $\theta$ of $\operatorname{SL}(2,E)$, where
$E={\mathbb{Q}}_{p}(\alpha)$ is a quadratic extension of ${\mathbb{Q}}_{p}$.
We show in Theorem 1.1 they are all the composition of an inner automorphism
$\iota_{A}$ of $\operatorname{GL}(2,E)$ with the identity or the field
conjugation automorphism on $E$ (i.e. $\sigma:E\to E$ given by
$\sigma(a+\alpha b)=a-\alpha b$), and we determine the matrices
$A\in\operatorname{GL}(2,E)$ for the inner automorphism $\iota_{A}$.

######  Theorem 1.1.

Let $E={\mathbb{Q}}_{p}(\alpha)$ be a quadratic extension of
${\mathbb{Q}}_{p}$. Then any abstract involution $\theta$ of
$\operatorname{SL}(2,E)$ is of the form $\theta=\iota_{A}\circ\gamma$ where:

  1. (1)

$\gamma=\sigma$ and $A\in\\{\left(\begin{smallmatrix}z&y\\\
1&-\sigma(z)\end{smallmatrix}\right),\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)\\}\subset\operatorname{GL}(2,E)$, with
$y\in{\mathbb{Q}}_{p},z,x\in E$, with $z\sigma(z)+y\neq 0$ and $x\sigma(x)=1,$

  2. (2)

$\gamma=\operatorname{Id}$ and $A=\left(\begin{smallmatrix}0&1\\\
a&0\end{smallmatrix}\right)$, with $a\in E^{*}/(E^{*})^{2}$, up to conjugacy.

We further compute the fixed point groups $H_{\theta}$ of the involutions
$\theta$ from Theorem 1.1 using the ends in the ideal boundary $\partial
T_{K}$ of the Bruhat–Tits tree $T_{K}$ of $\operatorname{SL}(2,K)$, where $K$
is finite field-extension of $E$, and thus of ${\mathbb{Q}}_{p}$, that is
chosen accordingly. We will show those fixed point groups are either trivial,
or compact, or $\operatorname{GL}(2,E)$-conjugate to either the diagonal in
$\operatorname{SL}(2,E)$, or to $\operatorname{SL}(2,{\mathbb{Q}}_{p})$.

In Section 4 we obtain a geometric interpretation of the fixed point groups
$H_{\theta_{a}}$ of the $k$-involutions computed in [HW_class] for
$\operatorname{SL}(2,k)$, when $k$ is a finite field-extension of
${\mathbb{Q}}_{p}$ (i.e. $\theta_{a}$ as in Theorem 1.1(2)); they are all of
the form

$H_{\theta_{a}}:=\\{\left(\begin{smallmatrix}x&y\\\
ay&x\end{smallmatrix}\right)\in\operatorname{SL}(2,F)\;|\;x^{2}-ay^{2}=1\\}.$

For a precise and technical statement see Corollary 4.8.

Next, we classify the rest of $H_{\theta}$ given by Theorem 1.1.

######  Theorem 1.2.

Let $E={\mathbb{Q}}_{p}(\alpha)$ be a quadratic extension of
${\mathbb{Q}}_{p}$, and $\theta$ as in Theorem 1.1(1). Then
$H_{\theta}:=\\{g\in\operatorname{SL}(2,E)\;|\;\theta(g)=g\\}$ is either:

  1. (1)

$H_{\theta}$ is $\operatorname{GL}(2,E)$-conjugate to
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$

  2. (2)

$H_{\theta}=B\operatorname{SL}(2,K^{\sigma})B^{-1}\cap\operatorname{SL}(2,E)$,
for some matrix $B\in\operatorname{GL}(2,K)$, where $K\neq E$ is some finite
field-extension of $E$ and $K^{\sigma}:=\\{x\in K\;|\;\sigma(x)=x\\}$ is the
maximal subfield of $K$ with that property. In particular, $H_{\theta}$ is
compact or trivial.

One of the strategies to compute Chabauty limits of the fixed point groups $H$
of involutions over $G$ is to employ the polar decomposition
$G=\mathcal{K}\mathcal{B}H$, where $\mathcal{K}$ is a compact subset of $G$,
and $\mathcal{B}$ is a union of split tori of $G$. If moreover $\mathcal{B}$
is a finite union of such tori, then it is enough to compute the desired
Chabauty limits under conjugation with a sequence of elements from some fixed
split torus in $\mathcal{B}$. We cannot apply directly [BenoistOh] as their
result is proven only for $k$-involutions and not for abstract involutions. In
Section 5 we prove a polar decomposition for $k$-involutions and abstract
involutions for the $p$-adic $\operatorname{SL}_{2}$. Along the way we provide
different, direct, and more geometric proofs than in [HW_class, HelWD,
Hel_k_invol] for the case of $\operatorname{SL}_{2}$. Below $\theta_{a}$
refers to the involutions from Theorem 1.1(2), and $\theta$ to the involutions
from Theorem 1.1(1).

######  Proposition 1.3 (The polar decomposition for various subgroups of
$\operatorname{SL}_{2}$).

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$ and $E=F(\alpha)$ be
a quadratic extension of $F$. Let $H\leq G$ be such that

$(G,H)\in\\{(\operatorname{SL}(2,F),H_{\theta_{a}}),(\operatorname{SL}(2,E),\operatorname{SL}(2,F)),(\operatorname{SL}(2,{\mathbb{Q}}_{p}(\alpha)),H_{\theta})\\}.$

Denote by $T_{G}$ the Bruhat–Tits tree of $G$ and by $T_{H}$ the (possibly
finite) $H$ -invariant subtree of $T_{G}$. Let $I_{H}$ be the number of
$H$-orbits in the ideal boundary $\partial T_{G}-\partial T_{H}$, and
$\xi_{i}$ a representative in each such orbit. Let $x_{0}\in T_{G}\cap T_{H}$
be a vertex, which for the pair
$(\operatorname{SL}(2,E),\operatorname{SL}(2,F))$ will be taken to be the
point $0$ as in the figures 1, 2. Then

$G=\mathcal{K}\mathcal{B}H$

where $\mathcal{K}$ is a compact subset of $G$, and
$\mathcal{B}=\\{\operatorname{Id}\\}\bigsqcup\limits_{i\in I_{H}}A_{i}$, where
$A_{i}:=\\{a_{i}^{n}\;|\;n\in{\mathbb{Z}}\\}$ with $a_{i}\in G$ a hyperbolic
element of translation length $2$ and with attractive endpoint in the
$H$-orbit of $\xi_{i}$.

Finally, in Sections 6, 7 ,8 we enumerate all Chabauty limits of various fixed
point groups of involutions over $\operatorname{SL}(2,k)$. The subgroups
$B_{k}^{+}$ below correspond to the upper triangular Borel subgroups of
$\operatorname{SL}(2,k)$. See the relevant sections for more precise
statements.

######  Theorem 1.4.

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$ and $E$ be any
quadratic extension of $F$. Then any Chabauty limit of
$\operatorname{SL}(2,F)$ inside $\operatorname{SL}(2,E)$ is
$\operatorname{SL}(2,E)$-conjugate to either $\operatorname{SL}(2,F)$, or to
the subgroup $\\{\left(\begin{smallmatrix}a-\alpha b&z\\\ 0&a+\alpha
b\end{smallmatrix}\right)\;|\;a,b\in F\text{ with
}a^{2}-\alpha^{2}b^{2}=1,z\in E\\}\leq B_{E}^{+}.$

######  Theorem 1.5.

Any Chabauty limit of $\operatorname{SL}(2,{\mathbb{R}})$ inside
$\operatorname{SL}(2,{\mathbb{C}})$ is
$\operatorname{SL}(2,{\mathbb{C}})$-conjugate to either
$\operatorname{SL}(2,{\mathbb{R}})$, or to the subgroup
$\\{\left(\begin{smallmatrix}a-ib&z\\\
0&a+ib\end{smallmatrix}\right)\;|\;a,b\in{\mathbb{R}}\text{ with
}a^{2}+b^{2}=1,z\in{\mathbb{C}}\\}\leq B_{{\mathbb{C}}}^{+}.$

######  Theorem 1.6.

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$, and
$H_{\theta_{a}}$ the fixed point group of an involution $\theta_{a}$ on
$\operatorname{SL}(2,F)$ as in Theorem 1.1(2). Then any Chabauty limit of
$H_{\theta_{a}}$ is either $\operatorname{SL}(2,F)$-conjugate to
$H_{\theta_{a}}$, or to the subgroup
$\\{\mu_{2}\cdot\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right)\;|\;x\in F\\}$ of the Borel
$B_{F}^{+}\leq\operatorname{SL}(2,F)$, where $\mu_{2}$ is the group of
$2^{nd}$ roots of unity in $F$.

### Acknowledgements

Ciobotaru was partially supported by the Institute of Mathematics of the
Romanian Academy (IMAR), Bucharest, and The Mathematisches Forschungsinstitut
Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics). She would
like to thank those two institutions for the perfect working conditions they
provide. As well, Ciobotaru is supported by the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 754513 and The Aarhus University Research Foundation. Leitner
acknowledges support from Afeka college of engineering. We thank Uri Bader,
Linus Kramer, Thierry Stulemeijer, Maneesh Thakur, and Alain Valette for
helpful discussions.

##  2\.  The Chabauty Topology

The Chabauty topology was introduced in 1950 by Claude Chabauty [Ch] on the
set of all closed subgroups of a locally compact group. The initial motivation
of Chabauty was to show that some sets of lattices of some locally compact
groups are relatively compact, and to generalise a criterion of Mahler about
lattices of ${\mathbb{R}}^{n}$. In the Chabauty topology the set of all closed
subgroups of a locally compact group is compact. This implies that any
sequence of closed subgroups admits a convergent subsequence, and so it has at
least one limit, called a Chabauty limit. Therefore, limits of groups, and
thus limits of geometries, are not empty notions, the real difficulty is not
proving their existence, but computing concretely which geometric types may be
obtained as limits.

For a locally compact topological space $X$, the set of all closed subsets of
$X$ is a compact topological space with respect to the Chabauty topology (see
Proposition 2.1). Given a family $\mathcal{F}$ of closed subsets of $X$, it is
natural to study the closure of $\mathcal{F}$ with respect to the Chabauty
topology, $\overline{\mathcal{F}}$, and determine whether or not elements in
the boundary $\overline{\mathcal{F}}-\mathcal{F}$ satisfy the same properties
as those in $\mathcal{F}$. We call elements of $\overline{\mathcal{F}}$ the
Chabauty limits of $\mathcal{F}$.

For a good introduction to Chabauty topology [Ch] see [CoPau, Harpe, GJT] or
[Haettel, Section 2] and the references therein. We briefly recall some facts
that are used in this paper.

For a locally compact topological space $X$ we denote by $\mathcal{F}(X)$ the
set of all closed subsets of $X$. $\mathcal{F}(X)$ is endowed with the
Chabauty topology where every open set is a union of finite intersections of
subsets of the form $O_{K}:=\\{F\in\mathcal{F}(X)\;|\;F\cap K=\emptyset\\}$,
where $K$ is a compact subset of $X$, or
$O^{\prime}_{U}:=\\{F\in\mathcal{F}(X)\;|\;F\cap U\neq\emptyset\\}$, where $U$
is an open subset of $X$. By [CoPau, Proposition 1.7, p. 58] the space
$\mathcal{F}(X)$ is compact with respect to the Chabauty topology. Moreover,
if $X$ is Hausdorff and second countable then $\mathcal{F}(X)$ is separable
and metrisable, thus Hausdorff (see [CEM, Proposition I.3.1.2]).

######  Proposition 2.1.

([CoPau, Proposition 1.8, p. 60], [CEM, Proposition I.3.1.3]) Suppose $X$ is a
locally compact metric space. A sequence of closed subsets
$\\{F_{n}\\}_{n\in{\mathbb{N}}}\subset\mathcal{F}(X)$ converges to
$F\in\mathcal{F}(X)$ if and only if the following two conditions are
satisfied:

  * 1)

For every $f\in F$ there is a sequence $\\{f_{n}\in
F_{n}\\}_{n\in{\mathbb{N}}}$ converging to $f$;

  * 2)

For every sequence $\\{f_{n}\in F_{n}\\}_{n\in{\mathbb{N}}}$, if there is a
strictly increasing subsequence $\\{n_{k}\\}_{k\in{\mathbb{N}}}$ such that
$\\{f_{n_{k}}\in F_{n_{k}}\\}_{k\in{\mathbb{N}}}$ converges to $f$, then $f\in
F$.

For a locally compact group $G$ we denote by $\mathcal{S}(G)$ the set of all
closed subgroups of $G$. By [CoPau, Proposition 1.7, p. 58] the space
$\mathcal{S}(G)$ is closed in $\mathcal{F}(G)$, with respect to the Chabauty
topology, and is compact. Moreover, Proposition 2.1 applied to a sequence of
closed subgroups $\\{H_{n}\\}_{n\in{\mathbb{N}}}\subset\mathcal{S}(G)$
converging to $H\in\mathcal{S}(G)$, yeilds a similar characterisation of
convergence in $\mathcal{S}(G)$.

Understanding the topology of the entire Chabauty space of closed subgroups of
a group is difficult, and is known only for very few cases. For example, it is
easy to see $\mathcal{S}({\mathbb{R}})\cong[0,\infty]$. Hubbard and Pourezza
[HP] show $\mathcal{S}({\mathbb{R}}^{2})\cong\mathbb{S}^{4}$, the
4-dimensional sphere, and Kloeckner [Kloeckner] shows that while
$\mathcal{S}({\mathbb{R}}^{n})$ is not a manifold for $n>2$, it is a
stratified space in the sense of Goresky–MacPherson, and is simply connected.
However a full description of $\mathcal{S}({\mathbb{R}}^{n})$ is yet to be
obtained. There are a few non-abelian groups G for which $\mathcal{S}(G)$ is
reasonably well understood, e.g. the Heisenberg group and some other low
dimensional examples [BHK, Htt_2], but for most $G$ the topology of
$\mathcal{S}(G)$ is quite complicated.

Various authors have made progress understanding the closure of certain
families of subgroups in $\mathcal{S}(G)$: abelian subgroups [Baik1, Baik2,
Haettel, Leitner_sl3, Leitner_sln], connected subgroups [LL], and lattices
[BLL, Wang].

In more recent years, $p$-adic Chabauty spaces have received attention.
Bourquin and Valette [BV] have described the homeomorphism type of
$\mathcal{S}({\mathbb{Q}}_{p}^{*})$. Cornulier [Cornulier] has characterized
several properties of $\mathcal{S}(G)$ for $G$ a locally compact abelian
group. Chabauty closures of certain families of groups acting on trees have
been studied by [CR, Stulemeijer], and there are several open questions about
the Chabauty topology for locally compact groups in [CM]. The authors have
studied limits of families of subgroups in
$\operatorname{SL}(n,{\mathbb{Q}}_{p})$: parahoric subgroups [CL] and Cartan
subgroups [CLV]. Finally, [GR] have studied compactifications of Bruhat-Tits
buildings.

##  3\. Background Material

Throughout this article we restrict to $p\neq 2$. Let $F$ be a finite field-
extension of ${\mathbb{Q}}_{p}$ and $E$ be any quadratic extension of $F$. Let
$k_{F},k_{E}$ be the residue fields of $F,E$, respectively, and
$\omega_{F},\omega_{E}$ be uniformizers of $F,E$, respectively. Recall
$k_{F}^{*}/(k_{F}^{*})^{2}=\\{1,S\\}$, for some non-square $S\in k_{F}^{*}$.
Then $F^{*}/(F^{*})^{2}=\\{1,\omega_{F},S,S\omega_{F}\\}$ ([Serre, Corollaries
to Theorems 3 and 4] or [Sally, page 41, Section 12]). If $E$ is _ramified_
then $E=F(\sqrt{\omega_{F}})$, or $E=F(\sqrt{S\omega_{F}})$ (where
$\omega_{F}\neq\omega_{E}$), or if $E$ is _unramified_ then $E=F(\sqrt{S})$
(where $\omega_{F}=\omega_{E}$). We choose the unique valuation $|\cdot|_{E}$
on $E$ that extends the given valuation $|\cdot|_{F}$ on $F$. Choose
$\alpha\in\\{\sqrt{\omega_{F}},\sqrt{S},\sqrt{S\omega_{F}}\\}$ and so
$E=F(\alpha)$. Notice each element $x\in E$ can be uniquely written as
$x=a+b\alpha$, with $a,b\in F$. For the ramified extensions we can consider
$\omega_{E}^{2}=\omega_{F}$. Let $\mathcal{O}_{F}:=\\{x\in F\;|\;|x|_{F}\leq
1\\}$ denote the ring of integers of $F$, then $\mathcal{O}_{F}$ is compact
and open in $F$. For $F={\mathbb{Q}}_{p}$ we have
$\mathcal{O}_{{\mathbb{Q}}_{p}}={\mathbb{Z}}_{p}$,
$\omega_{{\mathbb{Q}}_{p}}=p$, $k_{{\mathbb{Q}}_{p}}={\mathbb{F}}_{p}$,
${\mathbb{F}}_{p}^{*}/({\mathbb{F}}_{p}^{*})^{2}=\\{1,S_{p}\\}$.

