Quasi-inner automorphisms of Drinfeld modular groups

###### Abstract.

Let $A$ be the set of elements in an algebraic function field $K$ over
${\mathbb{F}}_{q}$ which are integral outside a fixed place $\infty$. Let
$G=GL_{2}(A)$ be a Drinfeld modular group. The normalizer of $G$ in
$GL_{2}(K)$, where $K$ is the quotient field of $A$, gives rise to
automorphisms of $G$, which we refer to as quasi-inner. Modulo the inner
automorphisms of $G$ they form a group $Quinn(G)$ which is isomorphic to
$\mathop{\mathrm{Cl}}(A)_{2}$, the $2$-torsion in the ideal class group
$\mathop{\mathrm{Cl}}(A)$.
The group $Quinn(G)$ acts on all kinds of objects associated with $G$. For
example, it acts freely on the cusps and elliptic points of $G$. If
${\mathcal{T}}$ is the associated Bruhat-Tits tree the elements of $Quinn(G)$
induce non-trivial automorphisms of the quotient graph
$G\backslash{\mathcal{T}}$, generalizing an earlier result of Serre. It is
known that the ends of $G\backslash{\mathcal{T}}$ are in one-one
correspondence with the cusps of $G$. Consequently $Quinn(G)$ acts freely on
the ends. In addition $Quinn(G)$ acts transitively on the those ends which are
in one-one correspondence with the vertices of $G\backslash{\mathcal{T}}$
whose stabilizers are isomorphic to $GL_{2}({\mathbb{F}}_{q})$.

2020 Mathematics Subject Classification: 11F06, 20E08, 20E36, 20G30

Keywords: Drinfeld modular group; quasi-inner automorphism; elliptic point;
cusp; quotient graph

##  1\. Introduction

Let $K$ be an algebraic function field of one variable with constant field
${\mathbb{F}}_{q}$, the finite field of order $q$. Let $\infty$ be a fixed
place of $K$ and let $\delta$ be its degree. The ring $A$ of all those
elements of $K$ which are integral outside $\infty$ is a Dedekind domain.
Denote by $K_{\infty}$ the completion of $K$ with respect to $\infty$ and let
$C_{\infty}$ be the $\infty$-completion of an algebraic closure of
$K_{\infty}$. The group $GL_{2}(K_{\infty})$ (and its subgroup $G=GL_{2}(A)$)
act as Möbius transformations on $C_{\infty}$, $K_{\infty}$ and hence
$\Omega=C_{\infty}\backslash K_{\infty}$, the Drinfeld upper halfplane. This
is part of a far-reaching analogy, initiated by Drinfeld [Dr], where
${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are replaced by
$K,K_{\infty},C_{\infty}$, respectively. The roles of the classical upper half
plane (in ${\mathbb{C}}$) and the classical modular group
$SL_{2}({\mathbb{Z}})$ are assumed by $\Omega$ and $G$, repspectively.
Modular curves, that is quotients of the complex upper half plane by finite
index subgroups of $SL_{2}({\mathbb{Z}})$, are an indispensable tool when
proving deep theorems about elliptic curves. Of similar importance in the
theory of Drinfeld $A$-modules of rank $2$ are Drinfeld modular curves, which
are (the “compactifications” of) the quotient spaces $H\backslash\Omega$,
where $H$ is a finite index subgroup of $G$. Consequently we refer to $G$ as a
Drinfeld modular group.
A complicating factor in this correspondence between $SL_{2}({\mathbb{Z}})$
and $G$ is that, while the genus of the former is zero, for different choices
of $K$ and $\infty$ the genus of $G$ can take many values. The simplest case,
where $K={\mathbb{F}}_{q}(t)$ and $A={\mathbb{F}}_{q}[t]$ (equivalently $g=0$
and $\delta=1$), has to date attracted most attention.
An element $\omega\in\Omega$ which is stabilized by a non-scalar matrix in $G$
is called elliptic. Let $E(G)$ be the set of all such elements. It is known
[Ge, p.50] that $E(G)\neq\emptyset$ if and only if $\delta$ is odd. Clearly
$G$ acts on $E(G)$ and the elements of the set of $G$-orbits,
$\mathop{\mathrm{Ell}}(G)=G\backslash E(G)=\\{G\omega:\omega\in E(G)\\}$, are
called the elliptic points of $G$. It is known [Ge, p.50] that
$\mathop{\mathrm{Ell}}(G)$ is finite.
In addition $G$ acts on ${\mathbb{P}}^{1}(K)=K\cup\\{\infty\\}$. (Here, of
course, $\infty$ refers to the one point compactification of $K$.) We refer to
the elements of ${\mathbb{P}}^{1}(K)$ as rational points. For each finite
index subgroup, $H$, of $G$ the elements of
$\mathop{\mathrm{Cusp}}(H)=H\backslash{\mathbb{P}}^{1}(K)$ are called the
cusps of $H$. Since $A$ is a Dedekind domain it is well-known that
$\mathop{\mathrm{Cusp}}(G)$ can be identified with $\mathop{\mathrm{Cl}}(A)$,
the ideal class group of $A$. As Möbius transformations $G$ acts without
inversion on ${\mathcal{T}}$, the Bruhat-Tits tree associated with
$GL_{2}(K_{\infty})$ and the ends of the quotient graph
$G\backslash{\mathcal{T}}$ are determined by $\mathop{\mathrm{Cusp}}(G)$ [Se,
Theorem 9, p.106].
Cusps and elliptic points are important for several reasons. If $H$ is a
finite index subgroup of $G$, the quotient space $H\backslash\Omega$ will,
after adding $\mathop{\mathrm{Cusp}}(H)$, be the $C_{\infty}$-analog of a
compact Riemann surface, which is called the Drinfeld modular curve associated
with $H$. Moreover, in the covering of Drinfeld modular curves induced by the
natural map $H\backslash\Omega\rightarrow G\backslash\Omega$ ramification can
only occur above the cusps and elliptic points of $G$. Also, for (classical
and Drinfeld) modular forms, analyticity at the cusps and elliptic points
requires special care.

This paper is a continuation and extension of [MS4] which is concerned with
the elliptic points of $G$. There the starting point [Ge, p.51] is the
existence of a bijection between $\mathop{\mathrm{Ell}}(G)$ and
$\ker\overline{N}$, where
$\overline{N}:\mathop{\mathrm{Cl}}(\widetilde{A})\rightarrow\mathop{\mathrm{Cl}}(A)$
is the norm map and $\widetilde{A}=A.{\mathbb{F}}_{q^{2}}$. It can be shown
[MS4] that $\mathop{\mathrm{Cl}}(\widetilde{A})_{2}\cap\ker\overline{N}$, the
$2$-torsion subgroup of $\ker\overline{N}$, is in bijection with
$\mathop{\mathrm{Ell}}(G)^{=}=\\{G\omega:\omega\in
E(G),\;G\omega=G\overline{\omega}\\}$, where $\overline{\omega}$, the
conjugate of $\omega$, is the image of $\omega$ under the Galois automorphism
of $K.{\mathbb{F}}_{q^{2}}/K$. (In [MS4] $\mathop{\mathrm{Ell}}(G)^{=}$ is
denoted by $\mathop{\mathrm{Ell}}(G)_{2}$.) Here we show that, when $\delta$
is odd, $\mathop{\mathrm{Cl}}(A)_{2}$ and the $2$-torsion in
$\ker\overline{N}$ are isomorphic. This is the starting point for this paper
where the principal focus of attention is the group
$\mathop{\mathrm{Cl}}(A)_{2}$ and its actions on various objects related to
$G$. Unless otherwise stated results hold for all $\delta$.

Let $g\in N_{GL_{2}(K)}(G)$, the normalizer of $G$ in $GL_{2}(K)$. Then $g$,
acting by conjugation, induces an automorphism $\iota_{g}$ of $G$ which we
refer to as quasi-inner. If $g\in G.Z(K)$ then $\iota_{g}$ reduces to an inner
automorphism. If $g\in N_{GL_{2}(K)}(G)\backslash G.Z(K)$ we call $\iota_{g}$
non-trivial. We denote the quotient group $N_{GL_{2}(K)}(G)/G.Z(K)$ by
$Quinn(G)$. It is well-known [Cr] that $Quinn(G)$ is isomorphic to
$\mathop{\mathrm{Cl}}(A)_{2}$. Hence $G$ has non-trivial quasi-inner
automorphisms if and only if $|\mathop{\mathrm{Cl}}(A)|$ is even. Now, as an
element of $GL_{2}(K)$, $\iota_{g}$ acts as a Möbius transformation on the
rational points and elliptic elements of $G$, as well as ${\mathcal{T}}$. In
particular $\overline{g(\omega)}=g(\overline{\omega})$. Since all of these
actions are trivial for scalar matrices they extend to actions of $Quinn(G)$
on $\mathop{\mathrm{Cusp}}(G)$, $\mathop{\mathrm{Ell}}(G)$ and the quotient
graph, $G\backslash{\mathcal{T}}$. In this paper we study of the (often
surprising) properties of these actions.

######  Theorem 1.1.

$Quinn(G)$ acts freely on
(i) $\mathop{\mathrm{Cusp}}(G)$,
(ii) $\mathop{\mathrm{Ell}}(G)$ ($\delta$ odd).

From the above it is clear that $Quinn(G)$ can be embedded as a subgroup
$\mathop{\mathrm{Ell}}(G)^{=}$ (resp. $\mathop{\mathrm{Cl}}(A)_{2}$) of
$\mathop{\mathrm{Ell}}(G)$ (resp. $\mathop{\mathrm{Cusp}}(G)$). We show that
the action of $Quinn(G)$ is equivalent to multiplication by the elements of
the subgroup. The “freeness” in this result follows immediately. Restricting
to these subsets yields stronger results.

######  Corollary 1.2.

$Quinn(G)$ acts freely and transitively on
(i) $\mathop{\mathrm{Cl}}(A)_{2}$,
(ii) $\mathop{\mathrm{Ell}}(G)^{=}$ ($\delta$ odd).

######  Corollary 1.3.

When $\delta$ is odd $Quinn(G)$ acts freely on
$\mathop{\mathrm{Ell}}(G)^{\neq}=\\{G\omega:G\omega\neq
G\overline{\omega}\\}$. Moreover, if $\ker\overline{N}$ has no element of
order $4$, then $Quinn(G)$ acts freely on

$\\{\\{G\omega,G\overline{\omega}\\}:G\omega\in\mathop{\mathrm{Ell}}(G)^{\neq}\\}.$

######  Theorem 1.4.

Every non-trivial element of $Quinn(G)$ determines an automorphism of
$G\backslash{\mathcal{T}}$ of order $2$ which preserves the structure of all
its vertex and edge stabilizers.

Serre [Se, Exercise 2 e), p. 117] states this result for the special case
$K={\mathbb{F}}_{q}(t)$ with $\delta$ even. Our result shows that in general
the quotient graph has symmetries of this type provided
$|\mathop{\mathrm{Cl}}(A)|$ is even. (In general this restriction is
necessary.)

We now list more detailed results on the action of $Quinn(G)$ on
$G\backslash{\mathcal{T}}$. Serre [Se, Theorem 9, p.106] has described the
basic structure of $G\backslash{\mathcal{T}}$. Its ends (i.e. the equivalence
classes of semi-infinite paths without backtracking) are in one-one
correspondence with the elements of $\mathop{\mathrm{Cl}}(A)$. To date the
only cases for which the precise structures of $G\backslash{\mathcal{T}}$ are
known are $g=0$, [Ma2], [KMS], and $g=\delta=1$, [Ta].

######  Theorem 1.5.

$Quinn(G)$ acts freely on the ends of $G\backslash{\mathcal{T}}$ and, in
addition, transitively on the ends of $G\backslash{\mathcal{T}}$ corresponding
to the elements of $\mathop{\mathrm{Cl}}(A)_{2}$,

We show that the ends corresponding to $\mathop{\mathrm{Cl}}(A)_{2}$ are in
one-one correspondence with those vertices whose stabilizers are isomorphic to
$GL_{2}({\mathbb{F}}_{q})$. (Each such vertex is “attached” to the
corresponding end.) It is known [MS3, Corollary 2.12] that if $G_{v}$ contains
a cyclic subgroup of order $q^{2}-1$ then
$G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}\;\mathrm{or}\;GL_{2}({\mathbb{F}}_{q})$.
The building map [Ge, p.41] extends to a map
$\lambda:\mathop{\mathrm{Ell}}(G)\rightarrow\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}})$.
This map leads to another action of $Quinn(G)$ on the quotient graph.

######  Theorem 1.6.

(a) $Quinn(G)$ acts freely and transitively on

$\\{\tilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong
GL_{2}({\mathbb{F}}_{q})\\}.$

(b) Suppose that $\delta$ is odd and that $\ker\overline{N}$ has no element of
order $4$. Then $Quinn(G)$ acts freely on

$\\{\tilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}\\}.$

As an illustration of our results, especially the existence of reflective
symmetries as in Theorem 1.4, we conclude with diagrams of two examples of
$G\backslash{\mathcal{T}}$ for each of which $g=\delta=1$, the so called
“elliptic” case. For these we make use of Takahashi’s paper [Ta]. Special
features of these cases include the following. For part (i) see [MS4, Theorem
5.1].

######  Corollary 1.7.

Suppose that $\delta=1$.

(i) The isolated (i.e. (graph) valency $1$) vertices of
$G\backslash{\mathcal{T}}$ are precisely those whose stabilizers are
isomorphic to $GL_{2}({\mathbb{F}}_{q})$ or ${\mathbb{F}}_{q^{2}}^{*}$.

(ii) If $\ker\overline{N}$ has no element of order $4$ then $Quinn(G)$ acts
freely on the isolated vertices of $G\backslash{\mathcal{T}}$.

