Quasi-inner automorphisms of Drinfeld modular groups

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 {\it 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 {\it quasi-inner}. Modulo the inner automorphisms of $G$ they form a group $Quinn(G)$ which is isomorphic to ${\mathrm Cl}(A)_2$, the $2$-torsion in the ideal class group ${\mathrm Cl}(A)$. 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 $G\setminus{\mathcal T}$, generalizing an earlier result of Serre. It is known that the ends of $G\setminus{\mathcal T}$ 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 those ends which are in one-one correspondence with the vertices of $G\setminus{\mathcal T}$ whose stabilizers are isomorphic to $GL_2({\mathbb F}_q)$.


Introduction
Let K be an algebraic function field of one variable with constant field F q , the finite field of order q.Let ∞ be a fixed place of K and let δ be its degree.The ring A of all those elements of K which are integral outside ∞ is a Dedekind domain.Denote by K ∞ the completion of K with respect to ∞ and let C ∞ be the ∞-completion of an algebraic closure of K ∞ .The group GL 2 (K ∞ ) (and its subgroup G = GL 2 (A)) act as Möbius transformations on C ∞ , K ∞ and hence Ω = C ∞ \K ∞ , the Drinfeld upper halfplane.This is part of a far-reaching analogy, initiated by Drinfeld [Dr], where Q, R, C are replaced by K, K ∞ , C ∞ , respectively.The roles of the classical upper half plane (in C) and the classical modular group SL 2 (Z) are assumed by Ω and G, repspectively.Modular curves, that is quotients of the complex upper half plane by finite index subgroups of SL 2 (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\Ω, 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 (Z) and G is that, while the genus of the former is zero, for different choices of K and ∞ the genus of G can take many values.The simplest case, where K = F q (t) and A = F q [t] (equivalently g = 0 and δ = 1), has to date attracted most attention.An element ω ∈ Ω 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) = ∅ if and only if δ is odd.Clearly G acts on E(G) and the elements of the set of G-orbits, Ell(G) = G\E(G) = {Gω : ω ∈ E(G)}, are called the elliptic points of G.It is known [Ge,p.50] that Ell(G) is finite.In addition G acts on P 1 (K) = K ∪ {∞}.(Here, of course, ∞ refers to the one point compactification of K.) We refer to the elements of P 1 (K) as rational points.For each finite index subgroup, H, of G the elements of Cusp(H) = H\P 1 (K) are called the cusps of H. Since A is a Dedekind domain it is well-known that Cusp(G) can be identified with Cl(A), the ideal class group of A. As Möbius transformations G acts without inversion on T , the Bruhat-Tits tree associated with GL 2 (K ∞ ) and the ends of the quotient graph G\T are determined by 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\Ω will, after adding Cusp(H), be the C ∞ -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\Ω → G\Ω 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 Ell(G) and ker N, where N : Cl( A) → Cl(A) is the norm map and A = A.F q 2 .It can be shown [MS4] that Cl( A) 2 ∩ ker N, the 2-torsion subgroup of ker N, is in bijection with Ell(G) = = {Gω : ω ∈ E(G), Gω = Gω}, where ω, the conjugate of ω, is the image of ω under the Galois automorphism of K.F q 2 /K.(In [MS4] Ell(G) = is denoted by Ell(G) 2 .)Here we show that, when δ is odd, Cl(A) 2 and the 2-torsion in ker N are isomorphic.This is the starting point for this paper where the principal focus of attention is the group Cl(A) 2 and its actions on various objects related to G. Unless otherwise stated results hold for all δ.
Let g ∈ N GL 2 (K) (G), the normalizer of G in GL 2 (K).Then g, acting by conjugation, induces an automorphism ι g of G which we refer to as quasi-inner.If g ∈ G.Z(K) then ι g reduces to an inner automorphism.If g ∈ N GL 2 (K) (G)\G.Z(K) we call ι 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 Cl(A) 2 .Hence G has non-trivial quasi-inner automorphisms if and only if | Cl(A)| is even.Now, as an element of GL 2 (K), ι g acts as a Möbius transformation on the rational points and elliptic elements of G, as well as T .In particular g(ω) = g(ω).Since all of these actions are trivial for scalar matrices they extend to actions of Quinn(G) on Cusp(G), Ell(G) and the quotient graph, G\T .In this paper we study of the (often surprising) properties of these actions.