We denote by $T_{F}$ the Bruhat–Tits tree for $\operatorname{SL}(2,F)$ whose
vertices are equivalence classes of $\mathcal{O}_{F}$-lattices in $F^{2}$ (for
its construction see [Serre]). The tree $T_{F}$ is a regular, infinite tree
with valence $|k_{F}|+1$ at every vertex. The boundary at infinity $\partial
T_{F}$ of $T_{F}$ is the projective space $P^{1}(F)\cong F\cup\\{\infty\\}$.
Moreover, the endpoint $\infty\in\partial T_{F}$ corresponds to the vector
$\big{[}\begin{smallmatrix}0\\\ 1\end{smallmatrix}\big{]}\in P^{1}(F)$. The
rest of the endpoints $\xi\in\partial T_{F}$ correspond to the vectors
$\big{[}\begin{smallmatrix}1\\\ x\end{smallmatrix}\big{]}\in P^{1}(F)$, where
$x\in F$.

To give a concrete example, the Bruhat–Tits tree of
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$ is the
$p+1=|{\mathbb{F}}_{p}|+1$-regular tree. The boundary at infinity $\partial
T_{{\mathbb{Q}}_{p}}$ of $T_{{\mathbb{Q}}_{p}}$ is the projective space
$P^{1}({\mathbb{Q}}_{p})={\mathbb{Q}}_{p}\cup\\{\infty\\}$. In the figures
below we give a concrete visualisation of the Bruhat–Tits tree of
$\operatorname{SL}(2,F)$ inside the $\operatorname{SL}(2,E)$, when
$F={\mathbb{Q}}_{p}$. In both pictures we have drawn the Bruhat–Tits tree for
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$ (red) inside the tree for
$\operatorname{SL}(2,E)$ (blue and red). We denote $x\in{\mathbb{Q}}_{p}$,
$y\in{\mathbb{Q}}_{p}^{*}$.

Figure 1. Unramified quadratic extension: Let $E={\mathbb{Q}}_{p}(\alpha)$
where $\alpha^{2}=S_{p}$ and $\alpha\not\in{\mathbb{Q}}_{p}$. In the tree for
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$ every vertex has $p+1$ neighbours. In
the tree for $\operatorname{SL}(2,E)$ every vertex has $p^{2}+1$ neighbours,
obtained by adding more edges (blue) to each vertex of the tree for
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$. Denote $a\in{\mathbb{F}}_{p}$ ,
$b\in{\mathbb{F}}_{p}^{*}$.  Figure 2. Ramified quadratic extension: Let
$E={\mathbb{Q}}_{p}(\alpha)$ where $\alpha^{2}\in\\{p,pS_{p}\\}$ and
$\alpha\not\in{\mathbb{Q}}_{p}$. In the tree for
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$ every vertex has $p+1$ neighbours. In
the tree for $\operatorname{SL}(2,E)$ every vertex also has $p+1$ neighbours,
obtained by adding $p+1$ blue edges from a vertex in the middle of each red
edge. Denote $a\in{\mathbb{F}}_{p}^{*}$, $b\in{\mathbb{F}}_{p}$.

In the next few paragraphs we summarize results from [HW_class] for
$k$-involutions of $\operatorname{SL}(2,k)$ when $k$ is a field of
characteristic not $2$. Let $\overline{k}$ be the algebraic closure of $k$.

Recall, a mapping
$\phi:\operatorname{SL}(2,\overline{k})\to\operatorname{SL}(2,\overline{k})$
is a _$k$ -automorphism_ (or equivalently, _an automorphism defined over $k$_)
if $\phi$ is a bijective rational k-homomorphism whose inverse is also a
rational $k$-homomorphism, [Hel_k_invol, Sec. 2.2]. An abstract automorphism
$\theta$ of $\operatorname{SL}(2,\overline{k})$ of order two is an _abstract
involution_ of $\operatorname{SL}(2,\overline{k})$. A _$k$ -involution_
$\theta$ of $\operatorname{SL}(2,\overline{k})$ is an involution defined over
$k$ of $\operatorname{SL}(2,\overline{k})$, and the restriction of $\theta$ to
$\operatorname{SL}(2,k)$ is a _$k$ -involution_ of $\operatorname{SL}(2,k)$.
An _abstract involution_ of $\operatorname{SL}(2,k)$ is an abstract
automorphism of $\operatorname{SL}(2,k)$ of order two. Given
$g\in\operatorname{SL}(2,\overline{k})$ denote by $\iota_{g}$ the inner
automorphism of $\operatorname{SL}(2,\overline{k})$ defined by
$x\mapsto\iota_{g}(x):=gxg^{-1}$.

The classification of the isomorphism classes of $k$-involutions of a
connected reductive algebraic group defined over $k$ is given in
[Hel_k_invol]. A simple characterization of the isomorphism classes of
$k$-involutions of $\operatorname{SL}(n,k)$ is given in [HelWD]. We record the
classification of $k$-involutions of $\operatorname{SL}(2,k)$:

######  Theorem 3.1.

[[HW_class] Theorem 1, Corollary 1, Corollary 2]. Every $k$-isomorphism class
of $k$-involution of $\operatorname{SL}(2,k)$ is of the form $\iota_{A}$ with
$A=\left(\begin{smallmatrix}0&1\\\
a&0\end{smallmatrix}\right)\in\operatorname{GL}(2,k)$. Two such
$k$-involutions $\iota_{A}$ with $A\in\\{\left(\begin{smallmatrix}0&1\\\
a&0\end{smallmatrix}\right),\left(\begin{smallmatrix}0&1\\\
b&0\end{smallmatrix}\right)\\}\subset\operatorname{GL}(2,k)$ of
$\operatorname{SL}(2,k)$ are conjugate if and only if $a$ and $b$ are in the
same square class of $k^{*}$. In particular, there are
$order(k^{*}/(k^{*})^{2})$ $k$-isomorphism classes of $k$-involutions of
$\operatorname{SL}(2,k)$.

######  Definition 3.2.

Given an involution $\theta$ of a group $G$ the _fixed point group_ of
$\theta$ is $H_{\theta}:=\\{x\in G\;|\;\theta(x)=x\\}$.

For $\theta$ a $k$-involution of $\operatorname{SL}(2,k)$ the quotient
$\operatorname{SL}(2,k)/H_{\theta}$ is called a _$k$ -symmetric variety_, and
much of the structure of $\operatorname{SL}(2,k)/H_{\theta}$ is determined by
$H_{\theta}$.

######  Proposition 3.3 ([HW_class] Section 3).

Let $\theta=\iota_{A}$, with $A=\left(\begin{smallmatrix}0&1\\\
a&0\end{smallmatrix}\right)\in\operatorname{GL}(2,k)$, be a $k$-involution of
$\operatorname{SL}(2,k)$. Then $H_{\theta}=\\{\left(\begin{smallmatrix}x&y\\\
ay&x\end{smallmatrix}\right)\in\operatorname{SL}(2,k)\;|\;x^{2}-ay^{2}=1\\}$.

######  Theorem 3.4 ([HW_class] Section 3.2).

Let $k={\mathbb{Q}}_{p}$, $\theta=\iota_{A}$ with
$A=\left(\begin{smallmatrix}0&1\\\ a&0\end{smallmatrix}\right)$ and
$\overline{a}\in{\mathbb{Q}}_{p}^{*}/({\mathbb{Q}}_{p}^{*})^{2}$. Then
$H_{\theta}$ is anisotropic if and only if $\bar{a}\neq\bar{1}$. If
$\bar{a}=\bar{1}$, then $H_{\theta}$ is isotropic and conjugate to the maximal
${\mathbb{Q}}_{p}$-split torus of $\operatorname{SL}(2,{\mathbb{Q}}_{p})$,
i.e. the diagonal subgroup of $\operatorname{SL}(2,{\mathbb{Q}}_{p})$.

######  Remark 3.5.

In the case of $2\times 2$ matrices the operation inverse composed with
transpose is given by an inner automorphism:

$\left(\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)^{T}\right)^{-1}=\left(\begin{matrix}a&c\\\
b&d\end{matrix}\right)^{-1}=\left(\begin{matrix}d&-c\\\
-b&a\end{matrix}\right)=\left(\begin{matrix}0&1\\\
-1&0\end{matrix}\right)\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\left(\begin{matrix}0&-1\\\ 1&0\end{matrix}\right).$

This is not true for matrices of higher rank.

Finally, we link the ${\mathbb{Q}}_{p}$-involutions of
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$ given by Theorem 3.1 with quadratic
forms over ${\mathbb{Q}}_{p}$.

######  Remark 3.6.

By [Serre, Corollary of Section 2.3], for $p\neq 2$, there are exactly 7
classes of quadratic forms of rank $2$ over ${\mathbb{Q}}_{p}$. We apply
Remark 3.5. As $\left(\begin{smallmatrix}1&0\\\
0&-a\end{smallmatrix}\right)\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&1\\\
a&0\end{smallmatrix}\right)=:A\in\operatorname{GL}(2,{\mathbb{Q}}_{p})$, an
involution $\theta=\iota_{A}$ of $\operatorname{SL}(2,{\mathbb{Q}}_{p})$ is
determined by the quadratic form of a symmetric matrix from $A$

$\iota_{A}(g)=AgA^{-1}=\left(\begin{matrix}1&0\\\
0&-a\end{matrix}\right)(g^{-1})^{T}\left(\begin{matrix}1&0\\\
0&-a\end{matrix}\right)^{-1}$

for every $g\in\operatorname{SL}(2,{\mathbb{Q}}_{p})$.

For $\operatorname{SL}(n,{\mathbb{Q}}_{p})$, with $n\geq 3$, the number of
${\mathbb{Q}}_{p}$-involutions is strictly larger than the number of the
quadratic forms of rank $n$ over ${\mathbb{Q}}_{p}$, the latter is given by
_outer_ ${\mathbb{Q}}_{p}$-involutions (see [HelWD, Section 4.1.7, Section
6]).

##  4\. Automorphisms and abstract automorphisms of
$\operatorname{SL}(2,{\mathbb{Q}}_{p}(\alpha))$

Let $k$ be a local field of characteristic not two and denote by
$\overline{k}$ the algebraic closure of $k$. The group of $k$-automorphisms of
$\operatorname{SL}(2,\overline{k})$ is denoted by
$\mathrm{Aut}_{k}(\operatorname{SL}(2,\overline{k}))$. If $k=\overline{k}$
then we just write $\mathrm{Aut}(\operatorname{SL}(2,\overline{k}))$, see
[Hel_k_invol, Sec. 2.2].

Denote the group of inner automorphisms of $\operatorname{SL}(2,\overline{k})$
by $\operatorname{Inn}(\operatorname{SL}(2,\overline{k}))$ and the group of
inner $k$-automorphisms of $\operatorname{SL}(2,\overline{k})$ by
$\operatorname{Inn}_{k}(\operatorname{SL}(2,\overline{k}))$. Then
$\operatorname{Inn}_{k}(\operatorname{SL}(2,\overline{k}))=\operatorname{Inn}(\operatorname{SL}(2,\overline{k}))\cap\mathrm{Aut}_{k}(\operatorname{SL}(2,\overline{k}))$,
see [Hel_k_invol, Sec. 2.2].

By Borel [Bo] we have
$\mathrm{Aut}(\operatorname{SL}(2,\overline{k}))=\operatorname{Inn}(\operatorname{SL}(2,\overline{k}))$.

We denote by $\mathrm{Aut}_{abstrc}(\operatorname{SL}(2,\overline{k}))$ the
group of all abstract bi-continuous automorphisms of
$\operatorname{SL}(2,\overline{k})$, and by $\mathrm{Aut}(k)$ the group of all
bi-continuous field automorphisms of $k$.

Abstract bi-continuous automorphisms of the $k$-rational points $G(k)$ of an
absolutely almost simple algebraic group $G$ defined over an infinite field
$k$ were described by Borel–Tits [BoTi], and also by Steinberg [Stein]. For
example, the group $\operatorname{SL}(n,k)$ is an absolutely almost simple,
simply connected group, and split over $k$.

By those result, for $G$ as mentioned above, we have that
$\mathrm{Aut}_{abstrc}(G(k))$ fits in the exact sequence

$1\to\mathrm{Aut}_{k}(G)\to\mathrm{Aut}_{abstrc}(G(k))\to\mathrm{Aut}(k).$

Let $\mathrm{Aut}_{G}(k)$ be the image of $\mathrm{Aut}_{abstrc}(G(k))$ in
$\mathrm{Aut}(k)$. When $G$ is a $k$-split connected reductive group, then
$\mathrm{Aut}(k)=\mathrm{Aut}_{G}(k)$. And when $G$ is $k$-split,
$\mathrm{Aut}_{abstrc}(G(k))$ splits as the semi-direct product
$\mathrm{Aut}_{k}(G)\rtimes\mathrm{Aut}_{G}(k)$, (see [BCL, Section 9.1],
[Stu, Introduction]).

Thus, for the particular case of $\operatorname{SL}(2,k)$ we have that
$\mathrm{Aut}_{abstrc}(\operatorname{SL}(2,k))=\mathrm{Aut}_{k}(\operatorname{SL}(2,\overline{k}))\rtimes\mathrm{Aut}(k)$.
In [Hel_k_invol, HW_class] only $k$-involutions of $\operatorname{SL}(2,k)$
are studied, i.e. involutions in
$\mathrm{Aut}_{k}(\operatorname{SL}(2,\overline{k}))$.

In order to obtain all abstract involutions of $\operatorname{SL}(2,k)$, thus
involutions in $\mathrm{Aut}_{abstrc}(\operatorname{SL}(2,k))$, it remains to
compute $\mathrm{Aut}(k)$ and then to combine with the $k$-automorphisms.

For $k={\mathbb{Q}}_{p}$, by [Con] we have
$\mathrm{Aut}({\mathbb{Q}}_{p})=\\{\operatorname{Id}\\}$, and so for
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$ we have only
${\mathbb{Q}}_{p}$-involutions and those are computed by the results in
[HW_class] recalled in Section 3 above.

For $k=E={\mathbb{Q}}_{p}(\alpha)$ a quadratic extension of
${\mathbb{Q}}_{p}$, to compute all the abstract involutions of
$\operatorname{SL}(2,E)$, we compute $\mathrm{Aut}(E)$ and then combine that
with $\mathrm{Aut}_{E}(\operatorname{SL}(2,\overline{{\mathbb{Q}}_{p}}))$ in
order to obtain all abstract involutions of $\operatorname{SL}(2,E)$.

######  Definition 4.1.

Let $k$ be a local field and $K$ a finite field-extension of $k$. Let
$\theta,\Psi\in\mathrm{Aut}_{abstrc}(\operatorname{SL}(2,k))$. We say that
$\theta$ and $\Psi$ are _$\operatorname{GL}(2,K)$ -conjugate_ if there is
$X\in\operatorname{Inn}(\operatorname{GL}(2,K))$, such that $X^{-1}\theta
X=\Psi$. In particular, there is a matrix $A\in\operatorname{GL}(2,K)$, such
that $X=\iota_{A}$, and this means $A^{-1}(\theta(AgA^{-1}))A=\Psi(g)$, for
every $g\in\operatorname{SL}(2,k)$.

######  Lemma 4.2.

Let $E$ be a quadratic extension of ${\mathbb{Q}}_{p}$, so
$E={\mathbb{Q}}_{p}(\alpha)$, where $\alpha\in
E\setminus\\{{\mathbb{Q}}_{p}\\}$ and $\alpha^{2}\in{\mathbb{Q}}_{p}$. Then
$\mathrm{Aut}(E)=\\{\operatorname{Id},\sigma\\}$ where $\sigma(a+\alpha
b)=a-\alpha b$.

###### Proof.

From [Schm] we know that a field which is complete with respect to two
inequivalent nontrivial norms (i.e., the two norms induce distinct nondiscrete
topologies) must be algebraically closed. A corollary is that a field which is
complete with respect to a nontrivial norm and which is not algebraically
closed has only one equivalence class of norm, $n$. So an automorphism
$\sigma$ of $E$ induces a norm $n_{\sigma}(x):=n(\sigma(x))$. But then
$n_{\sigma}$ must be equivalent to the unique norm $n$, so $n_{\sigma}$ is
some scalar multiple of $n$. Thus every automorphism of $E$ is continuous with
respect to the norm topology. Since ${\mathbb{Q}}$ is dense in
${\mathbb{Q}}_{p}$ and because any automorphism of $E$ is continuous and the
identity on ${\mathbb{Q}}$, one can deduce that any automorohism on $E$ is
trivial on ${\mathbb{Q}}_{p}$, thus Galois. Now as
$\alpha^{2}\in{\mathbb{Q}}_{p}$, we have that any automorphism of $E$ will
send $\alpha$ to $\pm\alpha$. Therefore
$\mathrm{Aut}(E)=\\{\operatorname{Id},\sigma\\}$, where
$\sigma:{\mathbb{Q}}_{p}(\alpha)\to{\mathbb{Q}}_{p}(\alpha)$, with
$x\in{\mathbb{Q}}_{p}\mapsto\sigma(x)=x$, and
$\alpha\mapsto\sigma(\alpha)=-\alpha$. ∎

######  Remark 4.3.

Let $k$ be a local field. By the results of Borel–Tits [BoTi] that are
recalled in [BCL, Theorem 9.1 v)] we know that
$\mathrm{Aut}_{abstrc}(\operatorname{SL}(2,k))$ acts continuously, properly
and faithfully on the Bruhat–Tits tree of $\operatorname{SL}(2,k)$. In the
particular case when $k=E$ is a quadratic extension of ${\mathbb{Q}}_{p}$, the
involution $\sigma$ is an automorphism of the Bruhat–Tits tree of
$\operatorname{SL}(2,E)$, as well as any abstract involution $\theta$ of
$\operatorname{SL}(2,E)$.