By looking at the stabilizers in $G$ of the objects discussed above we obtain
several statements about the action of $Quinn(G)$ on the conjugacy classes of
certain types of subgroups of $G$. (See Sections $3$ and $5$.)
For convenience we begin with a list of notations which will be used
throughout this paper.

${\mathbb{F}}_{q}$ | the finite field with $q=p^{n}$ elements;
---|---
$K$ | an algebraic function field of one variable with constant field ${\mathbb{F}}_{q}$;
$g$ | the genus of $K$;
$\infty$ | a chosen place of $K$;
$\delta$ | the degree of the place $\infty$;
$A$ | the ring of all elements of $K$ that are integral outside $\infty$;
$K_{\infty}$ | the completion of $K$ with respect to $\infty$;
$\Omega$ | Drinfeld’s half-plane;
${\mathcal{T}}$ | the Bruhat-Tits tree of $GL_{2}(K_{\infty})$;
$G$ | the Drinfeld modular group $GL_{2}(A)$;
$Gx$ | the orbit of $x$ under the action of $G$ on the object $x$;
$\widehat{G}$ |  $GL_{2}(K)$;
$Z(K)$ | the set of scalar matrices in $\widehat{G}$;
$Z$ |  $Z(K)\cap G$;
$\widetilde{K}$ | the quadratic constant field extension $K.{\mathbb{F}}_{q^{2}}$;
$\widetilde{A}$ |  $A.{\mathbb{F}}_{q^{2}}$, the integral closure of $A$ in $\widetilde{K}$;
$\mathop{\mathrm{Cl}}(R)$ | the ideal class group of the Dedekind ring $R$;
${\mathop{\mathrm{Cl}}}^{0}(F)$ | the divisor class group of degree $0$ of the function field $F$;
$\mathop{\mathrm{Cusp}}(G)$ |  $G\backslash{\mathbb{P}}^{1}(K)$, the set of cusps of $G$;
$E(A)$ | the set of elliptic elements of $G$:
$\mathop{\mathrm{Ell}}(G)$ |  $G\backslash E(A)$, the set of elliptic points of $G$;
$\overline{\omega}$ | the image of $\omega\in E(A)$ under the Galois automorphism of $\widetilde{K}/K$;
$\mathop{\mathrm{Ell}}(G)^{=}$ |  $\\{G\omega:\omega\in E(A),G\omega=G\overline{\omega}\\}$;
$\mathop{\mathrm{Ell}}(G)^{\neq}$ |  $\mathop{\mathrm{Ell}}(G)\backslash\mathop{\mathrm{Ell}}(G)^{=}$;
$S(s)$ | the stabilizer in a finite index subgroup $S$ (of $G$) of $s\in{\mathbb{P}}^{1}(K)$;
$G^{\omega}$ | the stabilizer in $G$ of $\omega\in C_{\infty}\backslash K$;
$S_{w}$ | the stabilizer in $S$ of $w\in\mathop{\mathrm{vert}}({\mathcal{T}})\cup\mathop{\mathrm{edge}}({\mathcal{T}})$;
$\mathcal{H}$ |  $\\{H\leq G:H\cong GL_{2}({\mathbb{F}}_{q})\\}$ ;
$\mathcal{C}$ |  $\\{C\leq G:C\cong{\mathbb{F}}_{q^{2}}^{*}\\}$ ;
$\mathcal{C}_{mf}$ |  $\\{C\in\mathcal{C}:C\;\mathrm{maximally\;finite\;in}\;$G$\\}$;
$\mathcal{C}_{nm}$ |  $\mathcal{C}\backslash\mathcal{C}_{mf}$;
$\mathcal{V}$ |  $\\{\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\in\mathcal{C}\\}$;

##  2\. Quasi-inner automorphisms

Let $F$ be any field containing $A$ (and hence $K$) and let $Z(F)$ denote the
set of scalar matrices in $GL_{2}(F)$. We are interested here in automorphisms
of $G$ arising from conjugation by a non-scalar element of $GL_{2}(F)$. We
first show this problem reduces to $N_{\widehat{G}}(G)$, the normalizer of $G$
in $\widehat{G}=GL_{2}(K)$. For each $x\in F$ we use $(x)$ as a shorthand for
the fractional ideal $Ax$.

######  Lemma 2.1.

Let $M_{0}\in GL_{2}(F)$ normalize $G$. Then

$M_{0}\in Z(F).N_{\widehat{G}}(G).$

###### Proof.

Let

$M_{0}=\left[\begin{array}[]{cc}\alpha&\beta\\\
\gamma&\delta\end{array}\right].$

Suppose that $\gamma\neq 0$. Replacing $M_{0}$ with $\gamma^{-1}M_{0}$ we may
assume that $\gamma=1$. Now

$NT(1)N^{-1}\in G,$

where $N=M_{0}^{\pm 1}$. It follows that $\mathop{\mathrm{det}}(M_{0})$,
$\alpha$, $\delta\in K$ and hence that
$\beta=\alpha\delta-\mathop{\mathrm{det}}(M_{0})\in K$. The proof for the case
where $\gamma=0$ is similar. ∎

We state a special case $(n=2)$ of a result of Cremona [Cr] .

######  Theorem 2.2.

Let

$M=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in\widehat{G}$

and define

${\mathfrak{q}}(M):=(a)+(b)+(c)+(d).$

Then $M\in N_{\widehat{G}}(G)$ if and only if

${\mathfrak{q}}(M)^{2}=(\varDelta),$

where $\varDelta=\mathop{\mathrm{det}}(M)$.

######  Corollary 2.3.

Let $M\in N_{\widehat{G}}(G)$ with $\varDelta=\mathop{\mathrm{det}}(M)$.

  * (i)

$\varDelta^{-1}M^{2}\in SL_{2}(A).$

  * (ii)

If $\varDelta\in A^{*}$, then $M\in G$.

###### Proof.

(i) By Theorem 2.2 every entry of $M^{2}$ is in
${\mathfrak{q}}(M)^{2}=(\varDelta)$.

For part (ii) let $x$ be any entry of $M$. Then $x^{2}\in A$ by Theorem 2.2
and so $x\in A$, since $A$ is integrally closed. ∎

Another important consequence [Cr] of Theorem 2.2 is the following.

######  Theorem 2.4.

The map $M\mapsto{\mathfrak{q}}(M)$ induces an isomorphism

$N_{\widehat{G}}(G)/Z(K).G\;\cong\;\mathrm{Cl}(A)_{2},$

where $\mathrm{Cl}(A)_{2}$ is the subgroup of all involutions in
$\mathrm{Cl}(A)$.

###### Proof.

This is another special case ($n=2$) of a result in [Cr]. If
$\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in N_{\widehat{G}}(G)$,
it can be shown [Cr, Remarks $2$] that

$(a)+(b)=(a)+(c)=(d)+(b)=(d)+(c)={\mathfrak{q}}(M).$

Consequently there is a map from $N_{\widehat{G}}(G)$ to
${\mathop{\mathrm{Cl}}}(A)_{2}$, which turns out to be an isomorphism. ∎

######  Definition 2.5.

An automorphism $\iota_{g}$ of $G$ is called quasi-inner if

$\iota_{g}(x)=gxg^{-1}\;(x\in G),$

for some $g\in N_{\widehat{G}}(G)$. We call $\iota_{g}$ non-trivial if
$g\notin Z(K).G$, i.e. if $\iota_{g}$ does not act like an inner automorphism.
We note that

$\iota_{g_{1}}=\iota_{g_{2}}\Leftrightarrow g_{1}g_{2}^{-1}\in Z(K).$

Finally we define

$Quinn(G):=N_{\widehat{G}}(G)/Z(K).G\cong\mathop{\mathrm{Cl}}(A)_{2}.$

So $Quinn(G)$ is the group of quasi-inner automorphisms modulo the inner ones.
We note that in particular all quasi-inner automorphisms of $G$ act like inner
automorphisms if $|\mathop{\mathrm{Cl}}(A)|$ is odd.

Let $\mathrm{Cl}^{0}(K)$ be the group of divisor classes of degree zero [St,
p.186]. It is known [Se, p.104] that the following exact sequence holds

$0\;\rightarrow\;{\mathop{\mathrm{Cl}}}^{0}(K)\;\rightarrow\;\mathop{\mathrm{Cl}}(A)\;\rightarrow\;{\mathbb{Z}}/\delta{\mathbb{Z}}\;\rightarrow\;0.$
(1)

Our next result is an immediate consequence of Theorem 2.4.

######  Corollary 2.6.

$G$ has non-trivial quasi-inner automorphisms if and only if

$|\mathrm{Cl}(A)|=\delta|\mathrm{Cl}^{0}(K)|\;is\;even.$

######  Example 2.7.

We illustrate the results of this section with the simplest case
$K={\mathbb{F}}_{q}(t)$, the rational function field over ${\mathbb{F}}_{q}$.
Then there exists a (monic) polynomial $\pi(t)\in{\mathbb{F}}_{q}[t]$, of
degree $\delta$, irreducible over ${\mathbb{F}}_{q}$, such that

$A=\left\\{\frac{f}{\pi^{m}}:f\in k[t],\;m\geq 0,\;\deg f\leq\delta
m\right\\}.$

It is known [St, Theorem 5.1.15, p.193] that here $\mathrm{Cl}^{0}(K)$ is
trivial so that

$\mathop{\mathrm{Cl}}(A)\cong{\mathbb{Z}}/\delta{\mathbb{Z}}.$

Hence $G$ has non-trivial quasi-inner automorphisms if and only if $\delta$ is
even. Hence here either $Quinn(G)$ is trivial or cyclic of order $2$.

For a specific illustration of Theorem 2.4 we restrict further to $\delta=2$.
In this case $\pi(t)=t^{2}+\sigma t+\tau$, where $\sigma\in{\mathbb{F}}_{q}$
and $\tau\in{\mathbb{F}}_{q}^{*}$. We begin with the $A$-ideal generated by
$\pi^{-1}$ and $t\pi^{-1}$ which is not principal. Let
$\pi(t)=tt^{\prime}+\tau$ and put

$g_{0}=\left[\begin{array}[]{lll}\tau&t\\\\[10.0pt]
-t^{\prime}&1\end{array}\right].$

Then by Theorem 2.2 $g_{0}\in N_{\widehat{G}}(G)$ and from Theorem 2.4 we see
$g_{0}\notin Z(K).G$. Hence $g_{0}$ provides a generator of
$\mathop{\mathrm{Cl}}(A)_{2}$.

######  Remark 2.8.

From the theory of Jacobian varieties we know that the $2$-torsion in
$\mathrm{Cl}^{0}(K)$ is bounded by $2^{2g}$, and even by $2^{g}$ if the
characteristic of $K$ is $2$ [Ro, Theorem 11.12]. Hence by the exact sequence
(1) it follows that $|Quinn(G)|=|\mathrm{Cl}(A)_{2}|\leq 2^{2g+1}$ (and $\leq
2^{g+1}$, when $\mathrm{char}(K)=2$).

In odd characteristic we can easily find examples with
$|\mathop{\mathrm{Cl}}(A)_{2}|=2^{2g}$, provided we are willing to accept a
big constant field. Given a function field $F$ of genus $g$ with constant
field ${\mathbb{F}}_{p^{r}}$, just pick $q=p^{rn}$ such that all $2$-torsion
points of $Jac(F)$ are ${\mathbb{F}}_{q}$-rational and consider
$K=F.{\mathbb{F}}_{q}$. Then
$\mathop{\mathrm{Cl}}^{0}(K)_{2}\cong({\mathbb{Z}}/2{\mathbb{Z}})^{2g}$.
Choosing a place $\infty$ of $K$ of odd degree $\delta$, from the exact
sequence (1), we see that $|\mathrm{Cl}(A)_{2}|=2^{2g}$.

Similarly in characteristic $2$ examples for which
$|\mathop{\mathrm{Cl}}(A)_{2}|=2^{g}$ can be found by choosing $F$ suitably,
namely $F$ has to be ordinary.
Whether for even $\delta$ one can reach the bound $2^{2g+1}$ (resp. $2^{g+1}$)
depends on whether or not the induced short exact sequence for the Sylow
$2$-subgroup of $\mathrm{Cl}(A)$ splits or not.

######  Definition 2.9.

Let $R,\;S$ be subgroups of a group $T$. We write

$R\sim S$

if and only if $R=S^{t}=tSt^{-1}$, for some $t\in T$. We put

$R^{T}=\left\\{R^{t}:t\in T\right\\}.$

Let $\mathcal{S}$ be a set of subgroups of $T$. We put

$\mathcal{S}^{G}=\left\\{S^{G}:S\in\mathcal{S}\right\\}.$

This paper is principally concerned with various actions of $Quinn(G)$. It is
appropriate at this point to describe in detail the most important of these.
Let $\iota_{g}$ be as above.