From the above it is clear that Quinn(G) can be embedded as a subgroup Ell(G) = (resp.Cl(A) 2 ) of Ell(G) (resp.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.
Theorem 1.4.Every non-trivial element of Quinn(G) determines an automorphism of G\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 = F q (t) with δ even.Our result shows that in general the quotient graph has symmetries of this type provided | Cl(A)| is even.(In general this restriction is necessary.)We now list more detailed results on the action of Quinn(G) on G\T .Serre [Se,Theorem 9,p.106] has described the basic structure of G\T .Its ends (i.e. the equivalence classes of semi-infinite paths without backtracking) are in one-one correspondence with the elements of Cl(A).To date the only cases for which the precise structures of G\T are known are g = 0, [Ma2], [KMS], and g = δ = 1, [Ta].
Theorem 1.5.Quinn(G) acts freely on the ends of G\T and, in addition, transitively on the ends of G\T corresponding to the elements of Cl(A) 2 , We show that the ends corresponding to Cl(A) 2 are in one-one correspondence with those vertices whose stabilizers are isomorphic to GL 2 (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 ∼ = F * q 2 or GL 2 (F q ).The building map [Ge,p.41]extends to a map λ : Ell(G) → vert(G\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 {ṽ ∈ vert(G\T ) : (b) Suppose that δ is odd and that ker N has no element of order 4. Then Quinn(G) acts freely on {ṽ ∈ vert(G\T ) : 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\T for each of which g = δ = 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].
(i) The isolated (i.e.(graph) valency 1) vertices of G\T are precisely those whose stabilizers are isomorphic to GL 2 (F q ) or F * q 2 .
(ii) If ker N has no element of order 4 then Quinn(G) acts freely on the isolated vertices of G\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.
F q the finite field with q = p n elements; K an algebraic function field of one variable with constant field F q ; g the genus of K; ∞ a chosen place of K; δ the degree of the place ∞; A the ring of all elements of K that are integral outside ∞; K ∞ the completion of K with respect to ∞; Ω Drinfeld's half-plane; T the Bruhat-Tits tree of GL 2 (K ∞ ); G the Drinfeld modular group GL 2 (A); Gx the orbit of x under the action of G on the object x; G GL 2 (K); the ideal class group of the Dedekind ring R; Cl 0 (F ) the divisor class group of degree 0 of the function field F ; Cusp(G) G\P 1 (K), the set of cusps of G; E(A) the set of elliptic elements of G: Ell(G) G\E(A), the set of elliptic points of G; ω the image of ω ∈ E(A) under the Galois automorphism of K/K; the stabilizer in a finite index subgroup S (of G) of s ∈ P 1 (K);

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 G (G), the normalizer of G in G = GL 2 (K).For each x ∈ F we use (x) as a shorthand for the fractional ideal Ax.
We state a special case (n = 2) of a result of Cremona [Cr] .Then M ∈ N G (G) if and only if where ∆ = det(M).
For part (ii) let x be any entry of M. Then x 2 ∈ A by Theorem 2.2 and so x ∈ A, since A is integrally closed.
Another important consequence [Cr] of Theorem 2.2 is the following.
Theorem 2.4.The map M → q(M) induces an isomorphism where Cl(A) 2 is the subgroup of all involutions in Cl(A).
Proof.This is another special case (n = 2) of a result in [Cr].If Consequently there is a map from N G (G) to Cl(A) 2 , which turns out to be an isomorphism.
Definition 2.5.An automorphism ι g of G is called quasi-inner if for some g ∈ N G (G).We call ι g non-trivial if g / ∈ Z(K).G, i.e. if ι g does not act like an inner automorphism.We note that

Finally we define
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 | Cl(A)| is odd.Let 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 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 Example 2.7.We illustrate the results of this section with the simplest case K = F q (t), the rational function field over F q .Then there exists a (monic) polynomial π(t) ∈ F q [t], of degree δ, irreducible over F q , such that It is known [St,Theorem 5.1.15,p.193]that here Cl 0 (K) is trivial so that Hence G has non-trivial quasi-inner automorphisms if and only if δ 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 δ = 2.In this case π(t) = t 2 + σt + τ , where σ ∈ F q and τ ∈ F * q .We begin with the A-ideal generated by π −1 and tπ −1 which is not principal.Let π(t) = tt ′ + τ and put Then by Theorem 2.2 g 0 ∈ N G (G) and from Theorem 2.4 we see g 0 / ∈ Z(K).G.Hence g 0 provides a generator of Cl(A) 2 .