Let us now compute the abstract involutions of $\operatorname{SL}(2,E)$.

######  Theorem 4.4.

Let $E={\mathbb{Q}}_{p}(\alpha)$ be a quadratic extension of
${\mathbb{Q}}_{p}$. Then any abstract involution $\theta$ of
$\operatorname{SL}(2,E)$ is of the form $\theta=\iota_{A}\circ\gamma$ where:

  1. (1)

$\gamma=\sigma$ and $A\in\\{\left(\begin{smallmatrix}z&y\\\
1&-\sigma(z)\end{smallmatrix}\right),\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)\\}\subset\operatorname{GL}(2,E)$, with
$y\in{\mathbb{Q}}_{p},z,x\in E$, with $z\sigma(z)+y\neq 0$ and $x\sigma(x)=1,$

  2. (2)

$\gamma=\operatorname{Id}$ and $A=\left(\begin{smallmatrix}0&1\\\
a&0\end{smallmatrix}\right)$, with $a\in E^{*}/(E^{*})^{2}$, up to conjugacy.

###### Proof.

By Borel–Tits [BoTi] any abstract automorphism of $\operatorname{SL}(2,E)$ is
written as $\beta\circ\gamma$, with
$\beta\in\mathrm{Aut}_{E}(\operatorname{SL}(2,\overline{{\mathbb{Q}}_{p}}))$
and $\gamma\in\mathrm{Aut}(E)=\\{\operatorname{Id},\sigma\\}$. By [HW_class,
Remark 2] every $E$-automorphism of $\operatorname{SL}(2,E)$ can be written as
the restriction to $\operatorname{SL}(2,E)$ of a inner automorphism of
$\operatorname{GL}(2,E)$. Thus, given $\theta$ an abstract automorphism of
$\operatorname{SL}(2,E)$ there exists $A=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{GL}(2,E)$ such that
$\theta=\iota_{A}\circ\gamma$. From now on assume that $\theta$ is an abstract
involution, thus $\theta^{2}=\operatorname{Id}$. Then, by [HW_class, Lemma 2],
$A\gamma(A)=\left(\begin{smallmatrix}a\gamma(a)+b\gamma(c)&a\gamma(b)+b\gamma(d)\\\
c\gamma(a)+d\gamma(c)&c\gamma(b)+d\gamma(d)\end{smallmatrix}\right)=q\operatorname{Id}$,
for some $q\in E^{*}$, and with $\gamma\in\\{\operatorname{Id},\sigma\\}$.
Thus we have

(1) $a\gamma(a)+b\gamma(c)=c\gamma(b)+d\gamma(d)\qquad\text{and}\qquad
c\gamma(a)+d\gamma(c)=0=a\gamma(b)+b\gamma(d).$

When $\gamma=\operatorname{Id}$ we can directly apply the results of
[HW_class, Section 1.3] and get that up to $\operatorname{SL}(2,E)$-conjugacy,
$\theta=\iota_{A}$, with $A=\left(\begin{smallmatrix}0&1\\\
t&0\end{smallmatrix}\right)$, with $t\in E^{*}/(E^{*})^{2}$.

We consider now the case $\gamma=\sigma$. By applying $\sigma$ to the second
equality of (1) and putting together all four equations, we get

$a(\sigma(b)-\sigma(c))=\sigma(d)(c-b)\qquad\text{and}\qquad\sigma(a)(b-c)=d(\sigma(c)-\sigma(b)).$

Case 1: $a\neq 0$ then $d\neq 0$ and
$\frac{a}{d}=\frac{\sigma(d)}{\sigma(a)}=x\neq 0$ , so $a=xd$ and
$\sigma(d)=x\sigma(a)$. By applying $\sigma$, we have also
$\sigma(a)=\sigma(x)\sigma(d)$. Then $\sigma(a)=\sigma(x)x\sigma(a)$ thus
getting $\sigma(x)x=1$. Replacing $a=xd$ in the first equality of (1) we get
$b\sigma(c)=c\sigma(b)$.

If $c=b=0$ then $A=\left(\begin{smallmatrix}xd&0\\\
0&d\end{smallmatrix}\right)$, and in fact one can just take
$A=\left(\begin{smallmatrix}x&0\\\ 0&1\end{smallmatrix}\right)$. Then
$A\sigma(A)=\left(\begin{smallmatrix}x\sigma(x)&0\\\
0&1\end{smallmatrix}\right)=q\operatorname{Id}$, implying $x\sigma(x)=1$.

As $b\sigma(c)=c\sigma(b)$ is symmetric in $b$ and $c$, we consider next that
$c\neq 0$, then $b=cy$, with $y\in{\mathbb{Q}}_{p}$. Thus
$A=\left(\begin{smallmatrix}xd&cy\\\ c&d\end{smallmatrix}\right)$ and so

$A\sigma(A)=\begin{pmatrix}d\sigma(d)+yc\sigma(c)&xdy\sigma(c)+cy\sigma(d)\\\
c\sigma(xd)+d\sigma(c)&cy\sigma(c)+d\sigma(d)\end{pmatrix}=\begin{pmatrix}q&0\\\
0&q\end{pmatrix}$

from which it follows

$xd\sigma(c)=-c\sigma(d)\qquad d\sigma(d)+yc\sigma(c)=q,\text{ implying
}q\in{\mathbb{Q}}_{p}.$

We have three equations

$xd\sigma(c)=-c\sigma(d)\qquad\sigma(x)\sigma(d)c=-\sigma(c)d\qquad
d\sigma(d)=q-yc\sigma(c).$

As by our assumption $c\neq 0$ and $d\neq 0$ then
$\frac{xd}{c}=\frac{-\sigma(d)}{\sigma(c)}=z\neq 0$. Rewriting our equations
to include $z$ we have

$xd=zc\qquad\sigma(x)\sigma(d)=\sigma(z)\sigma(c)\qquad-\sigma(d)=\sigma(c)z\qquad-d=c\sigma(z).$

Thus $-\sigma(x)z=\sigma(z)$ and we can write
$\sigma(x)=\frac{-\sigma(z)}{z}$. Returning to $A$ and substituting from above
$d=\frac{zc}{x}$, we have

$A=\begin{pmatrix}xd&cy\\\ c&d\end{pmatrix}=\begin{pmatrix}x\frac{zc}{x}&cy\\\
c&\frac{zc}{x}\end{pmatrix}=c\begin{pmatrix}z&y\\\ 1&-\sigma(z)\end{pmatrix}.$

And returning to our computation for $A\sigma(A)$ we see

$A\sigma(A)=\begin{pmatrix}z\sigma(z)+y&0\\\
0&y+\sigma(z)z\end{pmatrix}=\begin{pmatrix}q&0\\\ 0&q\end{pmatrix},\qquad
q,y\in{\mathbb{Q}}_{p}.$

If $b=0$, then $A=\left(\begin{smallmatrix}xd&0\\\
c&d\end{smallmatrix}\right)$, and we can reduce $A$ to
$A=\left(\begin{smallmatrix}x&0\\\ c&1\end{smallmatrix}\right)$, for some
$c\in E^{*}$. Taking

$A\sigma(A)=\begin{pmatrix}x\sigma(x)&0\\\
c\sigma(x)+\sigma(c)&1\end{pmatrix}=\begin{pmatrix}q&0\\\ 0&q\end{pmatrix}$

we obtain the condition $c\sigma(x)+\sigma(c)=0$. Since $x\sigma(x)=1$ this
completes the proof if $a\neq 0$.

Case 2: $a=0$ then $d=0$, and $A=\left(\begin{smallmatrix}0&b\\\
c&0\end{smallmatrix}\right)$, and from (1) $b\sigma(c)=c\sigma(b)$. So
$\frac{b}{c}=\frac{\sigma(b)}{\sigma(c)}=y\neq 0$, and thus
$y\in{\mathbb{Q}}_{p}^{*}$. Then $A=\left(\begin{smallmatrix}0&cy\\\
c&0\end{smallmatrix}\right)$, reducing to $A=\left(\begin{smallmatrix}0&y\\\
1&0\end{smallmatrix}\right)$ with $y\in{\mathbb{Q}}_{p}^{*}$. ∎

Now we compute the fixed point groups of the involutions from Theorem 4.4
using the ends of the Bruhat–Tits tree of $\operatorname{SL}(2,K)$, where $K$
is finite extension of $E$, and thus of ${\mathbb{Q}}_{p}$, that is chosen
suitably. We will show those fixed point groups are either trivial, or
compact, or $\operatorname{GL}(2,E)$-conjugate to either the diagonal subgroup
in $\operatorname{SL}(2,E)$, or to $\operatorname{SL}(2,{\mathbb{Q}}_{p})$.
Recall $E={\mathbb{Q}}_{p}(\alpha)$ is a quadratic extension of
${\mathbb{Q}}_{p}$ and $\mathcal{O}_{E}$ denotes the ring of integers of $E$.

We start with some easy lemmas.

######  Lemma 4.5.

Let $X\in\operatorname{SL}(2,E)$, resp., $X\in\operatorname{GL}(2,E)$. Let
$\theta=\iota_{A}\circ\sigma$ be an abstract involution of
$\operatorname{SL}(2,E)$, with $A\in\operatorname{GL}(2,E)$ as in Theorem 4.4.
Then $\iota_{X^{-1}}\circ\theta\circ\iota_{X}=\sigma$ if and only if
$A\sigma(X)=qX$, for some $q\in E^{*}$.

###### Proof.

By writing the equality $\iota_{X^{-1}}\theta\iota_{X}=\sigma$, we have

$\begin{split}\sigma(g)&=\iota_{X^{-1}}\theta\iota_{X}(g)=X^{-1}(A(\sigma(XgX^{-1}))A^{-1})X=X^{-1}A\sigma(X)\sigma(g)\sigma(X^{-1})A^{-1}X\\\
&=\iota_{X^{-1}A\sigma(X)}(\sigma(g)),\text{ for every
}g\in\operatorname{SL}(2,E).\end{split}$

Then by [HW_class, Lemma 2], we have $X^{-1}A\sigma(X)=q\operatorname{Id}$ for
some $q\in E^{*}$. The converse implication is trivial. ∎

######  Definition 4.6.

Let $T_{d}$ be a $d$-regular tree with $d\geq 3$. Denote by $\partial T_{d}$
the visual boundary or ends of $T_{d}$. An automorphism
$\theta_{1}\in\mathrm{Aut}(T_{d})$ is a _tree-involution_ if
$\theta_{1}^{2}=\operatorname{Id}\in\mathrm{Aut}(T_{d})$, in particular
$\theta_{1}$ is an elliptic automorphism of $T_{d}$.

We denote by $Fix_{T_{d}}(\theta_{1}):=\\{v\in T_{d}\;|\;\theta_{1}(v)=v\\}$,
and by $Fix_{\partial T_{d}}(\theta_{1}):=\\{\xi\in\partial
T_{d}\;|\;\theta_{1}(\xi)=\xi\\}$. Notice, $Fix_{T_{d}}(\theta_{1})$ is a
(finite or infinite) connected subtree of $T_{d}$, and when the latter is
infinite, we have $Fix_{\partial T_{d}}(\theta_{1})\subseteq\partial
Fix_{T_{d}}(\theta_{1})$.

Also, consider the following two subgroups of $\mathrm{Aut}(T_{d})$:

$\mathrm{Stab}_{\mathrm{Aut}(T_{d})}(Fix_{T_{d}}(\theta_{1})):=\\{g\in\mathrm{Aut}(T_{d})\;|\;g(Fix_{T_{d}}(\theta_{1}))=Fix_{T_{d}}(\theta_{1})\text{
setwise}\\}\text{ and }$ $\mathrm{Stab}_{\mathrm{Aut}(T_{d})}(Fix_{\partial
T_{d}}(\theta_{1})):=\\{g\in\mathrm{Aut}(T_{d})\;|\;g(Fix_{\partial
T_{d}}(\theta_{1}))=Fix_{\partial T_{d}}(\theta_{1})\text{ setwise}\\}.$

If moreover $Fix_{\partial T_{d}}(\theta_{1})$ is not the empty set we have

$\mathrm{Stab}_{\mathrm{Aut}(T_{d})}(Fix_{T_{d}}(\theta_{1}))\leq\mathrm{Stab}_{\mathrm{Aut}(T_{d})}(Fix_{\partial
T_{d}}(\theta_{1}))\text{ but they might not be equal.}$

######  Lemma 4.7.

Let $K$ be a finite field-extension of ${\mathbb{Q}}_{p}$. Let $\theta$ be an
abstract involution of $\operatorname{SL}(2,K)$, and let
$H_{\theta}:=\\{g\in\operatorname{SL}(2,K)\;|\;\theta(g)=g\\}$. Let
$\theta_{1}$ be the automorphism of the Bruhat–Tits tree $T_{K}$ of
$\operatorname{SL}(2,K)$ induced by $\theta$. Then $\theta_{1}$ is a tree-
involution of $T_{K}$ and

$H_{\theta}\leq\mathrm{Stab}_{\mathrm{Aut}(T_{K})}(Fix_{T_{K}}(\theta_{1}))\leq\mathrm{Stab}_{\mathrm{Aut}(T_{K})}(Fix_{\partial
T_{K}}(\theta_{1})).$

###### Proof.

Since $\theta^{2}=\operatorname{Id}$, and by Remark 4.3 we have that
$\theta_{1}^{2}=\operatorname{Id}$, thus $\theta_{1}$ is an involution in
$\mathrm{Aut}(T_{K})$ as claimed, and in particular an elliptic automorphism
of $T_{K}$.

Take now $g\in H_{\theta}$ and $x\in Fix_{T_{K}}(\theta_{1})$. Then
$\theta_{1}(g(x))=\theta(g)(\theta_{1}(x))=g(x)$, and thus $g(x)\in
Fix_{T_{K}}(\theta_{1})$, implying that
$g\in\mathrm{Stab}_{\mathrm{Aut}(T_{K})}(Fix_{T_{K}}(\theta_{1}))$ and the
conclusion follows. ∎

We first want to understand the case of the involution $\theta=\iota_{A}$ of
$\operatorname{SL}(2,E)$ from Theorem 4.4 given by the matrix
$A=\left(\begin{smallmatrix}0&1\\\ a&0\end{smallmatrix}\right)$, with $a\in
E^{*}/(E^{*})^{2}$. The fixed point group
$H_{\theta}\leq\operatorname{SL}(2,E)$ of $\theta$ is computed in [HW_class].
We give a geometric interpretation of $H_{\theta}$ using the ends of the
Bruhat–Tits tree of $\operatorname{SL}(2,K_{a})$, where $K_{a}:=E(\sqrt{a})$
is a quadratic extension of $E$. The ends of the tree $T_{K_{a}}$ are the
elements of the projective space $P^{1}K_{a}=\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}\bigsqcup\limits_{x\in
K_{a}}\big{[}\begin{smallmatrix}1\\\ x\end{smallmatrix}\big{]}$. Notice, the
action of $\theta=\iota_{A}$ on the Bruhat–Tits tree (and its boundary) of
$\operatorname{SL}(2,E)$, and respectively of $\operatorname{SL}(2,K_{a})$, is
the natural map induced by $A\in\operatorname{GL}(2,E)$ on the trees and their
respective boundaries.

Now we will use the ends of the Bruhat–Tits trees to understand the fixed
point groups of the abstract involutions.

######  Corollary 4.8.

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$,
$A=\left(\begin{smallmatrix}0&1\\\ a&0\end{smallmatrix}\right)$, with $a\in
F^{*}/(F^{*})^{2}$, and $\theta_{a}:=\iota_{A}$ the corresponding
$F$-involution of $\operatorname{SL}(2,F)$. Take $K_{a}:=F(\sqrt{a})$ a
quadratic field extension. Then the only solutions of the equation
$A(\xi)=\xi$ with $\xi\in P^{1}K_{a}$ are
$\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}$ and
$H_{\theta_{a}}=\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})\cap\operatorname{SL}(2,F)=\\{\left(\begin{smallmatrix}x&y\\\
ay&x\end{smallmatrix}\right)\in\operatorname{SL}(2,F)\;|\;x^{2}-ay^{2}=1\\}$.
Moreover,

  1. (1)

if $a=1$ then $\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\ \pm
1\end{smallmatrix}\big{]}$ and $H_{\theta_{a}}$ contains all the hyperbolic
elements of $\operatorname{SL}(2,F)$ with $\xi_{\pm}$ as their repelling and
attracting endpoints. In particular, $H_{\theta_{a}}$ is
$\operatorname{GL}(2,F)$-conjugate to the entire diagonal subgroup of
$\operatorname{SL}(2,F)$, thus $H_{\theta_{a}}$ is noncompact and abelian.

  2. (2)

if $a\neq 1$ then $\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}\in P^{1}K_{a}-P^{1}F$, and
$H_{\theta_{a}}$ is compact and abelian.