(i) It is clear that $GL_{2}(K_{\infty})$ acts on $\Omega$ as Möbius
transformations and that this action is trivial for all scalar matrices. Then
$\iota_{g}$ acts on $E(G)$ since, for all $\omega\in E(G)$,

$G^{g(\omega)}=(G^{\omega})^{g}(\leq G).$

Recall that $\mathop{\mathrm{Ell}}(G)=\\{G\omega:\omega\in E(G)\\}$. The map

$G\omega\mapsto Gg(\omega)$

extends naturally to a well-defined action of $Quinn(G)$ on
$\mathop{\mathrm{Ell}}(G)$.
(ii) Clearly $G$ acts as Möbius transformations on ${\mathbb{P}}^{1}(K)$ and
it is well-known that

$G\backslash{\mathbb{P}}^{1}(K)\leftrightarrow\mathop{\mathrm{Cl}}(A).$

As we shall see later from the structure of the quotient graph it follows
that, for all $k\in{\mathbb{P}}^{1}(K)$, $G(k)$ is infinite, metabelian.
Recall that $\mathop{\mathrm{Cusp}}(G)=\\{Gk:k\in{\mathbb{P}}^{1}(K)\\}$. As
before the map

$Gk\mapsto Gg(k)$

extends to a well-defined action of $Quinn(G)$ on $\mathop{\mathrm{Cusp}}(G)$.
(iii) Serre [Se, Chapter II, Section 1.1, p.67] uses lattice classes as a
model for the vertices and edges of ${\mathcal{T}}$. It is clear that
$GL_{2}(K_{\infty})$ acts naturally on these. In particular the scalar
matrices act trivially. The map

$Gw\mapsto Gg(w),$

where
$w\in\mathop{\mathrm{vert}}({\mathcal{T}})\cup\mathop{\mathrm{edge}}({\mathcal{T}})$,
extends to a well-defined action of $Quinn(G)$ on the quotient graph
$G\backslash{\mathcal{T}}$. Note that $G_{g(w)}=((G_{w}))^{g}\leq G$. We will
use this action to extend a result of Serre.
(iv) Suppose that

$\mathcal{S}=\left\\{H\leq G:H\cong GL_{2}({\mathbb{F}}_{q})\right\\}$

or $\mathcal{S}$ is a $G$-conjugacy closed subset of
$\mathcal{C}=\left\\{C\leq G:C\cong{\mathbb{F}}_{q^{2}}^{*}\right\\}$. Then
$Quinn(G)$ acts by conjugation on $\mathcal{S}^{G}$. We use these to define
actions of $Quinn(G)$ on significant subsets of
$\mathop{\mathrm{vert}}({\mathcal{T}})$.

##  3\. Action on vertex stabilizers

Almost all the results in this section hold for all $\delta$. We record the
important general properties of subgroups of vertex stabilizers.

######  Lemma 3.1.

(i) $G_{v}$ is finite, for all $v\in\mathop{\mathrm{vert}}({\mathcal{T}})$.

(ii) Let $S$ be a finite subgroup of $G$. Then

$S\leq G_{v_{0}},$

for some $v_{0}\in\mathop{\mathrm{vert}}({\mathcal{T}})$.

###### Proof.

See [Se, Proposition 2, p.76]. ∎

In this section we are concerned with subgroups of $G_{v}$ which contain a
cyclic subgroup of order $q^{2}-1$. We record the following result.

######  Lemma 3.2.

Suppose that $G_{v}$ contains a cyclic subgroup of order $q^{2}-1$. Then

$G_{v}\cong GL_{2}({\mathbb{F}}_{q})\;or\;G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}.$

###### Proof.

See [MS3, Corollaries 2.2, 2.4, 2.12]. ∎

In the first part of this section we look at the action of quasi-inner
automorphisms on the following set

$\mathcal{H}=\left\\{H\leq G:H\cong GL_{2}({\mathbb{F}}_{q})\right\\}.$

######  Lemma 3.3.

Let $H\in\mathcal{H}$. Then there exists
$v_{0}\in\mathop{\mathrm{vert}}({\mathcal{T}})$ for which

$H=G_{v_{o}}.$

###### Proof.

Follows from Lemmas 3.1 (ii) and 3.2. ∎

######  Remark 3.4.

(i) Every ${\mathcal{T}}$ contains a particular vertex $v_{s}$, usually
referred to as standard (after Serre), for which

$G_{v_{s}}=GL_{2}({\mathbb{F}}_{q}).$

See [Se, Remark 3), p.97]
(ii) On the other hand for the case $A={\mathbb{F}}_{q}[t]$ (equivalently
$g(K)=0,\;\delta=1$) it follows from Nagao’s Theorem [Se, Corollary, p.87]
that here $\mathop{\mathrm{vert}}({\mathcal{T}})$ has no stabilizer which is
cyclic of order $q^{2}-1$.

######  Lemma 3.5.

Let $H\in\mathcal{H}$. Then there exists a quasi-inner automorphism
$\kappa=\iota_{g}$ of $G$ such that

$H=\kappa(GL_{2}({\mathbb{F}}_{q})).$

###### Proof.

From the proofs of [MS3, Theorem 2.6, Corollary 2.8], as well as [MS3,
Corollary 2.12] it is clear that there exists

$g=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in
GL_{2}(\widetilde{K})$

such that

$H=g(GL_{2}({\mathbb{F}}_{q}))g^{-1}.$

We denote by $\overline{x}$ the image of $x\in\widetilde{K}$ under the
extension of the Galois automorphism of
${\mathbb{F}}_{q^{2}}/{\mathbb{F}}_{q}$ to $\widetilde{K}$. It is clear that
$gE_{ij}g^{-1}\in M_{2}(A)$, where $1\leq i,j\leq 2$ and so

$xy/\varDelta(=\overline{xy}/\overline{\varDelta})\in A,$

for all $x,y\in\\{a,b,c,d\\}$, where $\varDelta=\mathop{\mathrm{det}}(g)$.

Now we may assume without loss of generality that $c\neq 0$. Let
$z\in\\{a,b,d\\}$. Then

$c^{2}/\varDelta=\overline{c}^{2}/\overline{\varDelta},\;\;\;cz/\varDelta=\overline{cz}/\overline{\varDelta}.$

It follows that $z/c=\overline{z}/\overline{c}$ so that $z/c\in K$. We now
replace $g=M$ with $g_{0}=c^{-1}M$. Then by Theorem 2.2 the map
$\kappa_{0}:G\rightarrow G$ defined by $\kappa_{0}(x)=g_{0}xg_{0}^{-1}$ is a
quasi-inner automorphism of $G$. ∎

######  Lemma 3.6.

Let $\kappa_{0}=\iota_{g_{o}}$ be a non-trivial quasi-inner automorphism of
$G$ and let $H\in\mathcal{H}$. Then

$\kappa_{0}(H)\not\sim H.$

###### Proof.

By definition $g_{0}\in N_{\widehat{G}}(G)\backslash G.Z(K)$. Suppose to the
contrary that

$\kappa_{0}(H)=gHg^{-1}$

for some $g\in G$. Replacing $g_{0}$ with $g^{-1}g_{0}$ we may assume that
$g=1$. Now by Lemma 3.5 $H=\kappa_{0}^{\prime}(GL_{2}({\mathbb{F}}_{q}))$ for
some quasi-inner $\kappa_{0}^{\prime}=\iota_{g_{0}^{\prime}}$, say. It follows
that

$g_{1}(GL_{2}({\mathbb{F}}_{q}))g_{1}^{-1}=GL_{2}({\mathbb{F}}_{q}),$

where $g_{1}=(g_{0}^{\prime})^{-1}g_{0}g_{0}^{\prime}$. As
$N_{\widehat{G}}(G)/G.Z(K)$ is abelian this implies that

$g_{1}\equiv g_{0}\;(\mathop{\mathrm{mod}}Z(K).G)$

and so we may further assume that $g_{1}=g_{0}$. Let

$S_{p}=\left\\{T(a)=E_{12}(a):a\in{\mathbb{F}}_{q}\right\\}.$

Now $S_{p}$ is a Sylow $p$-subgroup of $GL_{2}({\mathbb{F}}_{q})$ and so from
the above

$g_{0}(S_{p})g_{0}^{-1}=h(S_{p})h^{-1},$

for some $h\in GL_{2}({\mathbb{F}}_{q})$. As above we may assume then that
$h=1$. It follows that $g_{0}$ “fixes” $\infty$ and so

$g_{0}=\left[\begin{array}[]{cc}\alpha&*\\\ 0&\beta\end{array}\right].$

By Corollary 2.3 (i) we note that

$(\mathop{\mathrm{det}}(g_{0})^{-1}\mathrm{tr}((g_{0})^{2})=\gamma+\gamma^{-1}\in
A,$

where $\gamma=\alpha\beta^{-1}$. Since $A$ is integrally closed it follows
that $\gamma\in A^{*}(={\mathbb{F}}_{q}^{*})$. Then we can replace $g_{0}$
with $\beta^{-1}g_{0}$ which belongs to $G$ by Corollary 2.3 (ii). Thus
$g_{0}\in Z(K).G$. ∎

######  Lemma 3.7.

Let $e\in\mathop{\mathrm{edge}}({\mathcal{T}})$ be incident with $v_{s}$. Then

$G_{e}\lneqq GL_{2}({\mathbb{F}}_{q}).$

###### Proof.

The edges attached to $v_{s}$ are parametrized by
${\mathbb{P}}^{1}({\mathbb{F}}_{q^{\delta}})$ and $GL_{2}({\mathbb{F}}_{q})$
acts on these as Möbius transformations. See [Se, Exercise 6), p.99].
If the edge corresponds to $f\in{\mathbb{F}}_{q^{\delta}}$ it is not fixed by
the translations in $GL_{2}({\mathbb{F}}_{q})$, and if it corresponds to
$\infty$, it is not fixed by ${0\ 1\choose 1\ 0}\in GL_{2}({\mathbb{F}}_{q})$.
∎

######  Proposition 3.8.

No edge of ${\mathcal{T}}$ can have a stabilizer isomorphic to
$GL_{2}({\mathbb{F}}_{q})$.

###### Proof.

For odd $\delta$ this follows from [MS3, Corollary 2.16]. We provide a proof
that holds for all $\delta$. Suppose to the contrary that there is an edge $e$
whose stabilizer is isomorphic to $GL_{2}({\mathbb{F}}_{q})$. Then by Lemma
3.2 the stabilizers of its terminal vertices are both $G_{e}$.
By Lemma 3.5 and the action of quasi-inner automorphisms on ${\mathcal{T}}$ we
can assume that

$G_{e}=GL_{2}({\mathbb{F}}_{q}).$

It follows that $GL_{2}({\mathbb{F}}_{q})$ stabilizes the geodesic from
$v_{s}$ to one of the terminal vertices of $e$ which includes $e$ and hence an
edge incident with $v_{s}$. This contradicts Lemma 3.7. ∎

######  Corollary 3.9.

Let $H\in\mathcal{H}$. Then there exists a unique vertex
$v\in\mathop{\mathrm{vert}}({\mathcal{T}})$ such that

$G_{v}=H.$

###### Proof.

Follows from Lemma 3.3 and Proposition 3.8. ∎

######  Remark 3.10.

Another interesting consequence of Lemma 3.5 and Proposition 3.8 is the
following. Suppose that $G_{v}\in\mathcal{H}$. Then there exists
$\kappa=\iota_{g}$ such that $\kappa(v)=v_{s}$ Since $\kappa$ is an
automorphism of ${\mathcal{T}}$ the action of $G_{v}$ on the $q^{\delta}+1$
edges of ${\mathcal{T}}$ incident with $v$ is identical to the action of
$GL_{2}({\mathbb{F}}_{q})$ on the edges of ${\mathcal{T}}$ incident with
$v_{s}$ as described in Lemma 3.7.

######  Definition 3.11.

By definition

$\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}})=\left\\{Gv:v\in\mathop{\mathrm{vert}}({\mathcal{T}})\right\\}.$

We put $\widetilde{v}=Gv$ and define its stabilizer

$G_{\widetilde{v}}=(G_{v})^{G}.$

We refer to $G_{\widetilde{v}}$ as being isomorphic to $G_{v}$.

######  Lemma 3.12.

There exists a bijection

$\mathcal{H}^{G}\longleftrightarrow\left\\{\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\in\mathcal{H}\right\\}.$

###### Proof.

Follows from Corollary 3.9 and the above. ∎

It is clear that $Quinn(G)$ acts on $\mathcal{H}^{G}$. Since $Z(K)$,
represented by scalar matrices, acts trivially on ${\mathcal{T}}$, it is also
clear that $Quinn(G)$ acts on $G\backslash{\mathcal{T}}$. We now come to the
principal result in this section which follows from Lemmas 3.5, 3.6 and 3.12.

######  Theorem 3.13.

$Quinn(G)$ acts freely and transitively on

  * (i)

the conjugacy classes of subgroups of $G$ which are isomorphic to
$GL_{2}({\mathbb{F}}_{q})$,

  * (ii)

the vertices of $G\backslash{\mathcal{T}}$ whose stabilizers are isomorphic to
$GL_{2}({\mathbb{F}}_{q})$ .

A special case of this result is provided by Corollary 2.6.

######  Corollary 3.14.

Suppose that $|\mathop{\mathrm{Cl}}(A)|$ is odd. Then

  * (i)

every subgroup $H$ of $G$ isomorphic to $GL_{2}({\mathbb{F}}_{q})$ is actually
conjugate in $G$ to $GL_{2}({\mathbb{F}}_{q})$,

  * (ii)

the only vertex in $G\setminus{\mathcal{T}}$ whose stabilizer is isomorphic to
$GL_{2}({\mathbb{F}}_{q})$ is $\widetilde{v_{s}}$, the image of the standard
vertex $v_{s}$.

##  4\. Action on elliptic points

Throughout this section we assume that $\delta$ is odd. Recall that

$\mathop{\mathrm{Ell}}(G)=\left\\{G\omega:\omega\in E(G)\right\\}$

denotes the elliptic points of the Drinfeld modular curve $G\backslash\Omega$.

######  Definition 4.1.

We define

$\mathop{\mathrm{Ell}}(G)^{=}=\left\\{G\omega:G\omega=G\overline{\omega}\right\\}\;\mathrm{and}\;\mathop{\mathrm{Ell}}(G)^{\neq}=\left\\{G\omega:G\omega\neq
G\overline{\omega}\right\\}.$

(In [MS4, Section 3] $\mathop{\mathrm{Ell}}(G)^{=}$ is denoted by
$\mathop{\mathrm{Ell}}(G)_{2}$.)
The action of an element of $GL_{2}(K_{\infty})$ on an element of $\Omega$
will always refer to its action as a Möbius transformation. We record the
following.