Remark 2.8.From the theory of Jacobian varieties we know that the 2-torsion in 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)| = |Cl(A) 2 | ≤ 2 2g+1 (and ≤ 2 g+1 , when char(K) = 2).In odd characteristic we can easily find examples with | 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 F p r , just pick q = p rn such that all 2-torsion points of Jac(F ) are F qrational and consider K = F.F q .Then Cl 0 (K) 2 ∼ = (Z/2Z) 2g .Choosing a place ∞ of K of odd degree δ, from the exact sequence (1), we see that |Cl(A) 2 | = 2 2g .Similarly in characteristic 2 examples for which | Cl(A) 2 | = 2 g can be found by choosing F suitably, namely F has to be ordinary.
Whether for even δ 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 Cl(A) splits or not.
Definition 2.9.Let R, S be subgroups of a group T .We write Let S be a set of subgroups of T .We put 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 ι g be as above.
(i) It is clear that GL 2 (K ∞ ) acts on Ω as Möbius transformations and that this action is trivial for all scalar matrices.Then ι g acts on E(G) since, for all ω ∈ E(G), extends naturally to a well-defined action of Quinn(G) on Ell(G).(ii) Clearly G acts as Möbius transformations on P 1 (K) and it is well-known that As we shall see later from the structure of the quotient graph it follows that, for all extends to a well-defined action of Quinn(G) on Cusp(G).
(iii) Serre [Se, Chapter II, Section 1.1, p.67] uses lattice classes as a model for the vertices and edges of T .It is clear that GL 2 (K ∞ ) acts naturally on these.In particular the scalar matrices act trivially.The map where w ∈ vert(T ) ∪ edge(T ), extends to a well-defined action of Quinn(G) on the quotient graph G\T .Note that G g(w) = ((G w )) g ≤ G.We will use this action to extend a result of Serre.
(iv) Suppose that Then Quinn(G) acts by conjugation on S G .We use these to define actions of Quinn(G) on significant subsets of vert(T ).

Action on vertex stabilizers
Almost all the results in this section hold for all δ.We record the important general properties of subgroups of vertex stabilizers.
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 In the first part of this section we look at the action of quasi-inner automorphisms on the following set Proof.Follows from Lemmas 3.1 (ii) and 3.2.
Remark 3.4.(i) Every T contains a particular vertex v s , usually referred to as standard (after Serre), for which See [Se, Remark 3), p.97] (ii) On the other hand for the case A = F q [t] (equivalently g(K) = 0, δ = 1) it follows from Nagao's Theorem [Se,Corollary,p.87]that here vert(T ) has no stabilizer which is cyclic of order q 2 − 1.
Lemma 3.5.Let H ∈ H. Then there exists a quasi-inner automorphism 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 We denote by x the image of x ∈ K under the extension of the Galois automorphism of , where 1 ≤ i, j ≤ 2 and so for all x, y ∈ {a, b, c, d}, where ∆ = det(g).Now we may assume without loss of generality that c = 0. Let z ∈ {a, b, d}.Then It follows that z/c = z/c so that z/c ∈ K.We now replace g = M with g 0 = c −1 M.
Then by Theorem 2.2 the map κ 0 : G → G defined by κ 0 (x) = g 0 xg −1 0 is a quasi-inner automorphism of G.
Lemma 3.6.Let κ 0 = ι go be a non-trivial quasi-inner automorphism of G and let Proof.By definition g 0 ∈ N G (G)\G.Z(K).Suppose to the contrary that for some g ∈ G. Replacing g 0 with g −1 g 0 we may assume that g = 1.Now by Lemma 3.5 and so we may further assume that g 1 = g 0 .Let Now S p is a Sylow p-subgroup of GL 2 (F q ) and so from the above g 0 (S p )g −1 0 = h(S p )h −1 , for some h ∈ GL 2 (F q ).As above we may assume then that h = 1.It follows that g 0 "fixes" ∞ and so where γ = αβ −1 .Since A is integrally closed it follows that γ ∈ A * (= F * q ).Then we can replace g 0 with β −1 g 0 which belongs to G by Corollary 2.3 (ii).Thus g 0 ∈ Z(K).G.