###### Proof.

We search for all the ends $\xi\in P^{1}K_{a}$ such that $A(\xi)=\xi$. This
means that we search for all vectors $\big{(}\begin{smallmatrix}1\\\
x\end{smallmatrix}\big{)}$ with $x\in K_{a}$, or
$\big{(}\begin{smallmatrix}0\\\ 1\end{smallmatrix}\big{)}$, such that:

$\left(\begin{matrix}0&1\\\ a&0\end{matrix}\right)\begin{pmatrix}1\\\
x\end{pmatrix}=c\begin{pmatrix}1\\\
x\end{pmatrix}\Longleftrightarrow\begin{pmatrix}x\\\
a\end{pmatrix}=\begin{pmatrix}c\\\
cx\end{pmatrix}\qquad\textrm{or}\qquad\left(\begin{matrix}0&1\\\
a&0\end{matrix}\right)\begin{pmatrix}0\\\ 1\end{pmatrix}=c\begin{pmatrix}0\\\
1\end{pmatrix}\Longleftrightarrow\begin{pmatrix}1\\\
0\end{pmatrix}=\begin{pmatrix}0\\\ c\end{pmatrix}$

for some  $c\in K_{a}^{*}$. One can see that the only solutions are given by
$c=x$ and $a=c^{2}$, implying that $\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}\in P^{1}K_{a}=P^{1}F(\sqrt{a})$ are the
only solutions of the equation $A(\xi)=\xi$.

Next, we claim that the involution $\theta_{a}=\iota_{A}$ is
$\operatorname{GL}(2,K_{a})$-conjugate to the involution $\iota_{B}$ of
$\operatorname{SL}(2,K_{a})$, where $B:=\left(\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\right)$. Indeed, taking

$C:=\left(\begin{matrix}1&-\frac{1}{\sqrt{a}}\\\
\sqrt{a}&1\end{matrix}\right)\in\operatorname{GL}(2,K_{a})\qquad\textrm{then}\qquad\iota_{C^{-1}}\circ\theta_{a}\circ\iota_{C}=\iota_{C^{-1}AC}=\iota_{B}.$

It is easy to see the fixed point group in $\operatorname{SL}(2,K_{a})$ of the
involution $\iota_{B}$ is $Diag(2,K_{a})$, the full diagonal subgroup of
$\operatorname{SL}(2,K_{a})$. Thus, from above, the fixed point group in
$\operatorname{SL}(2,K_{a})$ of the involution $\iota_{A}$ is the subgroup

$\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})=CDiag(2,K_{a})C^{-1}\leq\operatorname{SL}(2,K_{a})$

which is the stabilizer in $\operatorname{SL}(2,K_{a})$ of the bi-infinite
geodesic line $[\xi_{-},\xi_{+}]$ in the Bruhat–Tits tree
$T_{K_{a}}\cup\partial T_{K_{a}}$, and fixes pointwise the two ends
$\xi_{-},\xi_{+}$. Then
$H_{\theta_{a}}=\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})\cap\operatorname{SL}(2,F)$.

Notice, the geodesic line $[\xi_{-},\xi_{+}]$ appears in the Bruhat–Tits tree
$T_{F}\cup\partial T_{F}$ if and only if $a=1$. The rest of the Corollary is
an easy verification, and an application of Proposition 3.3. ∎

In the next few pages we will compute the fixed point groups $H_{\theta}$ for
abstract involutions $\theta=\iota_{A}\circ\sigma$.

Consider now $\theta=\iota_{A}\circ\sigma$, with
$A=\left(\begin{smallmatrix}x&0\\\ 0&1\end{smallmatrix}\right)$ and where
$x=x_{1}+\sigma x_{2}\in E={\mathbb{Q}}_{p}(\alpha)$ such that $x\sigma(x)=1$.
There are two cases:

$\begin{array}[]{cc}\textrm{Case 1: }x=\pm 1\Rightarrow x_{1}=\pm
1,x_{2}=0&\textrm{Case 2: }x\neq\pm 1\Rightarrow x_{2}\neq 0\\\
B=\left(\begin{matrix}\alpha&0\\\
0&1\end{matrix}\right)&B=\left(\begin{matrix}\frac{x_{1}+1}{x_{2}}+\alpha&0\\\
0&1\end{matrix}\right)\end{array}.$

Then in both cases we see that
$\iota_{B^{-1}}\circ\theta\circ\iota_{B}=\iota_{B^{-1}A\sigma(B)}\circ\sigma=\sigma$,
implying that $\theta=\iota_{A}\circ\sigma$ is
$\operatorname{GL}(2,E)$-conjugate to $\sigma$ via the map $\iota_{B^{-1}}$.
Thus in this case
$H_{\theta}:=\\{g\in\operatorname{SL}(2,E)\;|\;\theta(g)=g\\}$ is
$\operatorname{GL}(2,E)$-conjugate to $\operatorname{SL}(2,{\mathbb{Q}}_{p})$.

Finally, consider the general case where $\theta=\iota_{A}\circ\sigma$, with
$A=\left(\begin{smallmatrix}z&y\\\ 1&-\sigma(z)\end{smallmatrix}\right)$ and
where $y\in{\mathbb{Q}}_{p},z\in E$ such that $z\sigma(z)+y\neq 0$. Recall
that

$A\sigma(A)=c\operatorname{Id},\text{ for some }c\in E^{*}.$

Notice, the action of $\sigma$ on the Bruhat–Tits tree of
$\operatorname{SL}(2,E)$ is again the natural map induced by $\sigma$, as the
vertices of $T_{E}$ can be labeled with $a+\alpha b$, where
$a,b\in{\mathbb{Q}}_{p}$ (see figures 1, 2). In other words, $\sigma$ induces
an action on $T_{E}$ which fixes the subtree for $T_{{\mathbb{Q}}_{p}}$ and
acts as an involution on the remaining branches in
$T_{E}-T_{{\mathbb{Q}}_{p}}$. Then the involution $\theta_{1}$ of the
Bruhat–Tits tree of $\operatorname{SL}(2,E)$ induced by the involution
$\theta$ of $\operatorname{SL}(2,E)$ is just the map
$\theta_{1}=A\circ\sigma$. The same involution $\theta_{1}=A\circ\sigma$ acts
on the boundary $\partial T_{E}$ of $T_{E}$. Moreover, since
$A\in\operatorname{GL}(2,E)$, the map $A$ acts on the $\partial T_{E}$ as an
automorphism, and thus is a bijection on $\partial T_{E}$.

In order to classify the involutions $\theta$ up to
$\operatorname{GL}(2,E)$-conjugacy, we first we want to find all the solutions
of the equation $\theta_{1}(\xi)=A(\sigma(\xi))=\xi$, with $\xi\in\partial
T_{E}$. If there is no such solution, we will solve that equation for
$\xi\in\partial T_{K}$, where $K$ is an appropriate finite extension of $E$.
Notice that the actions of $A$ and $\sigma$ extend naturally to
$T_{K}\cup\partial T_{K}$, with $\sigma$ fixing pointwise any primitive
element in $K-E$. The action of $\sigma$ extends to $K$, since all the
extensions are algebraic and we have ${\mathbb{Q}}_{p}\subset E\subset K$.

Putting together the above remarks, it is enough to find all the solutions of
the equation $A(\sigma(A(\xi)))=A(\xi)$, for $\xi\in\partial T_{K}$ and $K$ a
finite extension of $E$, which is equivalent to finding all the $c\in K^{*}$
and $x\in K$ such that

(2) $A\sigma(A)\begin{pmatrix}1\\\ \sigma(x)\end{pmatrix}=\begin{pmatrix}1\\\
\sigma(x)\end{pmatrix}=cA\begin{pmatrix}1\\\
x\end{pmatrix},\;\;\textrm{or}\;\;\begin{pmatrix}0\\\
1\end{pmatrix}=cA\begin{pmatrix}0\\\ 1\end{pmatrix}$

where $K$ is an appropriate finite extension of $E$ that will be determined
below. First notice that any element $x\in K$ can be uniquely written as
$x=x_{1}+\alpha x_{2}$, with $x_{1},x_{2}$ elements in $K$ such that
$\sigma(x_{i})=x_{i}$.

Thus, writing

$\left(\begin{matrix}z&y\\\ 1&-\sigma(z)\end{matrix}\right)=A=A_{1}+\alpha
A_{2}=\left(\begin{matrix}z_{1}&y\\\
1&-z_{1}\end{matrix}\right)+\alpha\left(\begin{matrix}z_{2}&0\\\
0&z_{2}\end{matrix}\right),\textrm{ with }\xi=\begin{bmatrix}x_{1}+\alpha
x_{2}\\\ y_{1}+\alpha y_{2}\end{bmatrix},$

and $A_{1},A_{2}\in M_{2}({\mathbb{Q}}_{p})$, for $x_{1},x_{2},y_{1},y_{2}\in
K$ such that $\sigma(x_{i})=x_{i}$, $\sigma(y_{i})=y_{i}$. We want to solve
the equation for $\xi\in\partial T_{K}$:

(3) $c\begin{pmatrix}x_{1}-\alpha x_{2}\\\ y_{1}-\alpha
y_{2}\end{pmatrix}=(A_{1}+\alpha A_{2})\begin{pmatrix}x_{1}+\alpha x_{2}\\\
y_{1}+\alpha y_{2}\end{pmatrix}=A_{1}\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix}+\alpha^{2}A_{2}\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}+\alpha A_{1}\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}+\alpha A_{2}\begin{pmatrix}x_{1}\\\ y_{1}\end{pmatrix}$

where $c=c_{1}+\alpha c_{2}\in K^{*}$ and $\sigma(c_{i})=c_{i}$. This equation
(3) is equivalent to the following system of equations:

(4) $(A_{1}-c_{1}\operatorname{Id})\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix}=-\alpha^{2}(z_{2}+c_{2})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix},\;\;\;\;(A_{1}+c_{1}\operatorname{Id})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}=-(z_{2}-c_{2})\begin{pmatrix}x_{1}\\\ y_{1}\end{pmatrix}.$

Composing the second equality with the first:

(5) $(A_{1}^{2}-c_{1}^{2}\operatorname{Id})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}=(z_{1}^{2}+y-c_{1}^{2})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}=\alpha^{2}(z_{2}^{2}-c_{2}^{2})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}.$

We want solve this equation (5) in cases, comparing when the coefficients
$(z_{1}^{2}+y-c_{1}^{2})$ and $\alpha^{2}(z_{2}^{2}-c_{2}^{2})$ are equal to
each other, or zero.

Case 1: If $z_{1}^{2}+y-c_{1}^{2}\neq\alpha^{2}(z_{2}^{2}-c_{2}^{2})\neq 0$
then we must have $\big{(}\begin{smallmatrix}x_{2}\\\
y_{2}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$, and so also $\big{(}\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$. Thus there is no solution $\xi\in\partial T_{K}$
for the system (4), for any finite extension $K$.

Case 2.1: If $z_{1}^{2}+y-c_{1}^{2}\neq 0$, $z_{2}^{2}-c_{2}^{2}=0$, and
$z_{2}\neq 0\neq c_{2}$. Then $\big{(}\begin{smallmatrix}x_{2}\\\
y_{2}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$. If moreover, $z_{2}=-c_{2}$, from (4) we get that
$\big{(}\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$. If instead $z_{2}=c_{2}$, from (4) we get that
$(A_{1}-c_{1}\operatorname{Id})\big{(}\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$ and because
$det(A_{1}-c_{1}\operatorname{Id})=-(z_{1}^{2}-c_{1}^{2}+y)\neq 0$ we also get
$\big{(}\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$. Thus in this Case 2 there is no solution
$\xi\in\partial T_{K}$ for the system (4).

Case 2.2: If $z_{1}^{2}+y-c_{1}^{2}\neq 0$, and $z_{2}=0=c_{2}$. This means
that $A_{2}=\left(\begin{smallmatrix}0&0\\\ 0&0\end{smallmatrix}\right)$, and
so $A=A_{1}=\left(\begin{smallmatrix}z_{1}&y\\\
1&-z_{1}\end{smallmatrix}\right)\in\operatorname{GL}(2,{\mathbb{Q}}_{p})$,
with $A^{2}=(z_{1}^{2}+y)\operatorname{Id}$. The system of equations (4)
reduces to

(6) $(A_{1}-c_{1}\operatorname{Id})\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix},\;\;\;\;(A_{1}+c_{1}\operatorname{Id})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}.$

As the matrices $(A_{1}-c_{1}\operatorname{Id})$ and
$(A_{1}+c_{1}\operatorname{Id})$ are invertible, with determinant
$-(z_{1}^{2}-c_{1}^{2}+y)\neq 0$, we get $\big{(}\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}x_{2}\\\
y_{2}\end{smallmatrix}\big{)}$. Again there is no solution $\xi\in\partial
T_{K}$ for the system (4).

Case 3: If $z_{1}^{2}+y-c_{1}^{2}=0$ and $z_{2}^{2}-c_{2}^{2}\neq 0$. Then
from (5) we get $\big{(}\begin{smallmatrix}x_{2}\\\
y_{2}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$ and then from (4)
$\big{(}\begin{smallmatrix}x_{1}\\\
y_{1}\end{smallmatrix}\big{)}=\big{(}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{)}$. Again there is no solution $\xi\in\partial T_{K}$
for the system (4).

It is thus convenient to search for solutions $\xi\in\partial T_{K}$ in the
following two cases:

  1. (1)

$0\neq z_{1}^{2}+y-c_{1}^{2}=\alpha^{2}(z_{2}^{2}-c_{2}^{2})\neq 0$, and where
we take $K$ to be the minimal field extension of ${\mathbb{Q}}_{p}$ that
contains $c_{1},c_{2}$

  2. (2)

$z_{2}^{2}=c_{2}^{2}$, thus for $c_{2}\in{\mathbb{Q}}_{p}$, and
$z_{1}^{2}+y-c_{1}^{2}=0$ for some $c_{1}\in K$, where $K$ is a minimal
extension of $E$ in which $z_{1}^{2}+y$ is a square.

In each of the Cases 4, 5,1, 5,2 , 5.3, one checks
$B_{i}\in\operatorname{GL}(2,K)$ since $\det B_{i}\neq 0$ and also that
$A\sigma(B_{i})=(c_{1}+\alpha c_{2})B_{i}=cB_{i}$.

Case 4: If $0\neq z_{1}^{2}+y-c_{1}^{2}=\alpha^{2}(z_{2}^{2}-c_{2}^{2})\neq
0$. Then take the matrix

$B_{1}:=\left(\begin{matrix}\frac{y}{z_{2}-c_{2}}&\frac{z_{1}+c_{1}}{z_{2}-c_{2}}+\alpha\\\
\frac{-z_{1}+c_{1}}{z_{2}-c_{2}}+\alpha&\frac{1}{z_{2}-c_{2}}\end{matrix}\right).$

Case 5.1: If $z_{1}^{2}+y-c_{1}^{2}=0$, $z_{2}^{2}-c_{2}^{2}=0$, $y\neq 0$,
and $z_{2}=c_{2}\neq 0$. Then take the matrix

$B_{2}:=\left(\begin{matrix}-\frac{\alpha^{2}(z_{2}+c_{2})(z_{1}+c_{1})}{y}+\alpha&\alpha^{2}(z_{2}+c_{2})+(z_{1}+c_{1})+\alpha(z_{1}-c_{1})\\\
-\frac{\alpha(z_{1}+c_{1})}{y}&1+\alpha\end{matrix}\right).$

Case 5.2: If $z_{1}^{2}+y-c_{1}^{2}=0$, $z_{2}^{2}-c_{2}^{2}=0$, $y\neq 0$,
and $z_{2}=-c_{2}\neq 0$. Then take the matrix

$B_{3}:=\left(\begin{matrix}1-\alpha\frac{2z_{2}}{z_{1}+c_{1}}&(z_{1}+c_{1})+\alpha(z_{1}-c_{1}-2z_{2})\\\
\frac{1}{z_{1}+c_{1}}&1+\alpha\end{matrix}\right).$

Case 5.3: If $z_{1}^{2}+y-c_{1}^{2}=0$, $z_{2}=c_{2}=0$, and $y\neq 0$. This
means that $A_{2}=\left(\begin{smallmatrix}0&0\\\
0&0\end{smallmatrix}\right)$, and so
$A=A_{1}=\left(\begin{smallmatrix}z_{1}&y\\\
1&-z_{1}\end{smallmatrix}\right)\in\operatorname{GL}(2,{\mathbb{Q}}_{p})$.
Then take the matrix

$B_{4}:=\left(\begin{matrix}\alpha(z_{1}-c_{1})&1\\\
\alpha&\frac{c_{1}-z_{1}}{y}\end{matrix}\right).$

Case 5.4: If $z_{1}^{2}-c_{1}^{2}=0$, $z_{2}^{2}-c_{2}^{2}=0$, and $y=0$. This
means that $c=c_{1}+\alpha c_{2}\in E$ and $A=\left(\begin{smallmatrix}z&0\\\
1&-\sigma(z)\end{smallmatrix}\right)\in\operatorname{GL}(2,E)$. Since
multiplication by a constant in $E^{*}$ will give the same results, we can
multiply $A$ with $-\frac{1}{\sigma(z)}$ we get a matrix
$A=\left(\begin{smallmatrix}x&0\\\ w&1\end{smallmatrix}\right)$ such that
$x\sigma(x)=1$ and $w=-x\sigma(w)$, with $x,w\in E^{*}$.