######  Lemma 4.2.

Let $g\in N_{\widehat{G}}(G)$ and $\omega\in E(A)$. Then

  * (i)

$g(\omega)\in E(A)$,

  * (ii)

$\overline{g(\omega)}=g(\overline{\omega}).$

It is clear then that $Quinn(G)$ acts on both $\mathop{\mathrm{Ell}}(G)^{=}$
and $\mathop{\mathrm{Ell}}(G)^{\neq}$.
In this section our approach is based on [MS4, Sections 3,4]. We recall some
details.

######  Definition 4.3.

Let $I$ be an $A$-ideal (resp. $\widetilde{A}$-ideal). Then $[I]$ denotes the
image of $I$ in $\mathop{\mathrm{Cl}}(A)$ (resp.
$\mathop{\mathrm{Cl}}(\widetilde{A}))$.

Fix $\varepsilon\in{\mathbb{F}}_{q^{2}}\setminus{\mathbb{F}}_{q}$. By [MS4,
Theorem 2.5] any elliptic point $\omega$ of $G$ can be written as
$\omega=\frac{\varepsilon+s}{t}$ where $s,t\in A$ and $t$ divides
$(\overline{\varepsilon}+s)(\varepsilon+s)$ in $A$. Now let

$J_{\omega}=A(\varepsilon+s)+At.$

It is known [MS4, Lemmas 3.1, 3.2] that

  * (i)

$J_{\omega}$ is an $\widetilde{A}$-ideal.

  * (ii)

$J_{\omega}$ is independent of the choice of
$\varepsilon\in{\mathbb{F}}_{q^{2}}\setminus{\mathbb{F}}_{q}$.

  * (iii)

Let $\omega,\omega^{\prime}\in E(A)$. Then

$G\omega=G\omega^{\prime}\Longleftrightarrow[J_{\omega}]=[J_{\omega^{\prime}}]\;\;\;(\mathrm{in}\;\mathop{\mathrm{Cl}}(\widetilde{A})).$

Let $\alpha$ be the Galois automorphism of $\widetilde{K}/K$ (which extends
that of ${\mathbb{F}}_{q^{2}}/{\mathbb{F}}_{q}$). Let $k\in\widetilde{K}$.
Then the norm of $k$ is $k\overline{k}$, where $\overline{k}=\alpha(k)$. Now
$\alpha$ restricts to $\widetilde{A}$ and so acts on its ideals and hence its
ideal class group. For each $\widetilde{A}$-ideal, $J$, the norm of $J$,
$N(J)=A\cap(J\bar{J})$, which is an $A$-ideal. We now come to the norm map

$\overline{N}:\mathop{\mathrm{Cl}}(\widetilde{A})\rightarrow\mathop{\mathrm{Cl}}(A),$

where $\overline{N}([I])=[(I\bar{I})\cap A]$. Then

$[I]\in\mathrm{ker}\;\overline{N}\Longleftrightarrow(I\bar{I})\cap
A\;\mathrm{is\;a\;principal}\;A\mathrm{-ideal}.$

We restate [MS4, Theorem 3.4].

######  Theorem 4.4.

The map $\omega\mapsto[J_{\omega}]$ induces a one-one correspondence

$\mathop{\mathrm{Ell}}(G)\longleftrightarrow\mathrm{ker}\;\overline{N}.$

For each $\omega$ it is known that

  * (i)

$\overline{J_{\omega}}=J_{\overline{\omega}}$,

  * (ii)

$J_{\omega}J_{\overline{\omega}}$ is a principal $A$-ideal.

It follows that

$\mathrm{ker}\;\overline{N}=\left\\{[J_{\omega}]:[J_{\overline{\omega}}]=[J_{\omega}]^{-1}\right\\}.$

We recall from Theorem 2.4 that $Quinn(G)$ can be identified with
$\mathop{\mathrm{Cl}}(A)_{2}$. From this and Theorem 4.4 we are able to study
the action of $Quinn(G)$ on $\mathop{\mathrm{Ell}}(G)$. For this purpose we
require two further lemmas.

######  Lemma 4.5.

Let
$\iota:\mathop{\mathrm{Cl}}(A)\rightarrow\mathop{\mathrm{Cl}}(\widetilde{A})$
be the canonical map, where $\iota([I])=[I\widetilde{A}],\;(I\unlhd A).$ Then

  * (i)

$\iota$ is injective.

  * (ii)

$\left\\{[I]\in\mathop{\mathrm{Cl}}(\widetilde{A});[I]=[\overline{I}]\right\\}=\iota(\mathop{\mathrm{Cl}}(A))$.

###### Proof.

The analagous statements are known to hold for the canonical map from
$\mathop{\mathrm{Cl}}^{0}(K)\to\mathop{\mathrm{Cl}}^{0}(\widetilde{K})$. See
[Ro, Corollary to Proposition 11.10]. The results follow from the exact
sequence in Section 2, since $\delta$ is odd and the infinite place is inert
in $\widetilde{K}$. ∎

######  Lemma 4.6.

With the above notation, the $2$-torsion in $\mathop{\mathrm{Cl}}(A)$,

$\mathop{\mathrm{Cl}}(A)_{2}\cong\iota(\mathop{\mathrm{Cl}}(A)_{2})=(\mathrm{ker}\;\overline{N})_{2},$

the $2$-torsion in $\mathrm{ker}\;\overline{N}$.

###### Proof.

Let $[I]\in\mathop{\mathrm{Cl}}(A)_{2}$. Then $\iota([I])$ has order $2$ in
$\mathop{\mathrm{Cl}}(\widetilde{A})$ by Lemma 4.5. Now

$\overline{N}(\iota([I]))=\iota([I])\overline{\iota([I])}=(\iota([I]^{2})=1,$

by Lemma 4.5 (ii). Hence $\iota([I])\in\mathrm{ker}\;\overline{N}$. Conversely
let $[J]\in Cl(\widetilde{A})$ have order $2$ and lie in
$\mathrm{ker}\;\overline{N}$. Then $[J]^{2}=1$ and
$[J][\overline{J}]=\overline{N}([J])=1$. Hence $[J]=[\overline{J}]$ and so
$[J]\in\iota(\mathop{\mathrm{Cl}}(A)_{2})$ again by Lemma 4.5 (ii). ∎

Any element of $N_{\widehat{G}}(G)$ can be represented by a matrix

$M=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in\widehat{G}.$

By multiplying $M$ by a suitable scalar matrix we may assume that $a,b,c,d\in
A$. As before let

${\mathfrak{q}}(M):=(a)+(b)+(c)+(d).$

Then

  * (i)

${\mathfrak{q}}(M)^{2}=(\varDelta).$

  * (ii)

$(a)+(b)=(a)+(c)=(d)+(b)=(d)+(c)={\mathfrak{q}}(M).$

See Theorem 2.2 and [Cr, Remarks $2$]. In this way ${\mathfrak{q}}$ induces an
isomorphism from $Quinn(G)$ onto $\mathop{\mathrm{Cl}}(A)_{2}$ and so
$\iota\circ{\mathfrak{q}}$ provides an embedding of $Quinn(G)$ into
$\mathop{\mathrm{Cl}}(\widetilde{A})$. As before each $\omega\in E(G)$ can be
represented as $\omega=\frac{\varepsilon+s}{t}$ where $s,t\in A$ and $t$
divides $(\varepsilon^{q}+s)(\varepsilon+s)$ in $A$. The element $M$ acts as a
Möbius transformation on $\omega$ by multiplying the column vector
${\varepsilon+s\choose t}$ on the left by the matrix $M$. It follows that
$J_{M(\omega)}$ is the $\widetilde{A}$-ideal generated by
$a(\varepsilon+s)+bt$ and $c(\varepsilon+s)+dt$. Our next result, the most
important in this section, shows that the action of $Quinn(G)$ on
$\mathop{\mathrm{Ell}}(G)$ is equivalent to group multiplication in
$\mathrm{ker}\;\overline{N}$.

######  Theorem 4.7.

With the above notation,

$[J_{M(\omega)}]=[\iota({\mathfrak{q}}(M))J_{\omega}]=[\iota({\mathfrak{q}}(M))][J_{\omega}]\;\;\;(\mathrm{in}\;\ker\;\overline{N}).$

###### Proof.

From the above it is clear that
$J_{M(\omega)}\leq{\mathfrak{q}}(M)J_{\omega}$. Since $\widetilde{A}$ is a
Dedekind domain, there is an integral ideal $I_{1}$ of $\widetilde{A}$ such
that

$J_{M(\omega)}={\mathfrak{q}}(M)J_{\omega}I_{1}.$

By the same argument there exists an integral ideal $I_{2}$ of $\widetilde{A}$
with

$J_{M^{2}(\omega)}={\mathfrak{q}}(M)J_{M(\omega)}I_{2}={\mathfrak{q}}(M)^{2}J_{\omega}I_{1}I_{2}=\varDelta
J_{\omega}I_{1}I_{2}.$

On the other hand, from part (i) of Corollary 2.3 we see that
$J_{M^{2}(\omega)}=\varDelta J_{\omega}$. Hence $I_{1}=I_{2}=\widetilde{A}$
and the result follows. ∎

An immediate consequence is the following.

######  Corollary 4.8.

$Quinn(G)$ acts freely on $Ell(G)$. More precisely, a quasi-inner automorphism
that fixes an elliptic point in $G\setminus\Omega$ must necessarily be inner.

Since

$G\omega=G\overline{\omega}\Longleftrightarrow[J_{\overline{\omega}}]=[J_{\omega}]=[J_{\omega}]^{-1}$

we can identify $\mathop{\mathrm{Ell}}(G)^{=}$ with
$\iota(\mathop{\mathrm{Cl}}(A)_{2})\cong Quinn(G))$. Combining Lemma 4.6 and
Corollary 4.8 we obtain the following result.

######  Theorem 4.9.

$Quinn(G)$ acts freely and transitively on $\mathop{\mathrm{Ell}}(G)^{=}$.

Theorem 3.13 (ii), which holds for all $\delta$, provides an alternative proof
of Theorem 4.9. Applying the former for the case of odd $\delta$ the latter
then follows from the existence of a $Quinn(G)$-invariant one-one
correspondence between $\mathop{\mathrm{Ell}}(G)^{=}$ and
$\left\\{\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong
GL_{2}({\mathbb{F}}_{q})\right\\}$.

From the above it is clear that
$|\mathop{\mathrm{Ell}}(G)|=n_{E}|\mathop{\mathrm{Ell}}(G)^{=}|$, where

$n_{E}=|\ker\;\overline{N}:\iota(\mathop{\mathrm{Cl}}(A)_{2})|.$

It follows that
$|\mathop{\mathrm{Ell}}(G)^{\neq}|=(n_{E}-1)|\mathop{\mathrm{Ell}}(G)^{=}|$.

We recall that the building map [Ge, p.41] restricts to a map

$\lambda:E(G)\rightarrow\mathop{\mathrm{vert}}({\mathcal{T}}),$

for which $G^{\omega}\leq G_{\lambda(\omega)}$. Let $\kappa$ be a quasi-inner
automorphism. Then by [Ge, (iii), p.44]

$\lambda(\kappa(\omega))=\kappa(\lambda(\omega)).$

Then $\lambda$ induces a map

$\mathop{\mathrm{Ell}}(G)\mapsto\mathop{\mathrm{vert}}{{\mathcal{T}}}.$

By Lemma 4.2 (ii), Theorem 3.13 and [MS4, Proposition 3.4] this leads to two
$Quinn(G)$-invariant one-one correspondences.

$\mathop{\mathrm{Ell}}(G)^{=}\longleftrightarrow\left\\{\tilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong
GL_{2}({\mathbb{F}}_{q})\right\\},$
$\mathcal{G}=\left\\{\\{G\omega,G\overline{\omega}\\}:G\omega\neq
G\overline{\omega}\right\\}\longleftrightarrow\mathcal{V}=\left\\{\tilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}\right\\}.$

Note that $|\mathcal{G}|=\frac{1}{2}|\mathop{\mathrm{Ell}}(G)^{\neq}|$.

######  Lemma 4.10.

Let $G\omega\in\mathop{\mathrm{Ell}}(G)^{\neq}$ and $\kappa$ be a quasi-inner
automorphism represented by $M\in N_{\widehat{G}}(G)$. Then
$\kappa(G\omega)=G\overline{\omega}$ if and only if $[J_{\omega}]$ has order
$4$ in $\mathrm{ker}\;\overline{N}$ and
$\iota({\mathfrak{q}}(M))]=[J_{\omega}]^{2}$.

###### Proof.

Let $n>2$ be the order of $[J_{\omega}]$ in $\mathrm{ker}\;\overline{N}$. If
$\kappa(G\omega)=G\overline{\omega}$ then by Theorem 4.7

$[\iota({\mathfrak{q}}(M)][J_{\omega}]=[J_{\omega}]^{n-1}.$

Hence $[\iota({\mathfrak{q}}(M)]=[J_{\omega}]^{n-2}=[J_{\omega}]^{-2}$ and so
$n=4$. The converse is straightforward. ∎

The following is an immediate consequence.

######  Lemma 4.11.

Let $\widetilde{v}\in\mathcal{V}$ and let $\\{G\omega,G{\overline{\omega}}\\}$
be the corresponding elliptic element of $\widetilde{v}$. Then the length of
the orbit of $\widetilde{v}$ under the action of $Quinn(G)$ is
$\frac{1}{2}|Quinn(G)|$ if $[J_{\omega}]$ has order $4$ in
$\mathrm{ker}\;\overline{N}$ and $|Quinn(G)|$ otherwise.

######  Proposition 4.12.

Suppose that $|\mathop{\mathrm{Ell}}(G)^{=}|<|\mathop{\mathrm{Ell}}(G)|$. Then

  * (a)

$Quinn(G)$ acts transitively on $\mathop{\mathrm{Ell}}(G)^{\neq}$ if and only
if $n_{E}=2$.

  * (b)

$Quinn(G)$ acts transitively on $\mathcal{V}$ if and only if
$n_{E}\in\\{2,3\\}$.