Lemma 3.7.Let e ∈ edge(T ) be incident with v s .Then Proof.The edges attached to v s are parametrized by P 1 (F q δ ) and GL 2 (F q ) acts on these as Möbius transformations.See [Se, Exercise 6), p.99].If the edge corresponds to f ∈ F q δ it is not fixed by the translations in GL 2 (F q ), and if it corresponds to ∞, it is not fixed by 0 1 1 0 ∈ GL 2 (F q ).Proposition 3.8.No edge of T can have a stabilizer isomorphic to GL 2 (F q ).Proof.For odd δ this follows from [MS3,Corollary 2.16].We provide a proof that holds for all δ.Suppose to the contrary that there is an edge e whose stabilizer is isomorphic to GL 2 (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 T we can assume that It follows that GL 2 (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 ∈ H. Then there exists a unique vertex v ∈ vert(T ) such that 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 ∈ H. Then there exists κ = ι g such that κ(v) = v s Since κ is an automorphism of T the action of G v on the q δ + 1 edges of T incident with v is identical to the action of GL 2 (F q ) on the edges of T incident with v s as described in Lemma 3.7.
We put v = Gv and define its stabilizer We refer to G v as being isomorphic to G v .
Lemma 3.12.There exists a bijection Proof.Follows from Corollary 3.9 and the above.
It is clear that Quinn(G) acts on H G .Since Z(K), represented by scalar matrices, acts trivially on T , it is also clear that Quinn(G) acts on G\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 (F q ), (ii) the vertices of G\T whose stabilizers are isomorphic to GL 2 (F q ) .
A special case of this result is provided by Corollary 2.6.
image of the standard vertex v s .

Action on elliptic points
Throughout this section we assume that δ is odd.Recall that denotes the elliptic points of the Drinfeld modular curve G\Ω.(In [MS4, Section 3] Ell(G) = is denoted by Ell(G) 2 .) The action of an element of GL 2 (K ∞ ) on an element of Ω will always refer to its action as a Möbius transformation.We record the following.
It is clear then that Quinn(G) acts on both Ell(G) = and Ell(G) = .
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.A-ideal).Then [I] denotes the image of I in Cl(A) (resp.Cl( A)).
Fix ε ∈ F q 2 \ F q .By [MS4, Theorem 2.5] any elliptic point ω of G can be written as ω = ε+s t where s, t ∈ A and t divides (ε + s) Let α be the Galois automorphism of K/K (which extends that of F q 2 /F q ).Let k ∈ K. Then the norm of k is kk, where k = α(k).Now α restricts to A and so acts on its ideals and hence its ideal class group.For each A-ideal, J, the norm of J, N(J) = A ∩ (J J ), which is an A-ideal.We now come to the norm map where We restate [MS4,Theorem 3.4].
Theorem 4.4.The map ω → [J ω ] induces a one-one correspondence For each ω it is known that We recall from Theorem 2.4 that Quinn(G) can be identified with Cl(A) 2 .From this and Theorem 4.4 we are able to study the action of Quinn(G) on Ell(G).For this purpose we require two further lemmas.Proof.The analagous statements are known to hold for the canonical map from Cl 0 (K) → Cl 0 ( K). See [Ro,Corollary to Proposition 11.10].The results follow from the exact sequence in Section 2, since δ is odd and the infinite place is inert in K.
Lemma 4.6.With the above notation, the 2-torsion in Cl(A), See Theorem 2.2 and [Cr,Remarks 2].In this way q induces an isomorphism from Quinn(G) onto Cl(A) 2 and so ι • q provides an embedding of Quinn(G) into Cl( A).
As before each ω ∈ E(G) can be represented as ω = ε+s t where s, t ∈ A and t divides (ε q + s)(ε + s) in A. The element M acts as a Möbius transformation on ω by multiplying the column vector ε+s t on the left by the matrix M. It follows that J M (ω) is the A-ideal generated by a(ε + s) + bt and c(ε + s) + dt.Our next result, the most important in this section, shows that the action of Quinn(G) on Ell(G) is equivalent to group multiplication in ker N .
Theorem 4.7.With the above notation, Proof.From the above it is clear that J M (ω) ≤ q(M)J ω .Since A is a Dedekind domain, there is an integral ideal I 1 of A such that By the same argument there exists an integral ideal I 2 of A with On the other hand, from part (i) of Corollary 2.3 we see that J M 2 (ω) = ∆J ω .Hence I 1 = I 2 = 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 \ Ω must necessarily be inner.