By our assumption $z_{1}\neq 0$ or $z_{2}\neq 0$. There are a few final
subcases, here $x=x_{1}+\alpha x_{2}$, and $w=w_{1}+\alpha w_{2}$:

$\begin{array}[]{ccc}z_{1}z_{2}\neq 0&\Rightarrow x_{2}\neq 0,w_{2}\neq
0&B_{5}:=\left(\begin{matrix}\frac{1+x_{1}}{x_{2}}+\alpha&0\\\
\alpha\frac{w_{2}}{x_{2}}&1\end{matrix}\right)\\\ &&\\\
z_{1}z_{2}=0&\textrm{and }x=1,w=\alpha
w_{2}&B_{5}:=\left(\begin{smallmatrix}1&0\\\
\alpha\frac{w_{2}}{2}&1\end{smallmatrix}\right)\\\ &&\\\
z_{1}z_{2}=0&\textrm{and
}x=-1,w=w_{1}&B_{5}:=\left(\begin{smallmatrix}\alpha&\alpha\\\
1-\alpha\frac{w_{1}}{2}&-1-\alpha\frac{w_{1}}{2}\end{smallmatrix}\right).\end{array}$

In all those subcases we see $A\sigma(B_{5})=cB_{5}$.

In all the above cases we verified
$\iota_{B_{i}^{-1}}\circ\theta\circ\iota_{B_{i}}=\iota_{B_{i}^{-1}A\sigma(B_{i})}\circ\sigma=\sigma$,
impying that $\theta=\iota_{A}\circ\sigma$ is
$\operatorname{GL}(2,K)$-conjugated to $\sigma$ via the map
$\iota_{B_{i}^{-1}}$ (both viewed as abstract involutions of
$\operatorname{SL}(2,K)$), where $K$ is $E$ or a finite field-extension of
$E$.

If the matrix $B_{i}$ is in $\operatorname{GL}(2,E)$, then the fixed point
group associated with $\theta$ is the group

$H_{\theta}:=\\{g\in\operatorname{SL}(2,E)\;|\;\theta(g)=g\\}=B_{i}\operatorname{SL}(2,{\mathbb{Q}}_{p})B_{i}^{-1}$

that is $\operatorname{GL}(2,E)$-conjugate to
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$.

But if the matrix $B_{i}$ is in $\operatorname{GL}(2,K)$, where $K\neq E$ is a
finite field-extension of $E$, then the fixed point group associated with
$\theta$ viewed as an involution of $\operatorname{SL}(2,K)$, is the group

$H_{\theta}(K):=\\{g\in\operatorname{SL}(2,K)\;|\;\theta(g)=g\\}=B_{i}\operatorname{SL}(2,K^{\sigma})B_{i}^{-1}$

where $K^{\sigma}:=\\{x\in K\;|\;\sigma(x)=x\\}$ is the maximal subfield of
$K$ which does not contain $\alpha$. Then

$H_{\theta}:=\\{g\in\operatorname{SL}(2,E)\;|\;\theta(g)=g\\}=B_{i}\operatorname{SL}(2,K^{\sigma})B_{i}^{-1}\cap\operatorname{SL}(2,E).$

######  Lemma 4.9.

Consider one of the matrices $B_{i}\in\operatorname{GL}(2,K)$computed above,
where $K\neq E$ is a finite field-extension of $E$. Then $B_{i}(\partial
T_{K^{\sigma}})\cap\partial T_{E}=\\{\emptyset\\}$ and

$H_{\theta}=B_{i}\operatorname{SL}(2,K^{\sigma})B_{i}^{-1}\cap\operatorname{SL}(2,E)$

is compact or trivial.

###### Proof.

Take a matrix $B_{i}$ as above such that $c=c_{1}+\alpha c_{2}\in K-E$, i.e.
as in Cases 4, 5.1, 5.2, 5.3. Then the first observation is that the set of
all ends in $\partial T_{K}$ that are pointwise fixed by the involution
$\theta_{1}=A\circ\sigma$ on $T_{K}\cup\partial T_{K}$ induced from the
involution $\theta=\iota_{A}\circ\sigma$, is the set $B_{i}(\partial
T_{K^{\sigma}})$. In particular, for a representative
$\left(\begin{smallmatrix}x_{1}+\alpha x_{2}\\\ y_{1}+\alpha
y_{2}\end{smallmatrix}\right)$ of such an end
$\xi=\big{[}\begin{smallmatrix}x_{1}+\alpha x_{2}\\\ y_{1}+\alpha
y_{2}\end{smallmatrix}\big{]}\in B_{i}(\partial T_{K^{\sigma}})$, where
$x_{1},x_{2},y_{1},y_{2}\in K$ with $\sigma(x_{i})=x_{i}$,
$\sigma(y_{i})=y_{i}$, we have that $A(\sigma(\xi))=\xi$ and this becomes

$\theta_{1}\left(\begin{matrix}x_{1}+\alpha x_{2}\\\ y_{1}+\alpha
y_{2}\end{matrix}\right)=A\left(\sigma\left(\begin{matrix}x_{1}+\alpha
x_{2}\\\ y_{1}+\alpha
y_{2}\end{matrix}\right)\right)=A\left(\begin{matrix}x_{1}-\alpha x_{2}\\\
y_{1}-\alpha y_{2}\end{matrix}\right)=c\left(\begin{matrix}x_{1}+\alpha
x_{2}\\\ y_{1}+\alpha y_{2}\end{matrix}\right).$

Notice, our equations (3) and (4) above are derived from the equation
$A(\eta)=c\sigma(\eta)$, where $\xi=A(\eta)$.

Suppose that there is $\xi\in\partial T_{E}$ such that $A(\sigma(\xi))=\xi$.
Then this means that for a representative
$\left(\begin{smallmatrix}x_{1}+\alpha x_{2}\\\ y_{1}+\alpha
y_{2}\end{smallmatrix}\right)$ of $\xi$, with
$x_{1},x_{2},y_{1},y_{2}\in{\mathbb{Q}}_{p}$ we must have

(7) $(A_{1}-c_{1}\operatorname{Id})\begin{pmatrix}x_{1}\\\
y_{1}\end{pmatrix}=\alpha^{2}(z_{2}+c_{2})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix},\;\;\;\;(A_{1}+c_{1}\operatorname{Id})\begin{pmatrix}x_{2}\\\
y_{2}\end{pmatrix}=(z_{2}-c_{2})\begin{pmatrix}x_{1}\\\ y_{1}\end{pmatrix}.$

For Cases 5.1, 5.2, 5.3, a solution $\xi\in\partial T_{E}$ of the system (7)
will imply $c_{1}\in{\mathbb{Q}}_{p}$, and since in those cases
$c_{2}\in{\mathbb{Q}}_{p}$, in contradiction with the assumption that $c\in
K-E$.

For the Case 4, taking separately the two equalities of system (7), we will
get that

$c_{1}x_{1}+c_{2}x_{2}\alpha^{2},c_{1}y_{1}+c_{2}y_{2}\alpha^{2},c_{1}x_{2}+c_{2}x_{1},c_{1}y_{2}+c_{2}y_{1}\in{\mathbb{Q}}_{p}.$

By multiplying accordingly, we will get that
$c_{2}(x_{1}^{2}-\alpha^{2}x_{2}^{2})\in{\mathbb{Q}}_{p}$ and
$c_{1}(y_{1}^{2}-\alpha^{2}y_{2}^{2})\in{\mathbb{Q}}_{p}$, thus $c_{1},c_{2}$
must be in ${\mathbb{Q}}_{p}$, giving again a contradiction with our
assumption.

By the first part of the lemma we now know that $B_{i}(\partial
T_{K^{\sigma}})\cap\partial T_{E}=\\{\emptyset\\}$. This implies that the
intersection of the tree $B_{i}(T_{K^{\sigma}})$ with the tree $T_{E}$ is
either empty or a finite subtree of $T_{E}$. In both such cases, we must have
that $H_{\theta}$ stabilizes setwise a finite connected subtree of $T_{K}$,
implying that $H_{\theta}$ is compact or trivial. ∎

We have proved the following.

######  Theorem 4.10.

Let $E={\mathbb{Q}}_{p}(\alpha)$ be a quadratic extension of
${\mathbb{Q}}_{p}$. Consider any abstract involution
$\theta=\iota_{A}\circ\sigma$ of $\operatorname{SL}(2,E)$ as in Theorem 4.4,
where $A\in\\{\left(\begin{smallmatrix}z&y\\\
1&-\sigma(z)\end{smallmatrix}\right),\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)\\}\subset\operatorname{GL}(2,E)$, with
$y\in{\mathbb{Q}}_{p},z,x\in E$, with $z\sigma(z)+y\neq 0$ and $x\sigma(x)=1$.
Then one of the following holds for
$H_{\theta}:=\\{g\in\operatorname{SL}(2,E)\;|\;\theta(g)=g\\}$:

  1. (1)

$H_{\theta}$ is $\operatorname{GL}(2,E)$-conjugate to
$\operatorname{SL}(2,{\mathbb{Q}}_{p})$

  2. (2)

$H_{\theta}=B\operatorname{SL}(2,K^{\sigma})B^{-1}\cap\operatorname{SL}(2,E)$,
for some matrix $B\in\operatorname{GL}(2,K)$, where $K\neq E$ is some finite
field-extension of $E$ and $K^{\sigma}:=\\{x\in K\;|\;\sigma(x)=x\\}$ is the
maximal subfield of $K$ with that property. In particular, $H_{\theta}$ is
compact or trivial.

######  Remark 4.11.

We leave it as an open question to compute the possibilities for compact
$H_{\theta}$ that preserve a finite subtree.

##  5\. Polar decomposition of $\operatorname{SL}(2,E)$ with respect to
various subgroups

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$ and $E=F(\alpha)$ be
a quadratic extension of $F$. Denote by $\omega$ a uniformizer of $F$. It is
well known that $|E^{*}/(E^{*})^{2}|=4=|F^{*}/(F^{*})^{2}|$. Recall,
throughout this article we consider $p\neq 2$.

Inspired by techniques from representation theory and spherical varieties, we
first give an upper bound for the number of orbits of various subgroups of
$\operatorname{SL}(2,K)$ on the boundary $\partial T_{K}$, where $K$ is either
$F$ or $E$. Those results will be used to apply the various polar
decompositions proven in Proposition 5.5 and to compute Chabauty limits of
groups of involutions, and also to provide different, direct, and more
geometric proofs than in [HW_class, HelWD, Hel_k_invol] for the case of
$\operatorname{SL}_{2}$.

######  Lemma 5.1.

There are at most $5$ $\operatorname{SL}(2,F)$-orbits on the boundary
$\partial T_{E}\cong P^{1}E=E\cup\\{\infty\\}$:

  1. (1)

the $\operatorname{SL}(2,F)$-orbit of $\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$

  2. (2)

the $\operatorname{SL}(2,F)$-orbits of $\big{[}\begin{smallmatrix}1\\\
m\alpha\end{smallmatrix}\big{]}$, for each $m\in F^{*}/(F^{*})^{2}$, that
might coincide for different $m$’s.

###### Proof.

Indeed, consider the subgroup $B:=\\{\left(\begin{smallmatrix}d^{-1}&0\\\
c&d\end{smallmatrix}\right)\;|\;c\in F,\;d\in F^{*}\\}$ in
$\operatorname{SL}(2,F)$. The image of $\big{[}\begin{smallmatrix}1\\\
m\alpha\end{smallmatrix}\big{]}$ under $B$, is $\big{[}\begin{smallmatrix}1\\\
cd+d^{2}m\alpha\end{smallmatrix}\big{]}$ these cover the entire $\partial
T_{E}-\partial T_{F}$. ∎

######  Lemma 5.2.

Let $K$ be a finite field-extension of ${\mathbb{Q}}_{p}$. There are exactly
$6$ orbits of the diagonal subgroup
$Diag(K):=\\{\left(\begin{smallmatrix}d^{-1}&0\\\
0&d\end{smallmatrix}\right)\;|\;d\in K^{*}\\}\leq\operatorname{SL}(2,K)$ on
the boundary $\partial T_{K}$:

  1. (1)

the $Diag(K)$-orbit of $\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$ and the $Diag(K)$-orbit of
[$\begin{smallmatrix}0\\\ 1\end{smallmatrix}\big{]}$

  2. (2)

the $Diag(K)$-orbits of $\big{[}\begin{smallmatrix}1\\\
m\end{smallmatrix}\big{]}$, for each $m\in K^{*}/(K^{*})^{2}$.

###### Proof.

The subgroup $Diag(K)$ fixes pointwise the ends
$\big{[}\begin{smallmatrix}1\\\ 0\end{smallmatrix}\big{]}$ and
$\big{[}\begin{smallmatrix}0\\\ 1\end{smallmatrix}\big{]}$. The
$Diag(K)$-orbit of $\big{[}\begin{smallmatrix}1\\\ m\end{smallmatrix}\big{]}$,
for each $m\in K^{*}/(K^{*})^{2}$, are vectors of the form
$\big{[}\begin{smallmatrix}1\\\ d^{2}m\end{smallmatrix}\big{]}$, these cover
the entire boundary $\partial T_{E}-\\{\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}\\}$. ∎

######  Lemma 5.3.

Let $F$ be a finite field extension of ${\mathbb{Q}}_{p}$,
$A=\left(\begin{smallmatrix}0&1\\\ a&0\end{smallmatrix}\right)$, with $a\in
F^{*}/(F^{*})^{2}$, $a\neq 1$ and $\theta_{a}:=\iota_{A}$ the corresponding
$F$-involution of $\operatorname{SL}(2,F)$. Then
$H_{\theta_{a}}:=\\{g\in\operatorname{SL}(2,F)\;|\;\theta_{a}(g)=g\\}$ has at
most $8$ orbits on the boundary $\partial T_{F}$.

###### Proof.

Take $K_{a}:=F(\sqrt{a})$ to be the quadratic field extension of $F$
corresponding to $a$. By Corollary 4.8 we know that $H_{\theta_{a}}$ fixes
pointwise the ends $\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}$ in the boundary $\partial
T_{K_{a}}-\partial T_{F}$, and
$H_{\theta_{a}}=\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})\cap\operatorname{SL}(2,F)$.

By Lemma 5.2 applied to the subgroup
$\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})$, which is
$\operatorname{SL}(2,K_{a})$-conjugate to $Diag(2,K_{a})$, we know that there
are at most $4$ orbits of
$\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})$ on the
boundary $\partial T_{F}\subset\partial
T_{K_{a}}-\\{\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}\\}$. Note the only elements in
$Diag(2,K_{a})$ fixing $\xi\in\partial
T_{K_{a}}-\\{\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}\\}$ are $\pm\operatorname{Id}.$ We claim an element
$h\in\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})$ fixes
an end $\xi\in\partial T_{K_{a}}-\\{\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}\\}$ if and only if
$h=\pm\operatorname{Id}$. Indeed, this follows from the fact that

$\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})=CDiag(2,K_{a})C^{-1}=\bigl{\\{}\frac{1}{2}\left(\begin{matrix}b+\frac{1}{b}&\frac{1}{\sqrt{a}}(b-\frac{1}{b})\\\
\sqrt{a}(b-\frac{1}{b})&b+\frac{1}{b}\end{matrix}\right)\;|\;b\in
K_{a}^{*}\bigr{\\}},$

where $C=\left(\begin{smallmatrix}1&-\frac{1}{\sqrt{a}}\\\
\sqrt{a}&1\end{smallmatrix}\right)$. Suppose now that there is
$h\in\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})-H_{\theta_{a}}$,
and $\xi_{1}\neq\xi_{2}\in\partial T_{F}$ such that $h(\xi_{1})=\xi_{2}$. Then
by applying $\theta_{a}$ to the latter equality we also get
$h(\theta_{a}(\xi_{1}))=\theta_{a}(\xi_{2})$. Because the only ends in
$\partial T_{K_{a}}$ fixed by $\theta_{a}$ are $\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}$ we have that
$\xi_{1}\neq\theta_{a}(\xi_{1})$ and $\xi_{2}\neq\theta_{a}(\xi_{2})$.
Consider the representatives $x_{1},x_{2}$ of $\xi_{1},\xi_{2}$ in $F^{2}$,
respectively. Then there is some $w=w_{1}+\sqrt{a}w_{2}\in K_{a}^{*}$, with
$w_{1},w_{2}\in F$, such that $h(x_{1})=wx_{2}$, and so
$h(\theta_{a}(x_{1}))=w\theta_{a}(x_{2})$. Both of the matrices
$(x_{1},\theta_{a}(x_{1}))$ and $(wx_{2},w\theta_{a}(x_{2}))$ are invertible,
the first with entries in $F$. Then
$h(x_{1},\theta_{a}(x_{1}))=w(x_{2},\theta_{a}(x_{2}))$, and so

$h=w(x_{2},\theta_{a}(x_{2}))(x_{1},\theta_{a}(x_{1}))^{-1}\in\operatorname{SL}(2,K_{a})$

with
$(x_{2},\theta_{a}(x_{2}))(x_{1},\theta_{a}(x_{1}))^{-1}\in\operatorname{GL}(2,F)$.
By taking the determinant in the latter equality, we have
$w^{2}=w_{1}^{2}+aw_{2}^{2}+2w_{1}w_{2}\sqrt{a}\in F^{*}$. This implies that
either $w_{2}=0$ or $w_{1}=0$. If $w_{2}=0$ then $h\in\operatorname{SL}(2,F)$,
which is a contradiction with our assumption that $h\notin H_{\theta_{a}}$. If
$w_{1}=0$, then $w=w_{2}\sqrt{a}$, implying that

$h\in\sqrt{a}\operatorname{GL}(2,F)\cap CDiag(2,K_{a})C^{-1}.$

From the matrix form of $CDiag(2,K_{a})C^{-1}$, we have that
$h=\sqrt{a}\left(\begin{smallmatrix}b&c\\\ ac&b\end{smallmatrix}\right)$, with
$b,c\in F$ such that $b^{2}-ac^{2}=\frac{1}{a}$. If there exists such a
solution $(b,c)$ in $F$ to the latter equation, then any other solution is of
the form $(b+\sqrt{a}c)(x+\sqrt{a}y)$, with $x,y\in F$ such that
$x^{2}-ay^{2}=1$. Thus $h\in\sqrt{a}\left(\begin{smallmatrix}b&c\\\
ac&b\end{smallmatrix}\right)H_{\theta}$, with $b^{2}-ac^{2}=\frac{1}{a}$.