  * (c)

$Quinn(G)$ acts freely on $\mathcal{V}$ if and only if $n_{E}$ is odd.

  * (d)

$Quinn(G)$ acts freely and transitively on $\mathcal{V}$ if and only if
$n_{E}=3$.

###### Proof.

(a) Since $Quinn(G)$ acts freely on $\mathop{\mathrm{Ell}}(G)^{\neq}$ the
action is transitive if and only if
$|Quinn(G)|=|\mathop{\mathrm{Ell}}(G)^{=}|=|\mathop{\mathrm{Ell}}(G)^{\neq}|$
that is if $n_{E}=2$.
(b) If $Quinn(G)$ acts transitively on $\mathcal{V}$ then
$|\mathcal{G}|\leq|\mathop{\mathrm{Ell}}(G)^{=}|$ and so $n_{E}\in\\{2,3\\}$.
When $n_{E}=2$ (a) applies. When $n_{E}=3$ the two $Quinn(G)$-orbits
represented by $G\omega$ and $G\overline{\omega}$ are identified in
$\mathcal{G}$.
(c) By Lemma 4.10 the action of $Quinn(G)$ on $\mathcal{G}$ is not free if and
only if there exists $[J_{\omega}]$ of order $4$ and such an element exists if
and only if $n_{E}$ is even.
(d) follows from (b) and (c). ∎

######  Remark 4.13.

Suppose that $g(K)=g>0$. The $2$-torsion rank of an abelian variety of
dimension $g$ is bounded by $2g$. Applying this to
$\mathop{\mathrm{Cl}}^{0}(\widetilde{K})$ or
$\mathop{\mathrm{Cl}}(\widetilde{A})$ (and using the fact that $\delta$ is
odd) it follows that

$|\mathop{\mathrm{Ell}}(G)^{=}|\leq 2^{2g}.$

See [Ro, Chapter 11]. On the other hand by the Riemann Hypothesis for function
fields [St, Theorems 5.1.15(e), 5.2.1]

$|\mathop{\mathrm{Ell}}(G)|=L_{K}(-1)\geq(\sqrt{q}-1)^{2g}.$

If $n_{E}=2$ then

$2^{2g+1}\geq(\sqrt{q}-1)^{2g}.$

(a) If $q\geq 16$ (and $g>0$), then $Quinn(G)$ cannot act transitively on
$\mathop{\mathrm{Ell}}(G)^{\neq}$.
Another consequence follows using an identical argument.
(b) If $q\geq 23$ (and $g>0$), then $Quinn(G)$ cannot act transitively on
$\mathcal{V}$.

######  Remark 4.14.

It is known [MS3, Corollary 2.12, Theorem 5.1] that a vertex $\widetilde{v}$
of $G\backslash{\mathcal{T}}$ is isolated if and only if $\delta=1$ and
$G_{v}\cong GL_{2}({\mathbb{F}}_{q})$ or ${\mathbb{F}}_{q^{2}}^{*}$. Hence
when $\delta=1$ therefore Theorem 4.9, Proposition 4.12 and Remarks 4.13 can
be interpreted as statements about the action of $Quinn(G)$ on the isolated
vertices of $G\backslash{\mathcal{T}}$.

##  5\. Action on cyclic subgroups

Our focus of attention in this section are the subgroups of $G$ which are
cyclic of order $q^{2}-1$. As distinct from Section 3 some of the results
require $\delta$ to be odd.

######  Definition 5.1.

A finite subgroup $S$ of $G$ is maximally finite if every subgroup of $G$
which properly contains it is infinite.

######  Lemma 5.2.

Let $C$ be a cyclic subgroup of $G$ of order $q^{2}-1$ which is not maximally
finite. Then there exists $H\in\mathcal{H}$ which contains $C$. Moreover $H$
is unique if $\delta$ is odd.

###### Proof.

By Lemma 3.1 (ii) there exists $G_{v}$ which properly contains $C$. Hence
$G_{v}\in\mathcal{H}$ by Lemma 3.2.
Suppose now that $\delta$ is odd. If $H$ is not unique then

$C\leq G_{v_{1}}\cap G_{v_{2}},$

where $v_{1}\neq v_{2}$. It follows that $C$ fixes the geodesic in
${\mathcal{T}}$ joining $v_{1}$ and $v_{2}$, including all its edges. This
contradicts [MS3, Corollary 2.16]. ∎

######  Lemma 5.3.

Let $C,C_{0}$ be cyclic subgroups of order $q^{2}-1$ contained in some
$H\in\mathcal{H}$. Then $C,C_{0}$ are conjugate in $H$.

###### Proof.

By Lemma 3.5 we may assume that $H=GL_{2}({\mathbb{F}}_{q})$. This then
becomes a well-known result. In the absence of a suitable reference we sketch
a proof which lies within the context of this paper.
By the proof of [MS3, Theorem 2.6] (based on [MS3, Lemma 1.4]) it follows that

$C=F^{\mu}=\left\\{g\in GL_{2}({\mathbb{F}}_{q}):g(\mu)=\mu\right\\},$

for some $\mu\in{\mathbb{F}}_{q^{2}}\backslash{\mathbb{F}}_{q}$. Let
$C_{0}=F^{\mu_{0}}$.
Now $\mu_{0}=\alpha\mu+\beta$ for some $\alpha,\beta\in{\mathbb{F}}_{q}$,
where $\alpha\neq 0$. Then $C_{0}=g_{0}Cg_{0}^{-1}$, where

$g_{0}=\left[\begin{array}[]{cc}\alpha&\beta\\\ 0&1\end{array}\right].$

∎

######  Definition 5.4.

Let

$\mathcal{C}=\left\\{C\leq G:C,\mathrm{cyclic\;of\;order}\;q^{2}-1\right\\},$
$\mathcal{C}_{mf}=\left\\{C\in\mathcal{C}:C,\mathrm{maximally\;finite}\right\\}.$
$\mathcal{C}_{nm}=\mathcal{C}\backslash\mathcal{C}_{mf}.$

Clearly every automorphism of $G$ acts on both $\mathcal{C}_{mf}$ and
$\mathcal{C}_{nm}$.

######  Proposition 5.5.

The quasi-inner automorphisms act transitively on all cyclic subgroups of $G$
of order $q^{2}-1$ that are not maximally finite.

###### Proof.

Let $C\in\mathcal{C}_{nm}$. Then by Lemmas 3.5 and 5.2 there exists $g_{0}\in
N_{\widehat{G}}(G)$ such that

$C^{g_{0}}\in GL_{2}({\mathbb{F}}_{q}).$

The rest follows from Lemma 5.3. ∎

The next result follows from Proposition 5.5 and Theorem 3.13.

######  Proposition 5.6.

If $\delta$ is odd, $Quinn(G)$ acts freely and transitively on the conjugacy
classes (in $G$) of cyclic subgroups of $G$ of order $q^{2}-1$ that are not
maximally finite.

The restrictions on $\delta$ in Lemma 5.2 and Proposition 5.6 are necessary.

######  Example 5.7.

Consider the case where $g(K)=0,\;\delta=2$. This case is studied in detail in
[MS1, Section 3]. By the exact sequence in Section 2 it is known that here

$\mathop{\mathrm{Cl}}(A)=\mathop{\mathrm{Cl}}(A)_{2}\cong
Quinn(G)\cong{\mathbb{Z}}/2{\mathbb{Z}}.$

There exists a vertex $v_{0}$ adjacent to the standard vertex $v_{s}$ and
$g_{0}\in N_{\widehat{G}}(G)\backslash G$ such that

$G_{v_{0}}=GL_{2}({\mathbb{F}}_{q})^{g_{0}}\;\mathrm{and}\;G_{v_{s}}\cap
G_{v_{0}}\in\mathcal{C}_{nm}.$

Hence the restriction on $\delta$ in part of Lemma 5.2 is necessary.

It is known [MS1, Theorem 3.3] that in this case

$G=GL_{2}({\mathbb{F}}_{q})\mathop{\mbox{\Large$*$}}_{\quad{C}}GL_{2}({\mathbb{F}}_{q})^{g_{0}},$

where $C(=GL_{2}({\mathbb{F}}_{q})\cap
GL_{2}({\mathbb{F}}_{q})^{g_{0}})\in\mathcal{C}_{nm}$. It follows by Lemma 5.3
that there exists $g\in GL_{2}({\mathbb{F}}_{q})$ for which

$C^{g}=C^{g_{0}}.$

In this case therefore $Quinn(G)$, which is non-trivial, fixes $C^{G}$. The
restriction on $\delta$ in Proposition 5.6 is therefore necessary.

We conclude this section with some remarks about $\mathcal{C}_{mf}$.

######  Lemma 5.8.

Suppose that $\delta$ is odd. Then

$C\in\mathcal{C}_{mf}\Longleftrightarrow
C=G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}.$

###### Proof.

Suppose that $C=G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}$ and that
$C\in\mathcal{C}_{nm}$. Then by Lemmas 3.1 and 3.3 it follows that $C\leq
G_{v}\cap G_{v_{0}}$ for some $v_{0}\neq v$, which contradicts [MS3, Corollary
2.16]. The rest follows from Lemma 3.1. ∎

When $\delta$ is odd there is therefore a one-one correspondence

$(\mathcal{C}_{mf})^{G}\longleftrightarrow\mathcal{V}.$

For the case where $\delta$ is odd this shows that the results in Proposition
4.12 apply to the action of $Quinn(G)$ on $(\mathcal{C}_{mf})^{G}$.

######  Remark 5.9.

As a Möbius transformation every member of $G$ fixes an element of
$C_{\infty}$. Suppose now that $\delta$ is even and that $C$ is a cyclic
subgroup of order $q^{2}-1$ (maximally finite or not). Then from the proof of
[MS4, Proposition 2.3] it follows that $C$ fixes $\mu\in
K.{\mathbb{F}}_{q^{2}}\backslash K$. In this case however $\mu\in K_{\infty}$
as $\delta$ is even. So $\mu$, which is not in $\Omega$ and not in $K$, can
neither be an inner point nor a cusp of the Drinfeld modular curve
$G\backslash\Omega$. We refer to $\mu$ as pseudo-elliptic.
On the other hand suppose that $\delta$ is odd. Let $g$ be any element of
infinite order in $G$ and let $g$ fix $\lambda$. Then $\lambda\in
K_{\infty}\backslash K$.

##  6\. Action on cusps

As distinct from Section $4$ the results here hold for all $\delta$. Any
element of $\widehat{G}$ acts on ${\mathbb{P}}^{1}(K)=K\cup\\{\infty\\}$ as a
Möbius transformation. In this way $Quinn(G)$ acts on
$G\backslash{\mathbb{P}}^{1}(K)=\mathop{\mathrm{Cusp}}(G)$. Every element of
$\mathop{\mathrm{Cusp}}(G)$ can be represented in the form $(a:b)$, where
$a,b\in A$. Since $A$ is a Dedekind ring this gives rise to a one-one
correspondence

$\mathop{\mathrm{Cusp}}(G)\longleftrightarrow\mathop{\mathrm{Cl}}(A).$

Hence the action of $Quinn(G)$ on $\mathop{\mathrm{Cusp}}(G)$ translates to an
action of $\mathop{\mathrm{Cl}}(A)_{2}$ on $\mathop{\mathrm{Cl}}(A)$. The
principal result in this section is similar to but simpler than Theorem 4.7.
It translates this action into multiplication in the group
$\mathop{\mathrm{Cl}}(A)$. We sketch a proof.
We can represent any cusp, $c$, by an element $(x:y)\in{\mathbb{P}}^{1}(K)$,
where $x,y\in A$. Let

$J_{c}=xA+yA,$

and let $[J_{c}]$ be its image in $\mathop{\mathrm{Cl}}(A)$.
Now let $\kappa$ be a non-trivial element of $Quinn(G)$. Then as before by
Theorem 2.2 $\kappa$ can be represented by a matrix

$M=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\in\widehat{G},$

where we may assume that $a,b,c,d\in A$. Let ${\mathfrak{q}}(M)$ be the
$A$-ideal generated by $a,b,c,d$.
The action of $\kappa$ on $c$ is given by the action of $M$ multiplying the
column vector ${x\choose y}$ on the left by $M$. In this way

$J_{\kappa(c)}=J_{M(c)}=(ax+by)A+(cx+dy)A.$

######  Theorem 6.1.

Under the identification of $\mathop{\mathrm{Cusp}}(G)$ with
$\mathop{\mathrm{Cl}}(A)$ and $Quinn(G)$ with $\mathop{\mathrm{Cl}}(A)_{2}$
the action of $Quinn(G)$ on the cusps translates into multiplication in the
group $\mathop{\mathrm{Cl}}(A)$. More precisely

$[J_{\kappa(c)}]=[{\mathfrak{q}}(M)J_{c}]=[{\mathfrak{q}}(M)][J_{c}]\;\;\;(in\;\mathop{\mathrm{Cl}}(A)).$

###### Proof.

Since $A$ is a Dedekind domain there exists an $A$-ideal $I_{1}$ such that

$J_{M(c)}={\mathfrak{q}}(M)J_{c}I_{1}.$

By Corollary 2.3 (i) there exists an $A$-ideal $I_{2}$ with

$\varDelta
J_{c}=J_{M^{2}(c)}={\mathfrak{q}}(M)J_{M(c)}I_{2}={\mathfrak{q}}(M)^{2}J_{c}I_{1}I_{2}=\varDelta
J_{c}I_{1}I_{2},$

where $\varDelta=\mathop{\mathrm{det}}(M)$. Hence $I_{1}=I_{2}=A$ and the
result follows. ∎

As in the previous section we have the following immediate consequence.

######  Corollary 6.2.

If a non-trivial quasi-inner automorphism $\kappa$ fixes any cusp, then
$\kappa$ reduces to an inner automorphism. In particular, $Quinn(G)$ acts
freely on $\mathop{\mathrm{Cusp}}(G)$.