Theorem 3.13 (ii), which holds for all δ, provides an alternative proof of Theorem 4.9.Applying the former for the case of odd δ the latter then follows from the existence of a Quinn(G)-invariant one-one correspondence between Ell(G) = and From the above it is clear that where We recall that the building map [Ge,p.41]restricts to a map for which G ω ≤ G λ(ω) .Let κ be a quasi-inner automorphism.Then by [Ge, (iii), p.44] Then λ induces a map Ell(G) → vert T .
The following is an immediate consequence.
When n E = 2 (a) applies.When n E = 3 the two Quinn(G)-orbits represented by Gω and Gω are identified in G. (c) By Lemma 4.10 the action of Quinn(G) on G is not free if and only if there exists [J ω ] 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 Cl 0 ( K) or Cl( A) (and using the fact that δ is odd) it follows that See [Ro,Chapter 11].On the other hand by the Riemann Hypothesis for function fields [St,Theorems 5.1.15(e),5.2.1] If n E = 2 then 2 2g+1 ≥ ( √ q − 1) 2g .
Another consequence follows using an identical argument.(b) If q ≥ 23 (and g > 0), then Quinn(G) cannot act transitively on V.
Remark 4.14.It is known [MS3, Corollary 2.12, Theorem 5.1] that a vertex v of G\T is isolated if and only if δ = 1 and G v ∼ = GL 2 (F q ) or F * q 2 .Hence when δ = 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\T .

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 δ 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 ∈ H which contains C. Moreover H is unique if δ is odd.
Proof.By Lemma 3.1 (ii) there exists G v which properly contains C. Hence G v ∈ H by Lemma 3.2.Suppose now that δ is odd.If H is not unique then where v 1 = v 2 .It follows that C fixes the geodesic in 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 ∈ H. Then C, C 0 are conjugate in H.
Proof.By Lemma 3.5 we may assume that H = GL 2 (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 for some µ ∈ F q 2 \F q .Let C 0 = F µ 0 .Now µ 0 = αµ + β for some α, β ∈ F q , where α = 0. Then C 0 = g 0 Cg −1 0 , where Clearly every automorphism of G acts on both C mf and 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 ∈ C nm .Then by Lemmas 3.5 and 5.2 there exists The rest follows from Lemma 5.3.
The next result follows from Proposition 5.5 and Theorem 3.13.
Proposition 5.6.If δ 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 δ in Lemma 5.2 and Proposition 5.6 are necessary.
Example 5.7.Consider the case where g(K) = 0, δ = 2.This case is studied in detail in [MS1,Section 3].By the exact sequence in Section 2 it is known that here There exists a vertex v 0 adjacent to the standard vertex v s and Hence the restriction on δ in part of Lemma 5.2 is necessary.
It is known [MS1,Theorem 3.3] that in this case where C(= GL 2 (F q ) ∩ GL 2 (F q ) g 0 ) ∈ C nm .It follows by Lemma 5.3 that there exists g ∈ GL 2 (F q ) for which In this case therefore Quinn(G), which is non-trivial, fixes C G .The restriction on δ in Proposition 5.6 is therefore necessary.
We conclude this section with some remarks about C mf .
Lemma 5.8.Suppose that δ is odd.Then Then by Lemmas 3.1 and 3.3 it follows that C ≤ G v ∩ G v 0 for some v 0 = v, which contradicts [MS3,Corollary 2.16].The rest follows from Lemma 3.1.
When δ is odd there is therefore a one-one correspondence For the case where δ is odd this shows that the results in Proposition 4.12 apply to the action of Quinn(G) on (C mf ) G .Remark 5.9.As a Möbius transformation every member of G fixes an element of C ∞ .Suppose now that δ 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 µ ∈ K.F q 2 \K.In this case however µ ∈ K ∞ as δ is even.So µ, which is not in Ω and not in K, can neither be an inner point nor a cusp of the Drinfeld modular curve G\Ω.We refer to µ as pseudo-elliptic.On the other hand suppose that δ is odd.Let g be any element of infinite order in G and let g fix λ.Then λ ∈ K ∞ \K.

Action on cusps
As distinct from Section 4 the results here hold for all δ.Any element of G acts on P 1 (K) = K ∪ {∞} as a Möbius transformation.In this way Quinn(G) acts on G\P 1 (K) = Cusp(G).Every element of Cusp(G) can be represented in the form (a : b), where a, b ∈ A. Since A is a Dedekind ring this gives rise to a one-one correspondence Cusp(G) ←→ Cl(A).