We have proved that the only elements
$h\in\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})$ that
can act nontrivially on ends of $\partial T_{F}$ are from $H_{\theta_{a}}$ or
belong to the coset $\sqrt{a}\left(\begin{smallmatrix}b&c\\\
ac&b\end{smallmatrix}\right)H_{\theta}$, with $b^{2}-ac^{2}=\frac{1}{a}$. This
implies there are at most $8$ $H_{\theta_{a}}$-orbits in $\partial T_{F}$.
This proves the lemma. ∎

Let $E={\mathbb{Q}}_{p}(\alpha)$ be a quadratic extension of
${\mathbb{Q}}_{p}$. Consider any abstract involution
$\theta=\iota_{A}\circ\sigma$ of $\operatorname{SL}(2,E)$ as in Theorem 4.4,
where $A\in\\{\left(\begin{smallmatrix}z&y\\\
1&-\sigma(z)\end{smallmatrix}\right),\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)\\}\subset\operatorname{GL}(2,E)$, with
$y\in{\mathbb{Q}}_{p},z,x\in E$, with $z\sigma(z)+y\neq 0$ and $x\sigma(x)=1$.
Let $H_{\theta}:=\\{g\in\operatorname{SL}(2,E)\;|\;\theta(g)=g\\}$ and suppose
$H_{\theta}=B\operatorname{SL}(2,K^{\sigma})B^{-1}\cap\operatorname{SL}(2,E)$,
for some matrix $B\in\operatorname{GL}(2,K)$, where $K\neq E$ is some finite
field-extension of $E$ and $K^{\sigma}:=\\{x\in K\;|\;\sigma(x)=x\\}$ is the
maximal subfield of $K$ with that property.

Are there infinitely or finitely many $H_{\theta}$-orbits on the boundary
$\partial T_{E}$? We give a partial answer in the following remark.

######  Remark 5.4.

There may be an infinite number of $H_{\theta}$-orbits on the boundary
$\partial T_{E}$. To see this, note: By Lemma 4.9 we know that $B(\partial
T_{K^{\sigma}})\cap\partial T_{E}=\\{\emptyset\\}$. Notice that $K$ is a
quadratic extension of $K^{\sigma}$, i.e. $K=K^{\sigma}(\alpha)$. By Lemma 5.1
there are at most $5$ $\operatorname{SL}(2,K^{\sigma})$-orbits on the boundary
$\partial T_{K}$. By conjugating the groups
$\operatorname{SL}(2,K),\operatorname{SL}(2,K^{\sigma})$ with the matrix
$B\in\operatorname{GL}(2,K)$, one easily deduces there are at most $5$
$B\operatorname{SL}(2,K^{\sigma})B^{-1}$-orbits on the boundary $\partial
T_{K}$.

If we use the same computational trick as in Lemma 5.3 one can see that the
number of $H_{\theta}$-orbits on the boundary $\partial T_{E}$ might be
infinite. Indeed, suppose there is $h\in
B\operatorname{SL}(2,K^{\sigma})B^{-1}-H_{\theta}$ and
$\xi_{1},\xi_{2}\in\partial T_{E}$ such that $h(\xi_{1})=\xi_{2}$. By taking
representatives $x_{1},x_{2}$ of $\xi_{1},\xi_{2}$ in $E^{2}$, respectively,
we will have that $h(x_{1})=wx_{2}$, for some $w\in K$. Then apply the
involution $\theta$ and get $h(\theta(x_{1}))=\sigma(w)\theta(x_{2})$. Since
$B(\partial T_{K^{\sigma}})\cap\partial T_{E}=\\{\emptyset\\}$, we know that
$\theta(\xi_{i})\neq\xi_{i}$ and so the the matrices $(x_{i},\theta(x_{i}))$
are invertible and in $\operatorname{GL}(2,E)$. So
$h=(x_{2},\theta(x_{2}))\left(\begin{smallmatrix}w&0\\\
0&\sigma(w)\end{smallmatrix}\right)(x_{1},\theta(x_{1}))^{-1}\in\operatorname{SL}(2,K)$
and by taking the determinant we must have $w\sigma(w)\in E$ and thus
$w\sigma(w)\in{\mathbb{Q}}_{p}^{*}$, where $w\in K$. By considering squares in
${\mathbb{Q}}_{p}^{*}$, we restrict to the case when
$w\sigma(w)\in{\mathbb{Q}}_{p}^{*}/({\mathbb{Q}}_{p}^{*})^{2}$ with $w\in K$.
The latter has infinitely many solutions over $K$ and so there might be an
infinite number of left $H_{\theta}$-cosets in
$B\operatorname{SL}(2,K^{\sigma})B^{-1}$, implying a possibly infinite number
of $H_{\theta}$-orbits in $\partial T_{E}$.

To prove our results we need a polar decomposition of $\operatorname{SL}(2,E)$
with respect to $\operatorname{SL}(2,F)$. We cannot apply directly [BenoistOh]
as their result is proven only for $k$-involutions and not for abstract
involutions. We will keep the notation $\theta_{a}$ from Corollary 4.8, and
$\theta$ will refer to the involutions from Theorem 4.10(2). Here we give a
general result which covers $k$-involutions as well as abstract involutions
for $\operatorname{SL}(2,k)$ where $k$ is a non-Archimedean local field of
characteristic 0.

######  Proposition 5.5 (The polar decomposition for various subgroups of
$\operatorname{SL}_{2}$).

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$ and $E=F(\alpha)$ be
a quadratic extension of $F$. Let $H\leq G$ be such that

$(G,H)\in\\{(\operatorname{SL}(2,F),H_{\theta_{a}}),(\operatorname{SL}(2,E),\operatorname{SL}(2,F)),(\operatorname{SL}(2,{\mathbb{Q}}_{p}(\alpha)),H_{\theta})\\}.$

Denote by $T_{G}$ the Bruhat–Tits tree of $G$ and by $T_{H}$ the (possibly
finite) $H$-invariant subtree of $T_{G}$. Let $I_{H}$ be the number of
$H$-orbits in the ideal boundary $\partial T_{G}-\partial T_{H}$, and
$\xi_{i}$ a representative in each such orbit. Let $x_{0}\in T_{G}\cap T_{H}$
be a vertex, which for the pair
$(\operatorname{SL}(2,E),\operatorname{SL}(2,F))$ will be taken to be the
point $0$ as in the figures 1, 2. Then

$G=\mathcal{K}\mathcal{B}H$

where $\mathcal{K}$ is a compact subset of $G$, and
$\mathcal{B}=\\{\operatorname{Id}\\}\bigsqcup\limits_{i\in I_{H}}A_{i}$, where
$A_{i}:=\\{a_{i}^{n}\;|\;n\in{\mathbb{Z}}\\}$ with $a_{i}\in G$ a hyperbolic
element of translation length $2$ and with attractive endpoint in the
$H$-orbit of $\xi_{i}$.

###### Proof.

Let us make first some useful remarks. As the group $\operatorname{SL}(2)$
over a non-Archimedean local field acts by type-preserving automorphisms, thus
edge-transitively, on its Bruhat–Tits tree, our groups $G,H$ will do the same
on $T_{G}$, $T_{H}$, respectively. Denote by $V$ a fundamental domain of $H$
acting on $T_{H}$, such that $x_{0}\in V$. If $H=H_{\theta}$, or
$H=H_{\theta_{a}}$, with $a\neq 1$, then $T_{H}$ is a finite subtree, and we
can just consider $V=T_{H}$. In those cases the ideal boundary $\partial
T_{H}$ is $\\{\emptyset\\}$. If $H=\operatorname{SL}(2,F)$ or
$H=H_{\theta_{a}}$, with $a=1$, then $T_{H}$ is the Bruhat–Tits tree of
$\operatorname{SL}(2,F)$, or a bi-infinite geodesic line in $T_{G}$,
respectively. In those two cases the fundamental domain $V$ is an edge in
$T_{H}$. Moreover, the boundary $\partial T_{H}$ is the projective space
$P^{1}F$, and two endpoints of $\partial T_{E}$, respectively. Notice that for
the case when $G=\operatorname{SL}(2,E)$ and $H=\operatorname{SL}(2,F)$, the
edge $V$ is either an edge of $T_{E}$ (for $E$ unramified), or the union of
two consecutive edges of $T_{E}$ (for $E$ ramified).

For each of the $H$-orbits in $\partial T_{G}-\partial T_{H}$, thus for each
$i\in I_{H}$, we can choose a representative $\xi_{i}\in\partial
T_{G}-\partial T_{H}$ such that its projection $x_{i}$ on the tree $T_{H}$ is
in the fundamental domain $V$. Then $x_{i}\in V$ (which is viewed as a subset
of $T_{G}$), but not necesarily a vertex (e.g. this is the case for
$G=\operatorname{SL}(2,E),H=\operatorname{SL}(2,F)$ and $E=F(\alpha)$ a
ramified extension of $F$).

This means that the geodesic ray $[x_{i},\xi_{i})\subset T_{G}$ that starts
from $x_{i}\in V$ and with endpoint $\xi_{i}$ is entirely disjoint from the
tree $T_{H}$, except its basepoint $x_{i}$. Denote by $a_{i}$ a hyperbolic
element of $G$ with translation length $2$, translation axis containing the
geodesic ray $[x_{i},\xi_{i})$, and attracting endpoint $\xi_{i}$. Such an
element exists by the well-known properties of the group
$\operatorname{SL}(2)$ over a non-Archimedean local field, and its
corresponding Bruhat–Tits tree with the associated ideal boundary.

Let $g\in G$. If $g^{-1}(x_{0})\in T_{H}$, there is $h\in H$ such that
$hg^{-1}(x_{0})\in V$.

If $g^{-1}(x_{0})\in T_{G}\setminus T_{H}$, then let $y\in T_{H}$ be the
projection of $g^{-1}(x_{0})$ on the tree $T_{H}$; this projection $y$ is
unique. Then again there is $h\in H$ such that $h(y)\in V$, and the geodesic
$h([y,g^{-1}(x_{0})])$ is disjoint from $T_{H}$, except the point $h(y)$. By
left multiplying by an element $h^{\prime}$ in the $H$-stabilizer of $h(y)\in
V\subset T_{H}$, we can suppose that
$h^{\prime}h([y,g^{-1}(x_{0})])\subset[x_{i},\xi_{i})$ for some $i\in I_{H}$.
As we acted with type-preserving elements (i.e. $h,h^{\prime},g$ are all type-
preserving), there is $n\geq 1$ such that
$a_{i}^{-n}h^{\prime}hg^{-1}(x_{0})\in V$.

Let $\mathcal{K}$ to be set of all elements in $G$ that send the vertex
$x_{0}$ to one of vertices of $V\subset T_{G}$. Notice $\mathcal{K}$ is a
compact subset of $G$. Then for both cases $hg^{-1}(x_{0})\in V$, or
$a_{i}^{-n}h^{\prime}hg^{-1}(x_{0})\in V$, we have that
$hg^{-1},a_{i}^{-n}h^{\prime}hg^{-1}\in\mathcal{K}$. This implies that
$g^{-1}\in H\mathcal{B}\mathcal{K}$ and thus $g\in\mathcal{K}\mathcal{B}H$ as
required. ∎

######  Remark 5.6.

Notice Theorem 5.5 gives $\mathcal{B}$ as a union of a possibly infinite
number of $A_{i}$ that are pairwise non $H$-conjugate. Following results of
[Hel_k_invol, HelWD], the polar decomposition $\mathcal{K}\mathcal{B}H$ of
[BenoistOh] for $k$-involutions has a finite number of such $A_{i}$ in the
union $\mathcal{B}$. However, in the next section we use Lemmas 5.1, 5.2, 5.3
for the pairs
$(G,H)\in\\{(\operatorname{SL}(2,F),H_{\theta_{a}}),(\operatorname{SL}(2,E),\operatorname{SL}(2,F))\\}$
which give us a finite number of $A_{i}$ in our decomposition for
$\mathcal{B}$.

##  6\. Chabauty Limits of $\operatorname{SL}(2,F)$ inside
$\operatorname{SL}(2,E)$ for quadratic $E/F$

We will use the notation and conventions from Section 3. Let $F$ be a finite
field-extension of ${\mathbb{Q}}_{p}$ and $E$ be any quadratic extension of
$F$.

The stabiliser in $\operatorname{SL}(2,F)$, resp. $\operatorname{SL}(2,E)$, of
the endpoint $\big{[}\begin{smallmatrix}1\\\ 0\end{smallmatrix}\big{]}$ is the
Borel subgroup

$B_{F}^{+}:=\\{\left(\begin{smallmatrix}a&b\\\
0&a^{-1}\end{smallmatrix}\right)\;|\;b\in F,a\in
F^{\times}\\},\qquad\text{resp. }B_{E}^{+}:=\\{\left(\begin{smallmatrix}x&y\\\
0&x^{-1}\end{smallmatrix}\right)\;|\;y\in E,x\in E^{\times}\\}.$

The stabiliser in $\operatorname{SL}(2,F)$, resp. $\operatorname{SL}(2,E)$, of
the endpoint $\big{[}\begin{smallmatrix}0\\\ 1\end{smallmatrix}\big{]}$ is the
opposite Borel subgroup

$B_{F}^{-}:=\\{\left(\begin{smallmatrix}a&0\\\
b&a^{-1}\end{smallmatrix}\right)\;|\;b\in F,a\in
F^{\times}\\},\qquad\text{resp. }B_{E}^{-}:=\\{\left(\begin{smallmatrix}x&0\\\
y&x^{-1}\end{smallmatrix}\right)\;|\;y\in E,x\in E^{\times}\\}.$

Recall from Lemma 5.1 there are at most $5$ $\operatorname{SL}(2,F)$-orbits on
the boundary $\partial T_{E}\cong P^{1}E=E\cup\\{\infty\\}$: the
$\operatorname{SL}(2,F)$-orbit of $\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$, and the $\operatorname{SL}(2,F)$-orbits of
$\big{[}\begin{smallmatrix}1\\\ m\end{smallmatrix}\big{]}$, for each $m\in
F^{*}/(F^{*})^{2}$, that might coincide for two different $m$’s. By the polar
decomposition from Proposition 5.5 applied to the pair
$(G,H)=(\operatorname{SL}(2,E),\operatorname{SL}(2,F))$ and from Lemma 5.1,
the set $\mathcal{B}$ is a finite union of $A_{i}$. Since we want to compute
the Chabauty limits of $\operatorname{SL}(2,F)$ inside
$\operatorname{SL}(2,E)$, it is enough to compute the Chabauty limits of
$\operatorname{SL}(2,F)$ under conjugation by a sequence of elements from some
fixed $A_{i}\subset\mathcal{B}$. Because we want to choose a group
$A_{i}\subset\mathcal{B}$ such that the corresponding computations will be
easier, we rotate $\operatorname{SL}(2,F)$ in such a way that the chosen
$A_{i}$ is generated by the diagonal matrix
$\\{\left(\begin{smallmatrix}w_{E}&0\\\
0&w_{E}^{-1}\end{smallmatrix}\right)\\}$ of $\operatorname{SL}(2,E)$, which is
a hyperbolic element of translation length $2$ along the bi-infinite geodesic
line $(\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]})\subset T_{E}$. Such a rotation will affect the
Chabauty limits of $\operatorname{SL}(2,F)$ only up to
$\operatorname{SL}(2,E)$-conjugation.