######  Remark 6.3.

$Quinn(G)$ acts transitively on $\mathop{\mathrm{Cusp}}(G)$ if and only if
$\mathop{\mathrm{Cl}}(A)_{2}=\mathop{\mathrm{Cl}}(A)$.
From the exact sequence in Section $2$ a necessary condition for this is
$\delta\in\\{1,2\\}$. If $g(K)=0$, this condition is also sufficient, as then
$\mathrm{Cl}(A)\cong{\mathbb{Z}}/\delta{\mathbb{Z}}$.

But if $g(K)=g>0$, the action cannot be transitive for $q>9$ by an argument
very similar to that used in Remark 4.11. The inequality

$\frac{|\mathrm{Cl}^{0}(K)|}{|\mathrm{Cl}^{0}(K)_{2}|}\geq\frac{(\sqrt{q}-1)^{2g}}{2^{2g}}$

shows that for fixed $q>9$ the number of orbits of $Quinn(G)$ on
$\mathop{\mathrm{Cusp}}(G)$ tends to $\infty$ with $g(K)$.

The cusp $\infty$($={1\choose 0}$) corresponds to the principal $A$-ideals.
Its orbit under $Quinn(G)$ corresponds to the $2$-torsion in
$\mathop{\mathrm{Cl}}(A)$ and in the sense of Theorem 6.1 the action of
$Quinn(G)$ on it translates into $\mathop{\mathrm{Cl}}(A)_{2}$ acting on
itself by multiplication.
For every cusp $c$ represented by the ideal class $[J_{c}]$ in
$\mathop{\mathrm{Cl}}(A)$ there corresponds its (group) inverse $[J_{c}]^{-1}$
in $\mathop{\mathrm{Cl}}(A)$. We can partition $\mathop{\mathrm{Cl}}(A)$ thus

$Quinn(G)\leftrightarrow\mathop{\mathrm{Cl}}(A)_{2}=\\{[J_{c}]:[J_{c}]=[J_{c}]^{-1}\\},$
$\mathop{\mathrm{Cl}}(A)\backslash\mathop{\mathrm{Cl}}(A)_{2}=\\{[J_{c}]:[J_{c}]\neq[J_{c}]^{-1}\\}.$

Our next result follows from Theorem 6.1 analogous to the way Lemma 4.10
follows from Theorem 4.7.

######  Lemma 6.4.

A quasi-inner automorphism $\kappa$, represented by $M\in N_{\widehat{G}}(G)$,
maps the cusp $c$ corresponding to $[J_{c}]$ in
$\mathop{\mathrm{Cl}}(A)\backslash\mathop{\mathrm{Cl}}(A)_{2}$, to the cusp
corresponding to $[J_{c}]^{-1}$ if and only if $[J_{c}]$ has order $4$ and
$[J_{c}]^{2}={\mathfrak{q}}(M)$.

In the next section we will use the results in Sections $5$ and $6$, together
with Theorem 3.13 (ii), to examine in detail the action of $Quinn(G)$ on
$G\backslash{\mathcal{T}}$.

##  7\. Action on the quotient graph

The model used by Serre for ${\mathcal{T}}$ [Se, Chapter II, Section 1.1] is
based on two-dimensional so called lattice classes . Since every quasi-inner
automorphism, $\iota_{g}$, can be represented by a matrix in $\widehat{G}$ it
acts on ${\mathcal{T}}$ and hence $Quinn(G)$ acts on
$G\backslash{\mathcal{T}}$.

In this section we investigate the action of a quasi-inner automorphism on the
quotient graph $H\backslash{\mathcal{T}}$, where $H$ is a finite index
subgroup of $G$. In the process we extend a result of Serre [Se, Exercise
2(e), p.117] which motivated our interest in this question. We begin with a
detailed account of Serre’s classical description of
$G\backslash{\mathcal{T}}$. Serre’s original proof [Se, Theorem 9, p.106] is
based on the theory of vector bundles. For a more detailed version which
refers explicitly to matrices see [Ma1]. In addition we use the results the
previous sections to shed new light on the structure of
$G\backslash{\mathcal{T}}$.

######  Definition 7.1.

A ray $\mathcal{R}$ in a graph $\mathcal{G}$ is an infinite half-line, without
backtracking. In accordance with Serre’s terminology [Se, p.104] we call
$\mathcal{R}$ cuspidal if all its non-terminal vertices have valency $2$ (in
$\mathcal{G}$).

Let $\\{g_{1},\cdots,g_{s}\\}\subseteq\widehat{G}$, where $s\geq 1$, be a
complete system of representatives for $\mathop{\mathrm{Cl}}(A)_{2}(\cong
N_{\widehat{G}}(G)/G.Z(K))$. Let $c_{i}=g_{i}(\infty)\;(1\leq i\leq s)$. We
will assume that $c_{1}=\infty$. If
$\mathop{\mathrm{Cl}}(A)=\mathop{\mathrm{Cl}}(A)_{2}$, then
$\\{c_{1},\cdots,c_{s}\\}$ is a complete system of representatives for
$\mathop{\mathrm{Cl}}(A)$. If
$\mathop{\mathrm{Cl}}(A)\neq\mathop{\mathrm{Cl}}(A)_{2}$ we can find further
elements $h_{1},\cdots,h_{t}\in\widehat{G}$, where $t\geq 1$ so that

$\mathcal{S}=\\{c_{1},\cdots,c_{s},\;d_{1},\cdots,d_{t}\\}$

is a complete set of representatives for $\mathop{\mathrm{Cl}}(A)$, where
$d_{j}=h_{j}(\infty)\;\;(1\leq j\leq t)$.

######  Theorem 7.2.

There exists a complete system of representatives
$\mathcal{C}\;(\subseteq{\mathbb{P}}^{1}(K))$ for
$\mathop{\mathrm{Cusp}}(G)$(equivalently, $\mathop{\mathrm{Cl}}(A))$ of the
above type such that

$G\backslash{\mathcal{T}}=X\cup\left(\bigcup_{1\leq i\leq
s}\mathcal{R}(c_{i})\right)\left(\bigcup_{1\leq j\leq
t}\mathcal{R}(d_{j})\right),$

where

  * (i)

$X$ is finite,

  * (ii)

each $\mathcal{R}(c_{i}),\;\mathcal{R}(d_{j})$ is a cuspidal ray (in
$G\backslash{\mathcal{T}}$), whose only interesection with $X$ consists of a
single vertex,

  * (iii)

the $|\mathop{\mathrm{Cl}}(A)|$ cuspidal rays are pairwise disjoint.

Moreover if $\mathcal{R}(e)$ is any of these cuspidal rays then it has a lift,
$\overline{\mathcal{R}(e)}$, to ${\mathcal{T}}$ with the following properties.
Let
$\mathop{\mathrm{vert}}(\overline{\mathcal{R}(c)})=\\{v_{1},v_{2}\cdots\\}$.
Then

  * (i)

$G_{v_{i}}\leq G_{v_{i+1}},\;(i\geq 1),$

  * (ii)

$\bigcup_{i\geq 1}G_{v_{i}}=G(c),$

where $G(c)$ is the stabilizer (in $G$) of the cusp $c$.

For each $j$ let $\widetilde{d_{j}}$ be the element of
$\\{d_{1},\cdots,d_{t}\\}$ corresponding to $h_{j}^{-1}(\infty)$. We may
relabel the latter set as
$\\{d_{1},\widetilde{d_{1}},\cdots,d_{t^{\prime}},\widetilde{d_{t^{\prime}}}\\}$,
where $t^{\prime}=t/2$. We can use the results in Section 3 to elaborate on
the structure of the above cuspidal rays. We recall that

$\mathcal{H}=\\{H\leq G:H\cong GL_{2}({\mathbb{F}}_{q})\\}.$

######  Corollary 7.3.

For the above set of $|\mathop{\mathrm{Cl}}(A)|$ cuspidal rays

  * (i)

$\mathcal{R}_{1}=\\{\mathcal{R}(c_{1}),\cdots,\mathcal{R}(c_{s})\\}\leftrightarrow\\{\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\in\mathcal{H}\\}\leftrightarrow\mathop{\mathrm{Cl}}(A)_{2}.$

  * (ii)

$\mathcal{R}_{2}=\left\\{\mathcal{R}(d_{j}),\mathcal{R}(\tilde{d_{j}}):1\leq
j\leq
t^{\prime}\right\\}\leftrightarrow\mathop{\mathrm{Cl}}(A)\backslash\mathop{\mathrm{Cl}}(A)_{2}.$

###### Proof.

Let $\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}})$, where
$G_{v}\in\mathcal{H}$, and let $H\in\mathcal{H}$ be any representative of its
stabilizer. Then, for some unique $i$,

$H=gg_{i}(GL_{2}({\mathbb{F}}_{q}))(gg_{i})^{-1},$

where $g\in G$, by Lemmas 3.5 and 3.6. Now let $u$ be any unipotent element of
$H$. Then $u$ fixes $gg_{i}h(\infty)$, for some $h\in
GL_{2}({\mathbb{F}}_{q})$. It follows that

$u\in G(c)\Longleftrightarrow c=g^{\prime}c_{i},$

where $g^{\prime}\in G$. The rest follows from Corollary 3.9 together with
Theorem 3.13. ∎

######  Remark 7.4.

Let $\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}})$, where
$G_{v}\in\mathcal{H}$. Then it is shown in Corollary 7.3 that $\widetilde{v}$
is adjacent in $G\backslash{\mathcal{T}}$ to a vertex whose stabilizer (up to
conjugacy in $G$) is contained in $G(c_{i})$, for some unique $i$. In this way
$\widetilde{v}$ can be thought of as closer in $G\backslash{\mathcal{T}}$ to
$\mathcal{R}(c_{i})$ than to any other cuspidal ray. For the case $\delta=1$
(and only for this case) $\widetilde{v}$ is isolated in
$G\backslash{\mathcal{T}}$ by [MS3, Theorem 5.1]. As in Takahashi’s example
[Ta] such a $\widetilde{v}$ then appears as a “spike” next to its associated
cuspidal ray.

For each subgroup $H$ of $G$ we recall that the elements of
$H\backslash{\mathcal{T}}$ are

$\mathop{\mathrm{vert}}(H\backslash{\mathcal{T}})=\left\\{Hv:v\in\mathop{\mathrm{vert}}({\mathcal{T}})\right\\}\;\;\mathrm{and}\;\;\mathop{\mathrm{edge}}(H\backslash{\mathcal{T}})=\left\\{He:e\in\mathop{\mathrm{edge}}({\mathcal{T}})\right\\}.$

######  Definition 7.5.

Let $H,H^{*}$ be isomorphic subgroups of $G$. An isomorphism of graphs

$\phi:H\backslash{\mathcal{T}}\rightarrow H^{*}\backslash{\mathcal{T}},$

is said to be stabilizer invariant if the following condition holds.

For any
$w\in\mathop{\mathrm{vert}}({\mathcal{T}})\cup\mathop{\mathrm{edge}}({\mathcal{T}})$
let

$\phi(Hw)=H^{*}w^{*},$

(where $w^{*}\in\mathop{\mathrm{vert}}({\mathcal{T}})$ if and only if
$w\in\mathop{\mathrm{vert}}({\mathcal{T}})$). Then, for all $u\in H_{w}$ and
$u^{*}\in H^{*}w^{*}$

$H_{u}\cong H_{u^{*}}^{*}.$

As we shall see it is easy to find examples of isomorphisms of quotient graphs
which are not stabilizer invariant.

######  Theorem 7.6.

Let $\kappa=\iota_{g}$, where $g\in N_{\widehat{G}}(G)$ and let $H$ be a
subgroup of $G$. Then the map

$\overline{\kappa}_{H}:H\backslash{\mathcal{T}}\rightarrow\kappa(H)\backslash{\mathcal{T}},$

defined by

$\overline{\kappa}_{H}(Hw)=H^{\prime}w^{\prime},$

where $H^{\prime}=H^{g}=gHg^{-1}$, $w^{\prime}=g(w)$ and
$w\in\mathop{\mathrm{vert}}({\mathcal{T}})\cup\mathop{\mathrm{edge}}({\mathcal{T}})$,
defines a stabilizer invariant isomorphism of the quotient graphs

$\kappa(H)\backslash{\mathcal{T}}\cong H\backslash{\mathcal{T}}.$

###### Proof.

Note that $\overline{\kappa}_{H}$ is well defined since if
$\kappa(x)=g_{1}xg_{1}^{-1}$, where $g_{1}\in N_{\widehat{G}}(G)$ then
$gg_{1}^{-1}\in Z(K)$ and $Z_{\infty}$, the set of scalar matrrices in
$GL_{2}(K_{\infty})$, stabilizes every $w$. The rest is obvious (since $g$
acts on ${\mathcal{T}}$). ∎

Let $H$ be any finite index subgroup of $G$ and let $M$ be the largest normal
subgroup of $G$ contained in $H$. Then $N=M\cap M^{g}$ is the largest (finite
index) subgroup of $G$, contained in $H$, which is normalized by $G,\;Z(K)$
and $g$. (See Section $2$.)

######  Corollary 7.7.

Suppose that $\kappa$ is non-trivial (i.e. $g\notin G.Z(K)$). Let $N$ be a
finite index normal subgroup of $G$ normalized by $\kappa$. Then the map

$\overline{\kappa}_{N}:N\backslash{\mathcal{T}}\rightarrow
N\backslash{\mathcal{T}},$

defined as above, is a non-trivial stabilizer invariant automorphism whose
order $n$ is even. Moreover, if $Z\leq N$, then $n=2m$, where $m$ divides
$|G:N|$.