We apply this idea and choose the two endpoints
$\big{[}\begin{smallmatrix}1\\\
\alpha\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
2\alpha\end{smallmatrix}\big{]}\in\partial T_{E}-\partial T_{F}$. Notice

$\left(\begin{matrix}1&0\\\
\alpha&1\end{matrix}\right)\left[\begin{matrix}1\\\
0\end{matrix}\right]=\left[\begin{matrix}1\\\
\alpha\end{matrix}\right]\qquad\textrm{and}\qquad\left(\begin{matrix}(2\alpha)^{-1}&1\\\
0&2\alpha\end{matrix}\right)\left[\begin{matrix}0\\\
1\end{matrix}\right]=\left[\begin{matrix}1\\\ 2\alpha\end{matrix}\right].$

We conjugate $\operatorname{SL}(2,F)$ by
$\left(\begin{smallmatrix}(2\alpha)^{-1}&1\\\
0&2\alpha\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
\alpha&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}(2\alpha)^{-1}+\alpha&1\\\
2\alpha^{2}&2\alpha\end{smallmatrix}\right)$. We obtain

$\begin{split}&H:=\left(\begin{smallmatrix}(2\alpha)^{-1}+\alpha&1\\\
2\alpha^{2}&2\alpha\end{smallmatrix}\right)\operatorname{SL}(2,{\mathbb{Q}}_{p})\left(\begin{smallmatrix}2\alpha&-1\\\
-2\alpha^{2}&\alpha+(2\alpha)^{-1}\end{smallmatrix}\right)=\\\
&=\\{\left(\begin{smallmatrix}a+2a\alpha^{2}-2d\alpha^{2}+\alpha(2c-b-2b\alpha^{2})&\;\;\left((2\alpha)^{-1}+\alpha\right)(b((2\alpha)^{-1}+\alpha)+d-a)-c\\\
4c\alpha^{2}-4b\alpha^{4}+\alpha(4a\alpha^{2}-4d\alpha^{2})&-2a\alpha^{2}+2d\alpha^{2}+d+\alpha(b+2b\alpha^{2}-2c)\end{smallmatrix}\right)\;|\;a,b,c,d\in{\mathbb{Q}}_{p},\;ad-
bc=1\\},\end{split}$

the subtree $T_{H}:=\left(\begin{smallmatrix}(2\alpha)^{-1}+\alpha&1\\\
2\alpha^{2}&2\alpha\end{smallmatrix}\right)T_{F}\subset T_{E}$ and its ideal
boundary $\partial T_{H}:=\left(\begin{smallmatrix}(2\alpha)^{-1}+\alpha&1\\\
2\alpha^{2}&2\alpha\end{smallmatrix}\right)\partial T_{F}$ are invariant under
$H$. It is easy to see that the endpoints $\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}$ and $\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$ are not in $\partial T_{H}$, and the bi-infinite
geodesic line $(\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]})\subset T_{E}$ intersects the tree $T_{H}$ either in
a vertex, or an edge of $T_{E}$, depending on whether the extension $E$ is
unramified or ramified, respectively.

Let us recall a version of Hensel’s Lemma for finite extensions of
${\mathbb{Q}}_{p}$ that will be used below.

######  Lemma 6.1 (Hensel).

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$. Let $f(X)$ be a
polynomial with coefficients in $\mathcal{O}_{F}$. Suppose there is some
$a\in\mathcal{O}_{F}$ that satisfies:

$|f(a)|_{F}<|f^{\prime}(a)|_{F}^{2}.$

Then there is a unique $x\in\mathcal{O}_{F}$ such that $f(x)=0$ in
$\mathcal{O}_{F}$ and $|x-a|_{F}<|f^{\prime}(a)|_{F}$.

######  Proposition 6.2.

Take a sequence of matrices $\\{\left(\begin{smallmatrix}a_{n}&b_{n}\\\
c_{n}&d_{n}\end{smallmatrix}\right)\\}_{n\geq 1}\subset\operatorname{SL}(2,F)$
and the diagonal matrices $\\{\left(\begin{smallmatrix}\omega_{E}^{n}&0\\\
0&\omega_{E}^{-n}\end{smallmatrix}\right)\\}_{n\geq 1}$ that are hyperbolic
elements of $\operatorname{SL}(2,E)$. Then any limit in
$\operatorname{SL}(2,E)$ of

$\left(\begin{smallmatrix}a_{n}+2a_{n}\alpha^{2}-2d_{n}\alpha^{2}+\alpha(2c_{n}-b_{n}-2b_{n}\alpha^{2})&\;\;\omega_{E}^{2n}\left[\left((2\alpha)^{-1}+\alpha\right)(b_{n}((2\alpha)^{-1}+\alpha)+d_{n}-a_{n})-c_{n}\right]\\\
\omega_{E}^{-2n}\left[4c_{n}\alpha^{2}-4b_{n}\alpha^{4}+\alpha(4a_{n}\alpha^{2}-4d_{n}\alpha^{2})\right]&-2a_{n}\alpha^{2}+2d_{n}\alpha^{2}+d_{n}+\alpha(b_{n}+2b_{n}\alpha^{2}-2c_{n})\end{smallmatrix}\right)$

is of the form $\left(\begin{smallmatrix}a-\alpha b&0\\\ z&a+\alpha
b\end{smallmatrix}\right)$, with $a,b\in F$ and $a^{2}-\alpha^{2}b^{2}=1$,
$z\in E$. In particular, $z$ can take any value of $E$, and any solution
$a,b\in F$ of the equation $a^{2}-\alpha^{2}b^{2}=1$ can appear.

###### Proof.

To have a limit in $\operatorname{SL}(2,E)$, we need that in the Chabauty
topology

  1. (1)

$\lim\limits_{n\to\infty}a_{n}+2a_{n}\alpha^{2}-2d_{n}\alpha^{2}+\alpha(2c_{n}-b_{n}-2b_{n}\alpha^{2})=A=A_{1}+\alpha
A_{2}\in E$

  2. (2)

$\lim\limits_{n\to\infty}\omega_{E}^{2n}\left[\left((2\alpha)^{-1}+\alpha\right)(b_{n}((2\alpha)^{-1}+\alpha)+d_{n}-a_{n})-c_{n}\right]=B=B_{1}+\alpha
B_{2}\in E$

  3. (3)

$\lim\limits_{n\to\infty}\omega_{E}^{-2n}\left[4c_{n}\alpha^{2}-4b_{n}\alpha^{4}+\alpha(4a_{n}\alpha^{2}-4d_{n}\alpha^{2})\right]=C=C_{1}+\alpha
C_{2}\in E$

  4. (4)

$\lim\limits_{n\to\infty}-2a_{n}\alpha^{2}+2d_{n}\alpha^{2}+d_{n}+\alpha(b_{n}+2b_{n}\alpha^{2}-2c_{n})=D=D_{1}+\alpha
D_{2}\in E$

with all $A_{i},B_{i},C_{i},D_{i}\in F$. From (3) above, we have that
$\lim\limits_{n\to\infty}\omega_{E}^{-2n}\left[4c_{n}\alpha^{2}-4b_{n}\alpha^{4}\right]=C_{1}$
and
$\lim\limits_{n\to\infty}\omega_{E}^{-2n}\left[4a_{n}\alpha^{2}-4d_{n}\alpha^{2}\right]=C_{2}.$
This means that

$\omega_{E}^{2n}C_{1,n}:=c_{n}-b_{n}\alpha^{2}\textrm{ and
}\omega_{E}^{2n}C_{2,n}:=a_{n}-d_{n}\textrm{ with
}\lim\limits_{n\to\infty}C_{i,n}=C_{i}/(4\alpha^{2})\in F,$

implying that

$\lim\limits_{n\to\infty}\omega_{E}^{2n}C_{1,n}=\lim\limits_{n\to\infty}(c_{n}-b_{n}\alpha^{2})=0\textrm{
and
}\lim\limits_{n\to\infty}\omega_{E}^{2n}C_{2,n}=\lim\limits_{n\to\infty}(a_{n}-d_{n})=0.$

Moreover, adding (1) and (4), we have

$\begin{split}\lim\limits_{n\to\infty}(a_{n}+2a_{n}\alpha^{2}-2d_{n}\alpha^{2}-2a_{n}\alpha^{2}+2d_{n}\alpha^{2}+d_{n})&=\lim\limits_{n\to\infty}(a_{n}+d_{n})\\\
&=\lim\limits_{n\to\infty}(d_{n}+\omega_{E}^{2n}C_{2,n}+d_{n})\\\
&=\lim\limits_{n\to\infty}2d_{n}=A_{1}+D_{1}.\\\ \end{split}$

Thus
$a:=\lim\limits_{n\to\infty}a_{n}=\lim\limits_{n\to\infty}d_{n}=(A_{1}+D_{1})/2\in
F$. From the first terms of (1) and (4), we see $A_{1}=D_{1}.$

Adding the other terms from (1) and (4), we get

$\lim\limits_{n\to\infty}(2c_{n}-b_{n}-2b_{n}\alpha^{2}+b_{n}+2b_{n}\alpha^{2}-2c_{n})=0=A_{2}+D_{2}\Rightarrow
A_{2}=-D_{2}.$

From the above fact that $\lim\limits_{n\to\infty}(c_{n}-b_{n}\alpha^{2})=0$
and from the $\alpha$-term of (4) we get that

$\lim\limits_{n\to\infty}b_{n}=D_{2}\in F.$

Thus $b:=\lim\limits_{n\to\infty}b_{n}=D_{2}\in F$ and
$\lim\limits_{n\to\infty}c_{n}=\alpha^{2}D_{2}=\alpha^{2}b\in F$.

As $a_{n}d_{n}-c_{n}b_{n}=1$ and replacing
$a_{n}=d_{n}+\omega_{E}^{2n}C_{2,n}$ and
$c_{n}=b_{n}\alpha^{2}+\omega_{E}^{2n}C_{1,n}$, we get

$d_{n}^{2}+\omega_{E}^{2n}C_{2,n}d_{n}-\alpha^{2}b_{n}^{2}-\omega_{E}^{2n}C_{1,n}b_{n}=1\text{
implying }$
$1=\lim\limits_{n\to\infty}d_{n}^{2}+\omega_{E}^{2n}C_{2,n}d_{n}-\alpha^{2}b_{n}^{2}-\omega_{E}^{2n}C_{1,n}b_{n}=a^{2}-\alpha^{2}b^{2}.$

Recall that $a^{2}-\alpha^{2}b^{2}=1$ with $a,b\in F$ implies that
$a,b\in\mathcal{O}_{F}$ with $|a|_{F}=1$ and $|b|_{F}<1$.

In fact, (1), (2) and (4) above will become

(i):

$\lim\limits_{n\to\infty}a_{n}+2a_{n}\alpha^{2}-2d_{n}\alpha^{2}+\alpha(2c_{n}-b_{n}-2b_{n}\alpha^{2})=\lim\limits_{n\to\infty}a_{n}-\alpha
b_{n}=a-\alpha b=A$

(ii):

$\lim\limits_{n\to\infty}\omega_{E}^{2n}\left[\left((2\alpha)^{-1}+\alpha\right)(b_{n}((2\alpha)^{-1}+\alpha)+d_{n}-a_{n})-c_{n}\right]=0=B$

(iii):

$\lim\limits_{n\to\infty}-2a_{n}\alpha^{2}+2d_{n}\alpha^{2}+d_{n}+\alpha(b_{n}+2b_{n}\alpha^{2}-2c_{n})=\lim\limits_{n\to\infty}d_{n}+\alpha
b_{n}=a+\alpha b=D$.

We will use Hensel’s Lemma 6.1 to show any element $C\in E$ can be obtained in
(3). Take any $C=C_{1}+\alpha C_{2}\in E$, and any $b\in\mathcal{O}_{F}$, such
that $a^{2}-\alpha^{2}b^{2}=1$. Then for $n\geq 1$ large enough, we have that
$\omega_{E}^{2n}C_{2},\alpha^{2}b^{2}+\omega_{E}^{2n}C_{1}b\in\omega_{E}\mathcal{O}_{F}$.
Take

$f_{n}(X):=X^{2}+\omega_{E}^{2n}C_{2}X-\alpha^{2}b^{2}-\omega_{E}^{2n}C_{1}b-1,\text{
and }f^{\prime}_{n}(X)=2X+\omega_{E}^{2n}C_{2}.$

Then $f_{n}(X)\in\mathcal{O}_{F}[X]$,
$f_{n}(1)=1+\omega_{E}^{2n}C_{2}-\alpha^{2}b^{2}-\omega_{E}^{2n}C_{1}b-1\equiv
0(mod\;\omega_{E})$, and $f^{\prime}_{n}(1)=2+\omega_{E}^{2n}C_{2}\equiv
2(mod\;\omega_{E})$. Then by Hensel’s Lemma 6.1, there is
$d_{n}\in\mathcal{O}_{F}$ such that
$f_{n}(d_{n}):=d_{n}^{2}+\omega_{E}^{2n}C_{2}d_{n}-\alpha^{2}b^{2}-\omega_{E}^{2n}C_{1}b-1=0$,
and $d_{n}\equiv 1(mod\;\omega_{E})$. Then take
$a_{n}=d_{n}+\omega_{E}^{2n}C_{2}$ and
$c_{n}=b\alpha^{2}+\omega_{E}^{2n}C_{1}$, getting $a_{n}d_{n}-c_{n}b=1$. Thus
(3) becomes
$\lim\limits_{n\to\infty}\omega_{E}^{-2n}\left[4c_{n}\alpha^{2}-4b\alpha^{4}+\alpha(4a_{n}\alpha^{2}-4d_{n}\alpha^{2})\right]=4\alpha^{2}C=4\alpha^{2}(C_{1}+\alpha
C_{2})\in E$. Notice that up to extracting a subsequence and using the fact
that $d_{n}\in\mathcal{O}_{F}$, we have that
$\lim\limits_{n\to\infty}a_{n}=a$, and $a^{2}-\alpha^{2}b^{2}=1$. ∎

To summarize, we have just proved the following:

######  Theorem 6.3.

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$ and $E$ be any
quadratic extension of $F$. Then any Chabauty limit of
$\operatorname{SL}(2,F)$ in $\operatorname{SL}(2,E)$ is either an
$\operatorname{SL}(2,E)$-conjugate of $\operatorname{SL}(2,F)$, or an
$\operatorname{SL}(2,E)$-conjugate of the subgroup
$\\{\left(\begin{smallmatrix}a-\alpha b&z\\\ 0&a+\alpha
b\end{smallmatrix}\right)\;|\;a,b\in F\text{ with
}a^{2}-\alpha^{2}b^{2}=1,z\in E\\}\leq B_{E}^{+}.$

######  Remark 6.4.

Notice this proof does not depend on whether the extension is ramified or
unramified.

##  7\. Chabauty Limits of $\operatorname{SL}(2,{\mathbb{R}})$ inside
$\operatorname{SL}(2,{\mathbb{C}})$

Recall that $\operatorname{SL}(2,{\mathbb{R}})$ is the isometry group of the
real hyperbolic plane ${\mathbb{H}}^{2}$ and
$\operatorname{SL}(2,{\mathbb{C}})$ is the isometry group of real hyperbolic
3-space ${\mathbb{H}}^{3}$. Since ${\mathbb{C}}$ is a quadratic extension of
${\mathbb{R}}$, the situation mirrors Section 6. The boundary of
${\mathbb{H}}^{3}$ in the Poincaré ball model is the Riemann sphere, which may
be thought of as union of points $\\{\big{[}\begin{smallmatrix}1\\\
x\end{smallmatrix}\big{]}|x\in{\mathbb{C}}\\}\cup\\{\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}\\}$. Since $\operatorname{SL}(2,{\mathbb{R}})$ acts
on ${\mathbb{H}}^{3}$ and its boundary, one may easily compute that the
stabilizer in $\operatorname{SL}(2,{\mathbb{R}})$ of the endpoint
$\big{[}\begin{smallmatrix}1\\\ i\end{smallmatrix}\big{]}$ is the compact
subgroup $\\{\left(\begin{smallmatrix}a&b\\\
-b&a\end{smallmatrix}\right)\;|\;a^{2}+b^{2}=1\;a,b\in{\mathbb{R}}\\}$.

For $k={\mathbb{R}}$ or ${\mathbb{C}}$, the following stabilisers in
$\operatorname{SL}(2,k)$ are the Borel subgroups:

$\mathrm{Stab}_{\operatorname{SL}(2,k)}\left(\begin{bmatrix}1\\\
0\end{bmatrix}\right)=\\{\left(\begin{smallmatrix}a&b\\\
0&a^{-1}\end{smallmatrix}\right)|b\in k,a\in k^{\times}\\}=B^{+}_{k}$
$\mathrm{Stab}_{\operatorname{SL}(2,k)}\left(\begin{bmatrix}0\\\
1\end{bmatrix}\right)=\\{\left(\begin{smallmatrix}a&0\\\
b&a^{-1}\end{smallmatrix}\right)|b\in k,a\in k^{\times}\\}=B^{-}_{k}.$

Notice that with respect to the complex conjugation on ${\mathbb{C}}$, which
extends to an involution on $\operatorname{SL}(2,{\mathbb{C}})$, the group
$\operatorname{SL}(2,{\mathbb{R}})$ is the fixed point group under complex
conjugation. We apply the polar decomposition (see for example Proposition 2.4
of [HS]) to the pair
$(\operatorname{SL}(2,{\mathbb{C}}),\operatorname{SL}(2,{\mathbb{R}}))$. More
precisely, let $G$ be a semisimple Lie group with finite center and $H$ a
symmetric subgroup. Let $\mathcal{K}$ be the maximal compact subgroup of $G$
and $\mathcal{B}$ the exponential of a subalgebra of the maximal abelian
subalgebra that has a good intersection with the positive and negative
eigenspaces. Then:

######  Proposition 7.1 (Proposition 2.4 of [HS]).

For any $g\in G$ there exists $k\in\mathcal{K},b\in\mathcal{B},h\in H$ such
that $g=kbh$. Moreover $b$ is unique up to conjugation by the Weyl group
$W_{H\cap\mathcal{K}}$.

Alternatively, for the pair
$(\operatorname{SL}(2,{\mathbb{C}}),\operatorname{SL}(2,{\mathbb{R}}))$ one
can just notice that the $\operatorname{SL}(2,{\mathbb{R}})$-orbits on the
boundary of ${\mathbb{H}}^{3}$ are the
$\operatorname{SL}(2,{\mathbb{R}})$-orbits for $\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$, and $\big{[}\begin{smallmatrix}1\\\
i\end{smallmatrix}\big{]}$, respectively. Then proceed as in the proof of
Proposition 5.5.

As a consequence of the polar decomposition above, it suffices to consider
Chabauty limits of $\operatorname{SL}(2,{\mathbb{R}})$ inside
$\operatorname{SL}(2,{\mathbb{C}})$ by conjugating only by elements in
$\mathcal{B}$. Since we want to use the diagonal matrices in order to
conjugate and compute the Chabauty limits, we have to rotate/conjugate the
subgroup $\operatorname{SL}(2,{\mathbb{R}})$ such that the endpoints
$\big{[}\begin{smallmatrix}1\\\ 0\end{smallmatrix}\big{]}$ and
$\big{[}\begin{smallmatrix}0\\\ 1\end{smallmatrix}\big{]}$ will be transverse
to the ${\mathbb{H}}^{2}$-like slice corresponding to the rotated
$\operatorname{SL}(2,{\mathbb{R}})$.

There is an action on the endpoints

$\left(\begin{matrix}1&0\\\ i&1\end{matrix}\right)\begin{bmatrix}1\\\
0\end{bmatrix}=\begin{bmatrix}1\\\
i\end{bmatrix}\qquad\left(\begin{matrix}(2i)^{-1}&1\\\
0&2i\end{matrix}\right)\begin{bmatrix}0\\\ 1\end{bmatrix}=\begin{bmatrix}1\\\
2i\end{bmatrix}.$

Thus we conjugate the group $\operatorname{SL}(2,{\mathbb{R}})$ by the matrix
$\left(\begin{smallmatrix}(2i)^{-1}&1\\\
0&2i\end{smallmatrix}\right)\left(\begin{smallmatrix}1&0\\\
i&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}(2i)^{-1}+i&1\\\
-2&2i\end{smallmatrix}\right)$. We obtain

$\begin{split}&\left(\begin{smallmatrix}(2i)^{-1}+i&1\\\
-2&2i\end{smallmatrix}\right)\operatorname{SL}(2,{\mathbb{R}})\left(\begin{smallmatrix}2i&-1\\\
2&i+(2i)^{-1}\end{smallmatrix}\right)=\\\
&=\\{\left(\begin{smallmatrix}-a+2d+i(2c+b)&\;\;-c-\frac{b}{4}+\frac{i}{2}(d-a)\\\
-4c-4b+i(-4a+4d)&2a-d+i(-b-2c)\end{smallmatrix}\right)|a,b,c,d\in{\mathbb{R}},\;ad-
bc=1\\}.\end{split}$

######  Proposition 7.2.

Take a sequence of matrices $\\{\left(\begin{smallmatrix}a_{n}&b_{n}\\\
c_{n}&d_{n}\end{smallmatrix}\right)\\}_{n\geq
1}\subset\operatorname{SL}(2,{\mathbb{R}})$ and the diagonal matrices
$\\{\left(\begin{smallmatrix}e^{n}&0\\\
0&e^{-n}\end{smallmatrix}\right)\\}_{n\geq 1}$ that are hyperbolic elements of
$\operatorname{SL}(2,{\mathbb{C}})$. Then any limit in
$\operatorname{SL}(2,{\mathbb{C}})$ of

$\left(\begin{smallmatrix}-a_{n}+2d_{n}+i(2c_{n}+b_{n})&\;\;e^{2n}\left[-c_{n}-\frac{b_{n}}{4}+\frac{i}{2}(d_{n}-a_{n})\right]\\\
e^{-2n}\left[-4c_{n}-4b_{n}+i(-4a_{n}+4d_{n})\right]&2a_{n}-d_{n}+i(-b_{n}-2c_{n})\end{smallmatrix}\right)$

is of the form $\left(\begin{smallmatrix}a-ib&z\\\
0&a+ib\end{smallmatrix}\right)$, with $a,b\in{\mathbb{R}}$ with
$a^{2}+b^{2}=1$, and $z\in{\mathbb{C}}$. In particular, $z$ can be any complex
number, and any solution $a,b\in{\mathbb{R}}$ of the equation $a^{2}+b^{2}=1$
may appear.

###### Proof.

The proof follows the same computations as in Proposition 6.2. Just to
summarize the computations one can easily obtain that:
$\lim\limits_{n\to\infty}d_{n}-a_{n}=0,\lim\limits_{n\to\infty}2d_{n}-a_{n}$
converge in ${\mathbb{R}}$, and $\lim\limits_{n\to\infty}4c_{n}+b_{n}=0$,
$\lim\limits_{n\to\infty}2c_{n}+b_{n}$ converge in ${\mathbb{R}}$. Combining
those facts, we have that
$\lim\limits_{n\to\infty}a_{n}=-\lim\limits_{n\to\infty}d_{n}$,
$\lim\limits_{n\to\infty}c_{n}=-1/4\lim\limits_{n\to\infty}b_{n}$, all in
${\mathbb{R}}$. This implies

$\lim\limits_{n\to\infty}e^{-2n}\left[-4c_{n}-4b_{n}+i(-4a_{n}+4d_{n})\right]=0$
$\lim\limits_{n\to\infty}-a_{n}+2d_{n}+i(2c_{n}+b_{n})=a-ib,\textrm{ and
}\lim\limits_{n\to\infty}2a_{n}-d_{n}+i(-b_{n}-2c_{n})=a+ib,$

with $a^{2}+b^{2}=1$ and $a,b\in{\mathbb{R}}$.

To prove we can obtain any matrix of the form
$\left(\begin{smallmatrix}a-ib&z\\\ 0&a+ib\end{smallmatrix}\right)$, take any
$b$ and $z=z_{1}+iz_{2}\in{\mathbb{C}}$ with the required properties. Then:

  1. (1)

if $b^{2}=1$ then we have
$\lim\limits_{n\to\infty}a_{n}=-\lim\limits_{n\to\infty}d_{n}=0$. Then take
$d_{n}:=z_{2}e^{-2n}+a_{n},b_{n}:=-4(e^{-2n}z_{1}+c_{n})$ and solve the
equation
$1=a_{n}d_{n}-c_{n}d_{n}=a_{n}^{2}+4c_{n}^{2}+c_{n}4z_{1}e^{-2n}+a_{n}z_{2}e^{-2n}$
for $c_{n}$. As $a_{n}$ and $e^{-2n}$ will converge to zero, there are real
solutions for $c_{n}$.

  2. (2)

if $b^{2}\neq 1$ take
$c_{n}:=b/2,d_{n}:=z_{2}e^{-2n}+a_{n},b_{n}:=-4(e^{-2n}z_{1}+b/2)$ and solve
the equation
$1=a_{n}d_{n}-c_{n}d_{n}=a_{n}^{2}+a_{n}z_{2}e^{-2n}+2bz_{1}e^{-2n}+b^{2}$ for
$a_{n}$. When $e^{-2n}$ is very small, there are real solutions for $a_{n}$.

∎

Geometrically, we see that we can translate the slice ${\mathbb{H}}^{2}$ by
hyperbolic elements to limit to any point in the boundary at infinity of
${\mathbb{H}}^{3}$. To summarize, we have proved the following:

######  Theorem 7.3.

Any Chabauty limit of $\operatorname{SL}(2,{\mathbb{R}})$ inside
$\operatorname{SL}(2,{\mathbb{C}})$ is
$\operatorname{SL}(2,{\mathbb{C}})$-conjugate to either
$\operatorname{SL}(2,{\mathbb{R}})$, or to the subgroup
$\\{\left(\begin{smallmatrix}a-ib&z\\\
0&a+ib\end{smallmatrix}\right)\;|\;a,b\in{\mathbb{R}}\text{ with
}a^{2}+b^{2}=1,z\in{\mathbb{C}}\\}\leq B_{{\mathbb{C}}}^{+}.$

######  Remark 7.4.

Notice this computation is not covered by [CDW] since they only compute limits
inside of $\operatorname{PGL}(n,{\mathbb{R}})$. However, the result can easily
be deduced from their Theorem 4.1, which holds for real and complex symmetric
subgroups of semi-simple Lie groups.

##  8\. Chabauty Limits of Symmetric Subgroups of $\operatorname{SL}(2,F)$

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$ with ring of
integers denoted by $\mathcal{O_{F}}$, $\omega_{F}$ a uniformizer, and $k_{F}$
the residue field of $F$. Recall $|F^{*}/(F^{*})^{2}|=4$. Let
$A=\left(\begin{smallmatrix}0&1\\\ a&0\end{smallmatrix}\right)$, with $a\in
F^{*}/(F^{*})^{2}=\\{1,\omega_{F},S,\omega_{F}S\\}$, with $S$ a non-square in
$k_{F}^{*}$, and $\theta_{a}:=\iota_{A}$ the corresponding $F$-involution of
$\operatorname{SL}(2,F)$ (see [HW_class, Section 1.3]). Take
$K_{a}:=F(\sqrt{a})$ the field extension of $F$ corresponding to $a$; if $a=1$
then $F=K_{a}$, and if $a\neq 1$ then $K_{a}$ is a quadratic extension of $F$.
Then by Corollary 4.8 we know that $A$ fixes pointwise only the two endpoints
$\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}\in P^{1}K_{a}$. Take

$H_{\theta_{a}}:=\\{g\in\operatorname{SL}(2,F)\;|\;\theta_{a}(g)=g\\},$

and together with the results of [HW_class, Section 3] we have:

$H_{\theta_{a}}=\mathrm{Fix}_{\operatorname{SL}(2,K_{a})}(\\{\xi_{-},\xi_{+}\\})\cap\operatorname{SL}(2,F)=\\{\left(\begin{smallmatrix}x&y\\\
ay&x\end{smallmatrix}\right)\in\operatorname{SL}(2,F)\;|\;x^{2}-ay^{2}=1\\}.$

To compute the Chabauty limits in the general case of
$H_{\theta_{a}}\leq\operatorname{SL}(2,F)$ we use the polar decomposition from
Proposition 5.5 applied to the pair
$(G,H)=(\operatorname{SL}(2,F),H_{\theta_{a}})$. By Lemmas 5.2, 5.3 the
corresponding $\mathcal{B}$ is a finite union of groups $\mathcal{A}_{i}$. As
in Section 6 it is enough to compute the Chabauty limits of $H_{\theta_{a}}$
under conjugation by a sequence of elements from some fixed
$\mathcal{A}\subset\mathcal{B}$. We choose a group
$\mathcal{A}\subset\mathcal{B}$ such that the corresponding computations will
be easier. Notice that $Diag(2,F)\leq\operatorname{SL}(2,F)$ has fixed
endpoints $\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}\in\partial T_{F}$ (which are different from
$\xi_{\pm}:=\big{[}\begin{smallmatrix}1\\\
\pm\sqrt{a}\end{smallmatrix}\big{]}\in P^{1}K_{a}$), and it is transverse to
the bi-infinite geodesic line $(\big{[}\begin{smallmatrix}1\\\
-\sqrt{a}\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
\sqrt{a}\end{smallmatrix}\big{]})\subset T_{K_{a}}$. Thus, up to
$H_{\theta_{a}}$-conjugacy, we can take $Diag(2,F)\subset\mathcal{B}$. So, we
choose $\mathcal{A}$ to be generated by the diagonal matrix
$\\{\left(\begin{smallmatrix}w_{F}&0\\\
0&w_{F}^{-1}\end{smallmatrix}\right)\\}$ of $\operatorname{SL}(2,F)$, which is
a hyperbolic element of translation length $2$ along the bi-infinite geodesic
line $(\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]})\subset T_{F}$. This procedure affects the Chabauty
limits of $H_{\theta_{a}}$ only up to $\operatorname{SL}(2,F)$-conjugation.

Thus, consider a sequence $\\{\left(\begin{smallmatrix}x_{n}&y_{n}\\\
y_{n}a&x_{n}\end{smallmatrix}\right)\\}_{n\geq 1}\subset H_{\theta_{a}}$ and a
sequence $\\{\left(\begin{smallmatrix}\omega_{F}^{n}&0\\\
0&\omega_{F}^{-n}\end{smallmatrix}\right)\\}_{n\geq 1}\subset\mathcal{A}$ such
that

$\begin{pmatrix}\omega_{F}^{n}&0\\\
0&\omega_{F}^{-n}\end{pmatrix}\begin{pmatrix}x_{n}&y_{n}\\\
y_{n}a&x_{n}\end{pmatrix}\begin{pmatrix}\omega_{F}^{-n}&0\\\
0&\omega_{F}^{n}\end{pmatrix}=\begin{pmatrix}x_{n}&\omega_{F}^{2n}y_{n}\\\
\omega_{F}^{-2n}y_{n}a&x_{n}\end{pmatrix}\xrightarrow{n\to\infty}\begin{pmatrix}x&y\\\
z&t\end{pmatrix}\in\operatorname{SL}(2,F).$

Then we must have that $\lim\limits_{n\to\infty}x_{n}=x=t\in F$. As well, we
have $\lim\limits_{n\to\infty}\omega_{F}^{-2n}y_{n}=z/a\in F$ and
$\lim\limits_{n\to\infty}\omega_{F}^{2n}y_{n}=y\in F$. Writing
$C_{n}:=\omega_{F}^{-2n}y_{n}$ we get $y_{n}=\omega_{F}^{2n}C_{n}$, and
because $\lim\limits_{n\to\infty}C_{n}=z/a\in F$, one obtains
$\lim\limits_{n\to\infty}y_{n}=0\in F$, so
$\lim\limits_{n\to\infty}\omega_{F}^{2n}y_{n}=0=y\in F$. Since
$\left(\begin{smallmatrix}x&y\\\
z&t\end{smallmatrix}\right)\in\operatorname{SL}(2,F)$ with $x=t$ and $y=0$, we
get $x^{2}=1$, implying that $x\in\mu_{2}$ which denotes the group of $2^{nd}$
roots of unity in $F$.

So far we have proven that a Chabauty limit of $H_{\theta_{a}}$ is contained
in the subgroup $\\{\mu_{2}\cdot\left(\begin{smallmatrix}1&0\\\
z&1\end{smallmatrix}\right)|z\in F\\}$, which is condition 2 of Proposition
2.1 for Chabauty convergence.

It remains to show condition (1) of Proposition 2.1. To show equality, fix
some $z\in F$ and take $y_{n}:=\frac{z}{a}\omega_{F}^{2n}$, thus
$\lim\limits_{n\to\infty}y_{n}=0$. In particular, for every $n$ large enough
$y_{n}\in\omega_{F}\mathcal{O}_{F}$. Then, for each such $n$ we apply Hensel’s
Lemma 6.1 to $f_{n}(x)=x^{2}-ay_{n}^{2}-1$, where $f_{n}(1)\equiv
0\;(\text{mod }\omega_{F})$ and $f_{n}^{\prime}(1)=2\not\equiv 0\;(\text{mod
}\omega_{F})$. So there is a solution $x_{n}\in\mathcal{O}_{F}$ with
$f_{n}(x_{n})=0$ and $x_{n}\equiv 1\;(\text{mod }\omega_{F})$. In particular,
we have $x_{n}^{2}-ay_{n}^{2}=1$,
$\lim\limits_{n\to\infty}\omega_{F}^{-2n}y_{n}a=z\in F$,
$\lim\limits_{n\to\infty}\omega_{F}^{2n}y_{n}=0\in F$, and
$\lim\limits_{n\to\infty}x_{n}=x$, and $x=1$. By taking $f_{n}(\omega_{F}-1)$
and $f_{n}^{\prime}(\omega_{F}-1)$, one obtains $x=-1$. Thus we have proved:

######  Theorem 8.1.

Let $F$ be a finite field-extension of ${\mathbb{Q}}_{p}$. Let
$A=\left(\begin{smallmatrix}0&1\\\ a&0\end{smallmatrix}\right)$, with $a\in
F^{*}/(F^{*})^{2}$, and $\theta_{a}:=\iota_{A}$ the corresponding
$F$-involution of $\operatorname{SL}(2,F)$. Take
$H_{\theta_{a}}:=\\{g\in\operatorname{SL}(2,F)\;|\;\theta_{a}(g)=g\\}$. Then
any Chabauty limit of $H_{\theta_{a}}$ is either
$\operatorname{SL}(2,F)$-conjugate to $H_{\theta_{a}}$, or to the subgroup
$\\{\mu_{2}\cdot\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right)\;|\;x\in F\\}$ of the Borel
$B_{F}^{+}\leq\operatorname{SL}(2,F)$, where $\mu_{2}$ is the group of
$2^{nd}$ roots of unity in $F$.

######  Remark 8.2.

In the case $F={\mathbb{Q}}_{p}$ and $a=1$, we have
$H_{\theta_{a}}=CDiag(2,{\mathbb{Q}}_{p})C^{-1}$ where
$C=\left(\begin{smallmatrix}\frac{1}{2}&-1\\\
\frac{1}{2}&1\end{smallmatrix}\right)$. So $H_{\theta_{a}}$ is a maximal
${\mathbb{Q}}_{p}$-split torus in $\operatorname{SL}(2,{\mathbb{Q}}_{p})$,
(see [HW_class, Section 3] or Corollary 4.8). Then results of [CLV] (using
different methods than in this article) give the same result as Theorem 8.1.

## References