###### Proof.

To prove that $\overline{\kappa}_{N}$ is non-trivial it suffices to prove that
$\overline{\kappa}_{G}$ is not the identity map. There exists
$v_{0}\in\mathop{\mathrm{vert}}({\mathcal{T}})$ for which (non-central)
$G_{v_{0}}\leq G(\infty)$ [Ma1, Lemma 3.2]. Suppose to the contrary that
$\overline{\kappa}_{G}$ fixes $Gv_{0}$. Then there exists $g_{0}\in G$ such
that $g^{\prime}=gg_{0}\in G(\infty)$ which implies that

$g^{\prime}=\left[\begin{array}[]{cc}a&b\\\ 0&c\end{array}\right].$

We may assume that $a,b,c\in A$. By Theorem 2.2, together with an argument
used in the proof of Theorem 2.4, it follows that

$a^{2}A=c^{2}A=acA.$

Hence $a,c\in{\mathbb{F}}_{q}$. Thus $g^{\prime}\in G$ and so $g\in G.Z(K)$.
For the second part $n$ is the smallest $n(>0)$ such that $g^{n}\in N.Z(K)$.
Now $g^{2}\in G.Z(K)$ by Corollary 2.3(i). If $n$ is odd then $g\in G.Z(K)$.
Hence $n=2m$ is even. In addition when $Z\leq N$ $m$ divides
$|G.Z(K):N.Z(K)|=|G:N|$. ∎

A special case of Corollary 7.7, combined with Corollary 2.6, is the
following.

######  Corollary 7.8.

Suppose that $|\mathop{\mathrm{Cl}}(A)|=|\mathop{\mathrm{Cusp}}(G)|$ is even.
Then there exists a stabilizer invariant automorphism of
$G\backslash{\mathcal{T}}$ of order $2$.

Serre [Se, Exercise 2(e), p.117] states this result for the case $g(K)=0$
(i.e. $K={\mathbb{F}}_{q}(t)$) and $\delta$ even. The restriction here is
necessary. For the case $g(K)=0,\;\delta=1$, in which case
$A={\mathbb{F}}_{q}[t]$ and $|\mathop{\mathrm{Cl}}(A)|=1$, it is known by
Nagao’s Theorem [Se, Corollary, p.87] that $G\backslash{\mathcal{T}}$ is a
cuspidal ray whose terminal vertex is isolated. Here then the only (graph)
automorphism is trivial.
Corollary 7.8 shows that $Quinn(G)$ acts non-trivially on
$G\backslash{\mathcal{T}}$. This extends to an action on its cuspidal rays
which we now describe. We use the notation of Theorem 7.2.

######  Definition 7.9.

Let $\mathcal{R}_{1},\mathcal{R}_{2}$ be rays in a graph $\mathcal{G}$. We
write

$\mathcal{R}_{1}\sim\mathcal{R}_{2}$

if and only if
$|\mathcal{R}_{i}\backslash\mathcal{R}_{1}\cap\mathcal{R}_{2}|<\infty\;(i=1,2)$.
This a well-known equivalence relation. The equivalence class containing the
ray $\mathcal{R}$ is called the end (of $\mathcal{G}$) determined by
$\mathcal{R}$. In the notation of Theorem 7.2 we denote by $\mathcal{E}(e)$
the end (in $G\backslash{\mathcal{T}}$) determined by $\mathcal{R}(e)$, where
$e=c_{i},d_{j}$.

Now let $\kappa=\iota_{g}$, where $g\in N_{\widehat{G}}(G\backslash G.Z(K)$,
be a non-trivial quasi-inner automorphism and let $\widehat{\kappa}$ be the
corresponding (non-trivial) element of $Quinn(G)$. Now fix $e\in\mathcal{S}$.
Let $e^{*}=\kappa(e)$. Then by Corollary 6.2 $e\neq e^{*}$ and we may assume
that $e^{*}\in\mathcal{S}$. As in Theorem 7.2

$\mathop{\mathrm{vert}}(\mathcal{R}(e))=\\{\widetilde{v}_{1},\widetilde{v}_{2},\cdots\\}\;\mathrm{and}\;\mathop{\mathrm{vert}}(\mathcal{R}(e^{*}))=\\{\tilde{v}_{1}^{*},\tilde{v}_{2}^{*},\cdots\\}.$

Recall that

$\bigcup_{i\geq 1}G_{v_{i}}=G(e),$

and that $G_{v_{i}}\leq G_{v_{i+1}}\;(i\geq 1)$. In addition it is known [MS3,
Theorem 2.1 (a)] that there exists a normal subgroup $N_{i}$ of $G_{v_{i}}$
such that

$G_{v_{i}}/N_{i}\cong{\mathbb{F}}_{q}^{*}\times{\mathbb{F}}_{q}^{*},$

where $N_{i}\cong V_{i}^{+}$, the additive group of a
${\mathbb{F}}_{q}$-vector space of dimension $n_{i}$. It is also known that
$n_{i}<n_{i+1}$. Corresponding results hold for $\mathcal{R}(e^{*})$. Now let

$m_{X}=\max\\{|G_{v}|:v\in\mathop{\mathrm{vert}}(X)\\}.$

(Recall that $X$ is finite.). Now choose any $m>m_{X}$. By the definition of
graph automorphism $\overline{\kappa_{G}}$ (determined by the non-trivial
element $\widehat{\kappa}$ of $Quinn(G)$, together with Theorem 7.2, there
exists $n>m_{X}$ such that

$\overline{\kappa}_{G}:\tilde{v}_{m+i}\mapsto\tilde{v}_{n+i}^{*}\;,$

for all $i\geq 0$. This gives rise to a map

$\widehat{\kappa}:\mathcal{E}(e)\mapsto\mathcal{E}(e^{*}),$

which in turn defines a $Quinn(G)$-action on the ends defined by the cuspidal
rays in $G\backslash{\mathcal{T}}$. (Theorem 7.2.) Since this action coincides
precisely with the action of $Quinn(G)$ on $\mathop{\mathrm{Cusp}}(G)$ the
following result is an immediate consequence of Theorem 3.13(ii), Corollary
6.2 and Lemma 6.4.

######  Corollary 7.10.

With the notation of Theorem 7.2,

  * (i)

$Quinn(G)$ acts (simultaneously) freely and transitively on

$\\{\mathcal{E}(c_{1}),\cdots,\mathcal{E}(c_{s})\\}\;\mathrm{and}\;\\{\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong
GL_{2}({\mathbb{F}}_{q})\\},$

  * (ii)

$Quinn(G)$ acts freely on

$\left\\{\mathcal{E}(d_{j}),\mathcal{E}(\tilde{d_{j}}):1\leq j\leq
t^{\prime}\right\\},$

  * (iii)

$Quinn(G)$ acts on

$\left\\{\\{\mathcal{E}(d_{j}),\mathcal{E}(\tilde{d_{j}})\\}:1\leq j\leq
t^{\prime}\right\\}.$

  * (iv)

Under the action of $Quinn(G)$ some $\mathcal{E}(d_{j})$ is mapped to
$\mathcal{E}(\tilde{d_{j}})$ if and only if $d_{j}$ has order $4$ in
$\mathop{\mathrm{Cl}}(A)$.

We recall from Proposition 4.12 that when $\delta$ is odd $Quinn(G)$ also acts
on
$\\{\widetilde{v}\in\mathop{\mathrm{vert}}(G\backslash{\mathcal{T}}):G_{v}\cong{\mathbb{F}}_{q^{2}}^{*}\\}$.

Our final result in this section concerns the action of $N_{\widehat{G}}(G)$
on ${\mathcal{T}}$. it is known [Se, Corollary, p.75] that $G$ acts without
inversion (on the edges) of ${\mathcal{T}}$.

######  Proposition 7.11.

Suppose that $\delta$ is odd. Then every $\iota_{g}$ acts without inversion on
${\mathcal{T}}$ and hence on every quotient graph $H\backslash{\mathcal{T}}$.

###### Proof.

As in Theorem 2.2 we can represent $\iota_{g}$ with a matrix $M$ in
$\widehat{G}$ and we can assume that all its entries lie in $A$. Let
$\varDelta=\mathop{\mathrm{det}}(M)$. Then the $A$-ideal generated by
$\varDelta$ is the square of an ideal in $A$, again by Theorem 2.2. It follows
that, for all places $v\neq v_{\infty}$, $v(\varDelta)$ is even.
By the product formula then $\delta v_{\infty}(\varDelta)$ and hence
$v_{\infty}(\varDelta)$ is even. The result follows from [Se, Corollary,
p.75]. ∎

######  Example 7.12.

To conclude this section we consider the case where $g(K)=0$ and
$\delta(K)=2$. We recall that there exists a quadratic polynomial
$\pi\in{\mathbb{F}}_{q}[t]$, irreducible over ${\mathbb{F}}_{q}$, such that

$A=\left\\{\frac{f}{\pi^{m}}:f\in{\mathbb{F}}_{q}[t],\;m\geq 0,\deg f\leq
2m\right\\}.$

In this case it is known that
$\mathop{\mathrm{Cl}}(A)_{2}=\mathop{\mathrm{Cl}}(A)\cong
Quinn(G)\cong{\mathbb{Z}}/2{\mathbb{Z}}$. It is well-known that
$G\backslash{\mathcal{T}}$ is a doubly infinite line, without backtracking.
See [Se, 2.4.2 (a), p.113] and, for a more detailed description, [MS1, Section
3]. It is known that $G\backslash{\mathcal{T}}$ lifts to a doubly infinite
line $\mathcal{D}$ in ${\mathcal{T}}$ which we now describe in detail. For
some $g_{0}\in N_{\widehat{G}}(G)\backslash G.Z(K)$,
$\mathop{\mathrm{vert}}(\mathcal{D})=\\{v_{0},v_{0}^{*},\;v_{1},v_{1}^{*},\cdots\\}$,
where

  * (i)

$v_{i}^{*}=g_{0}(v_{i})\;(i\geq 0)$,

  * (ii)

$G_{v_{i}^{*}}=(G_{v_{i}})^{g_{0}}\;(i\geq 0))$,

  * (iii)

$G_{v_{0}}=GL_{2}({\mathbb{F}}_{q})$,

  * (iv)

for each $i\geq 1$

$G_{v_{i}}=\left\\{\left[\begin{array}[]{lll}\alpha&c\pi^{-i}\\\\[10.0pt]
0&\beta\end{array}\right]:\alpha,\beta\in{\mathbb{F}}_{q}^{*},\deg c\leq
2i\right\\}.$

Then $\mathcal{D}$ maps onto (and is isomorphic to) $G\backslash{\mathcal{T}}$
which has the following structure.

$\overline{v_{2}}$$\overline{v_{1}}$$\overline{v_{0}}$$\overline{v_{0}^{*}}$$\overline{v_{1}^{*}}$$\overline{v_{2}^{*}}$.

The action of the (essentially only) non-trivial quasi-inner automorphism of
$G\backslash{\mathcal{T}}$ (represented by $g_{0}$) is given by

$\overline{v_{i}}\leftrightarrow\overline{v_{i}^{*}}\;(i\geq 0).$

We note two features of $\mathcal{D}$ which are of interest relevant to this
section.
(i) From the structure of $\mathcal{D}$ it is clear that the non-trivial
quasi-inner automorphism determined by $g_{0}$ inverts the edge joining
$v_{0}$ and $v_{0}^{*}$, which shows that the restriction on $\delta$ in
Proposition 7.11 is necessary.
(ii) For this case there is only one stabilizer invariant involution. However
the graph $G\backslash{\mathcal{T}}$ has many automorphisms. Infinitely many
examples include translations (which have infinite order) and reflections in
any vertex (which are involutions).

##  8\. Two instructive examples

We conclude with two examples which demonstrate how our results apply to the
structure of the quotient graph $G\backslash{\mathcal{T}}$. Both are elliptic
function fields $K/{\mathbb{F}}_{q}$. We record some of their basic
properties.

######  Definition 8.1.

A function field $K/{\mathbb{F}}_{q}$ is elliptic [St, p.217] if $g(K)=1$ and
$K$ has a place $\infty$ of degree $1$.

######  Theorem 8.2.

Suppose that $K/{\mathbb{F}}_{q}$ is elliptic. Then

  * (i)

$K={\mathbb{F}}_{q}(x,y),$

where $x,y$ satisfy a (smooth) Weierstass equation $F(x,y)=0$ with

$F(x,y)=y^{2}+a_{1}xy+a_{3}y-x^{3}-a_{2}x^{2}-a_{4}x-a_{6}\in{\mathbb{F}}_{q}[x,y].$

  * (ii)

$\mathop{\mathrm{Cl}}^{0}(K)(\cong\mathop{\mathrm{Cl}}(A))$ is isomorphic to
$E({\mathbb{F}}_{q})$, the group of ${\mathbb{F}}_{q}$-rational points,
$\\{(\alpha,\beta)\in{\mathbb{F}}_{q}\times{\mathbb{F}}_{q}:F(\alpha,\beta)=0\\}\cup\\{(\infty,\infty)\\}$.
Here the group operation is point addition $\oplus$ according to the chord-
tangent law.

###### Proof.

For (i) see [St, Proposition 6.1.2]. For (ii) see [St, Propositions 6.1.6,
6.1.7]. ∎

Here a rational point $(a,b)\in E({\mathbb{F}}_{q})$ corresponds to the ideal
class of $A(x-a)+A(y-b)$. We also require some “elliptic” properties of
$\widetilde{K}=K.{\mathbb{F}}_{q^{2}}$ (which is a constant field extension of
$K$).

######  Corollary 8.3.

Suppose that $K/{\mathbb{F}}_{q}$ is elliptic. Then
$\widetilde{K}/{\mathbb{F}}_{q^{2}}$ is also elliptic and defined by the same
Weierstrass equation.

###### Proof.

From the above $\widetilde{K}={\mathbb{F}}_{q^{2}}(x,y)$, where $F(x,y)=0$.
The rest follows from [St, Proposition 6.1.3]. ∎

With our choice of infinite place we have

$A={\mathbb{F}}_{q}[x,y]\;\;\mathrm{and}\;\;\widetilde{A}={\mathbb{F}}_{q^{2}}[x,y],$

where $x$ and $y$ satisfy the Weierstrass equation $F(x,y)=0$. In an analagous
way

$\mathrm{Cl}(\widetilde{A})\cong\mathrm{Cl}^{0}(\widetilde{K})\cong
E({\mathbb{F}}_{q^{2}}).$

We recall that the image of any $\alpha\in{\mathbb{F}}_{q^{2}}$ under the
Galois automorphism of ${\mathbb{F}}_{q^{2}}/{\mathbb{F}}_{q}$ is denoted by
$\overline{\alpha}$. For each rational point $P=(\alpha,\beta)\in
E({\mathbb{F}}_{q^{2}})$ we put
$\overline{P}=(\overline{\alpha},\overline{\beta})$.

######  Corollary 8.4.

Suppose that $K/{\mathbb{F}}_{q}$ is elliptic. Under the identifications of
$\mathop{\mathrm{Cl}}^{0}(\widetilde{K})$ (resp.
$\mathop{\mathrm{Cl}}^{0}(K)$) with $E({\mathbb{F}}_{q^{2}})$ (resp.
$E({\mathbb{F}}_{q})$) the norm map
$N:\mathop{\mathrm{Cl}}^{0}(\widetilde{K})\rightarrow\mathop{\mathrm{Cl}}^{0}(K)$
translates to a map $N_{E}:E({\mathbb{F}}_{q^{2}})\rightarrow
E({\mathbb{F}}_{q})$ defined by

$N_{E}(P)=P\oplus\overline{P},$

so that

$P\in\ker N_{E}\Leftrightarrow\overline{P}=-P.$

Takahashi [Ta] has described in detail the quotient graph for an elliptic
function field over any field of constants. In all cases
$G\backslash{\mathcal{T}}$ is a tree. Since $\delta=1$, for the case of a
finite field of constants, the isolated vertices of $G\backslash{\mathcal{T}}$
are precisely those whose stabilizer is isomorphic to
$GL_{2}({\mathbb{F}}_{q})$ or ${\mathbb{F}}_{q^{2}}^{*}$ by [MS3, Theorem
5.1]. For each cusp $c\in\mathop{\mathrm{Cl}}(A)_{2}$ the cuspidal ray
$\mathcal{R}(c)$ in $G\backslash{\mathcal{T}}$ has attached to its terminal
vertex (appearing as a “spike”) an isolated vertex with stabilizer isomorphic
to $GL_{2}({\mathbb{F}}_{q})$. The remaining cuspidal rays consist of
$\frac{1}{2}|\mathop{\mathrm{Cl}}(A)\backslash\mathop{\mathrm{Cl}}(A)_{2}|$
inverse pairs $\\{\mathcal{R}(c),\mathcal{R}(c^{-1})\\}$ which share a
terminal vertex (appearing in $G\backslash{\mathcal{T}}$ as the “prongs” of a
“fork”).
In both our examples $q=7$ in which case the Weierstrass equation can be
assumed to take the short form

$y^{2}=f(x)=x^{3}+ax+b,$

where $a,b\in{\mathbb{F}}_{q}$ and $f(x)$ has no repeated roots.

######  Example 8.5.

$K={\mathbb{F}}_{7}(x,y),\;A={\mathbb{F}}_{7}[x,y]$ with $y^{2}=x^{3}-3x$
It can be easily shown that

$E({\mathbb{F}}_{7})=\\{(\infty,\infty),(0,0),(2,\pm 3),(3,\pm 2),(6,\pm
3)\\}.$

Since $E$ is in short Weierstrass form the $8$ points are listed as (additive)
inverse pairs. In particular $(0,0)$ is the only such $2$-torsion point. It
follows that
$Quinn(G)\cong\mathop{\mathrm{Cl}}(A)_{2}\cong{\mathbb{Z}}/2{\mathbb{Z}}$ and
hence that $\mathop{\mathrm{Cl}}(A)\cong{\mathbb{Z}}/8{\mathbb{Z}}$. Let
$\kappa$ be a non-trivial quasi-inner automorphism of $G$ representing the
non-trivial element of $Quinn(G)$. In $E({\mathbb{F}}_{7})$ $\kappa$ is
represented by $(0,0)$ and, by Theorem 6.1, its action on
$\mathop{\mathrm{Cusp}}(G)$ is determined by its action (via point addition
$\oplus$) in $E({\mathbb{F}}_{7})$. In a diagram of $G\backslash{\mathcal{T}}$
as described in [Ta] we wish to ensure that its involution provided by
$\kappa$, Corollary 7.8, is given by the reflection in the vertical axis. We
begin by labelling appropriately its $8$ cuspidal rays (corresponding to
$E({\mathbb{F}}_{7})$). By Corollary 6.2 $\kappa$ acts freely on these. By
Corollary 7.3 (i) it is clear that $\kappa$ interchanges the cusps
$(\infty,\infty)$ and $(0,0)$. Attached to each of these is a ”spike”
consisting of an isolated vertex whose stabilizer is isomorphic to
$GL_{2}({\mathbb{F}}_{7})$. Since $\kappa$ is a graph automorphism it
interchanges these vertices, namely, $g_{1}$ and $g_{2}$. By means of the
duplication formula [Si, p.53] it is easily checked that the rational
$4$-torsion points are $(2,\pm 3)$. Then $\kappa$ interchanges $(2,3)$ and
$(2,-3)$ by Lemma 6.4. For the remaining cusps $\kappa$ interchanges $(3,\pm
2)$ and $(6,\pm 3)$. To make this more precise we use the addition formulae
[Si, p.53] which show that $(0,0)\oplus(3,2)=(6,3)$. Hence $\kappa$
interchanges $(3,2)$ and $(6,3)$ by Theorem 6.1.
There remain the isolated vertices $1$, $4$ and $5$ each of whose stabilizer
is isomorphic to ${\mathbb{F}}_{49}^{*}$. We deal with these via their
connection with elliptic points. We recall from Theorem 4.4 and the above that
there exists a one-one correspondence

$\mathop{\mathrm{Ell}}(G)\leftrightarrow\ker N_{E}=\\{(\alpha,\beta)\in
E({\mathbb{F}}_{49}):(\overline{\alpha},\overline{\beta})=(\alpha,-\beta)\\},$

since the Weierstrass equation is in short form. Now let $i$ denote one of the
$2$ square roots of $-1$ in ${\mathbb{F}}_{q^{2}}^{*}$. Then

$N_{E}=\\{(\rho,\epsilon i)\in
E({\mathbb{F}}_{49}):\rho,\epsilon\in{\mathbb{F}}_{q}\\}.$

We conclude then that

$\mathop{\mathrm{Ell}}(G)\leftrightarrow\\{(\infty,\infty),(0,0),(1,\pm
3i),(4,\pm 2i),(5,\pm 3i)\\}.$

Here $\mathop{\mathrm{Ell}}(G)$ is identified with a subgroup of
$E({\mathbb{F}}_{49})$ listed as (additive) inverse pairs. Since there is only
one $2$-torsion point
$\mathop{\mathrm{Ell}}(G)\cong{\mathbb{Z}}/8{\mathbb{Z}}$. (In this case
$|\mathop{\mathrm{Cl}}(A)|=|\mathop{\mathrm{Ell}}(G)|$. However this not a
general feature. For this particular $K$ its $L$-polynomial is
$L_{K}(t)=1+7t^{2}$ so that $L_{K}(1)=L_{K}(-1)$.) As with
$\mathop{\mathrm{Cusp}}(G)$ the free action (Corollary 4.8) of $Quinn(G)$ on
$\mathop{\mathrm{Ell}}(G)$ is represented by the action of $(0,0)$ in $N_{E}$
(by point addition).
By identifications in Section $4$ the pairs $(1,\pm 3i),(4,\pm 2i),(5,\pm 3i)$
correspond with the vertices $1$, $4$ and $5$, respectively. By means of the
duplication formula it is readily verified that the two points of order $4$ in
$\mathop{\mathrm{Ell}}(G)$ are $5\pm 3i$. By Lemma 4.10 it follows that
$\kappa$ fixes vertex $5$ and that $\kappa$ interchanges vertices $1,4$. For a
more precise version of the latter statement we note that
$(0,0)\oplus(1,3i)=(4,2i)$ and so $(0,0)\oplus(1,-3i)=(4,-2i)$.
It is of interest to use Theorem 2.2 to construct a matrix $M$ which
represents $\kappa$. We begin with the $A$-ideal, $Ax+Ay$ whose square is
$Ax$. In determining a possible $M$ we recall from the proof of Theorem 2.4
the observation of Cremona [Cr] that every row and column of $M$ generates
${\mathfrak{q}}(M)$. Two possibilities which arise are

$M=\left[\begin{array}[]{cc}y&x^{2}\\\
x&y\end{array}\right]\;\mathrm{and}\;\left[\begin{array}[]{cc}y&-x^{2}\\\
x&-y\end{array}\right].$

The latter is simpler since its square is a scalar matrix.

c52$\infty$03614$g_{1}$$g_{2}$

$(\infty,\infty)$$(0,0)$$(3,2)$$(3,-2)$$(6,3)$$(6,-3)$$(2,3)$$(2,-3)$

######  Example 8.6.

$K={\mathbb{F}}_{7}(x,y),\;A={\mathbb{F}}_{7}[x,y]$ with $y^{2}=x^{3}-x$
It is easily verified that

$E({\mathbb{F}}_{7})=\\{(\infty,\infty),(0,0),(1,0),(6,0),(4,\pm 2),(5,\pm
1)\\},$

listed as (additive) inverse pairs. The $2$-torsion points are
$(0,0),(1,0),(6,0)$ and so
$Quinn(G)\cong\mathop{\mathrm{Cl}}(A)_{2}\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}$
and
$\mathop{\mathrm{Cusp}}(G)\cong\mathop{\mathrm{Cl}}(A)\cong{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/4{\mathbb{Z}}$.
Let the non-trivial quasi-inner automorphisms
$\kappa_{0},\kappa_{1},\kappa_{6}$ represent $(0,0),(1,0),(6,0)$,
respectively, where $\kappa_{0}=\kappa_{1}\kappa_{6}$. In the diagram
representing $G\backslash{\mathcal{T}}$ we label the $8$ cusps with the above
rational points in such a way that (i) the action of $\kappa_{6}$ is the
reflection about the vertical axis (ii) the action of $\kappa_{1}$ is
reflection about the horizontal axis and (iii) (consequently) the action of
$\kappa_{0}$ is a rotation of $180$ degrees about the “central” vertex $c$.
There are $4$ vertices whose stabilizers are isomorphic to
$GL_{2}({\mathbb{F}}_{7})$ which appear as “spikes” attached to the $4$ cusps
given by the $2$-torsion points in $E({\mathbb{F}}_{7})$ and so $\kappa_{6}$,
$\kappa_{1}$ and $\kappa_{0}$ interchange the vertex pairs
$\\{g_{1},g_{2}\\}$, $\\{g_{1},g_{4}\\}$ and $\\{g_{1},g_{3}\\}$,
respectively.
In $\mathop{\mathrm{Cl}}(A)$ there are $4$ points of order $4$, namely $(4,\pm
2)$ and $(5,\pm 1)$, and it is easily verified that the square of each is
$(1,0)$. By Lemma 6.4 it follows that $\kappa_{1}$ interchanges the cusps
$(4,2),(4,-2)$ as well as $(5,1),(5,-1)$. On the other hand $\kappa_{6}$
interchanges the pairs $(4,\pm 2)$ and $(5,\pm 1)$. In more detail
$\kappa_{6}$ maps $(5,1)$ to $(4,-2)$, since $(6,0)\oplus(5,1)=(4,-2)$.
There remain two vertices $2$ and $3$ whose stabilizers are cyclic order
$q^{2}-1$. As in the previous example we consider the elliptic function field
$\widetilde{K}=K.{\mathbb{F}}_{49}={\mathbb{F}}_{49}(x,y):y^{2}=x^{3}-x$. As
before let $i$ denote one of the square roots of $-1$ in ${\mathbb{F}}_{49}$.
It can be verified that

$\mathop{\mathrm{Ell}}(G)\leftrightarrow
N_{E}=\\{(\infty,\infty),((0,0),((1,0),(6,0),(2,\pm i),(3,\pm 2i)\\},$

listed as additive inverse pairs in $E({\mathbb{F}}_{49})$. As before
$|\mathop{\mathrm{Cl}}(A)|=|\mathop{\mathrm{Ell}}(G)|=8$. (Again this is
purely coincidental because $L_{K}(t)=1+7t^{2}$.) By correspondences discussed
in Section 4 the $2$ vertices of interest here correspond to the pairs $(2,\pm
i)$ and $(3,\pm 2i)$. It is easily verified that the squares of all $4$ of
these points are $(6,0)$. It follows from Lemma 4.10 that $\kappa_{6}$ fixes
$2$ and $3$. On the other hand $(1,0)\oplus(2,i)=(3,-2i)$ and so $\kappa_{1}$
interchanges $2$ and $3$. Finally using Theorem 2.2 the following matrices
$M_{0},\;M_{1},\;M_{6}=M_{0}M_{1}$ represent
$\kappa_{0},\kappa_{1},\kappa_{6}$, respectively,

$M_{0}=\left[\begin{array}[]{cc}y&-x^{2}\\\
x&-y\end{array}\right]\;\mathrm{and}\;M_{1}=\left[\begin{array}[]{cc}y&-(x-1)(x+2)\\\
x-1&-y\end{array}\right].$

c23$\infty$6451$g_{4}$

0$g_{3}$

$g_{2}$$g_{1}$

$(\infty,\infty)$$(6,0)$$(1,0)$$(0,0)$$(4,2)$$(4,-2)$$(5,-1)$$(5,1)$

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