Hence the action of Quinn(G) on Cusp(G) translates to an action of Cl(A) 2 on 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 Cl(A).We sketch a proof.We can represent any cusp, c, by an element (x : y) ∈ P 1 (K), where x, y ∈ A. Let and let [J c ] be its image in Cl(A).Now let κ be a non-trivial element of Quinn(G).Then as before by Theorem 2.2 κ can be represented by a matrix where we may assume that a, b, c, d ∈ A. Let q(M) be the A-ideal generated by a, b, c, d.
The action of κ on c is given by the action of M multiplying the column vector x y on the left by M. In this way Theorem 6.1.Under the identification of Cusp(G) with Cl(A) and Quinn(G) with Cl(A) 2 the action of Quinn(G) on the cusps translates into multiplication in the group Cl(A).More precisely Proof.Since A is a Dedekind domain there exists an A-ideal I 1 such that By Corollary 2.3 (i) there exists an A-ideal I 2 with where ∆ = 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 κ fixes any cusp, then κ reduces to an inner automorphism.In particular, Quinn(G) acts freely on Cusp(G).
From the exact sequence in Section 2 a necessary condition for this is δ ∈ {1, 2}.If g(K) = 0, this condition is also sufficient, as then Cl(A) ∼ = Z/δZ.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 shows that for fixed q > 9 the number of orbits of Quinn(G) on Cusp(G) tends to ∞ with g(K).
The cusp ∞(= 1 0 ) corresponds to the principal A-ideals.Its orbit under Quinn(G) corresponds to the 2-torsion in Cl(A) and in the sense of Theorem 6.1 the action of Quinn(G) on it translates into Cl(A) 2 acting on itself by multiplication.For every cusp c represented by the ideal class [J c ] in Cl(A) there corresponds its (group) inverse [J c ] −1 in Cl(A).We can partition Cl(A) thus Our next result follows from Theorem 6.1 analogous to the way Lemma 4.10 follows from Theorem 4.7.
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\T .

Action on the quotient graph
The model used by Serre for T [Se, Chapter II, Section 1.1] is based on two-dimensional so called lattice classes .Since every quasi-inner automorphism, ι g , can be represented by a matrix in G it acts on T and hence Quinn(G) acts on G\T .In this section we investigate the action of a quasi-inner automorphism on the quotient graph H\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\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\T .Definition 7.1.A ray R in a graph G is an infinite half-line, without backtracking.In accordance with Serre's terminology [Se,p.104]we call R cuspidal if all its nonterminal vertices have valency 2 (in G).
is a complete set of representatives for Cl(A), where Theorem 7.2.There exists a complete system of representatives C (⊆ P 1 (K)) for Cusp(G)(equivalently, Cl(A)) of the above type such that ) is a cuspidal ray (in G\T ), whose only interesection with X consists of a single vertex, (iii) the | Cl(A)| cuspidal rays are pairwise disjoint.
Moreover if R(e) is any of these cuspidal rays then it has a lift, R(e), to T with the following properties.Let vert(R(c where G(c) is the stabilizer (in G) of the cusp c.
For each j let d j be the element of {d 1 , • • • , d t } corresponding to h −1 j (∞).We may relabel the latter set as {d 1 , d 1 , • • • , d t ′ , d t ′ }, where t ′ = t/2.We can use the results in Section 3 to elaborate on the structure of the above cuspidal rays.We recall that Proof.Let v ∈ vert(G\T ), where G v ∈ H, and let H ∈ H be any representative of its stabilizer.Then, for some unique i, where g ∈ G, by Lemmas 3.5 and 3.6.Now let u be any unipotent element of H.
Then u fixes gg i h(∞), for some h ∈ GL 2 (F q ).It follows that where g ′ ∈ G.The rest follows from Corollary 3.9 together with Theorem 3.13.
Remark 7.4.Let v ∈ vert(G\T ), where G v ∈ H. Then it is shown in Corollary 7.3 that v is adjacent in G\T to a vertex whose stabilizer (up to conjugacy in G) is contained in G(c i ), for some unique i.In this way v can be thought of as closer in G\T to R(c i ) than to any other cuspidal ray.For the case δ = 1 (and only for this case) v is isolated in G\T by [MS3,Theorem 5.1].As in Takahashi's example [Ta] such a v then appears as a "spike" next to its associated cuspidal ray.(iv) Under the action of Quinn(G) some E(d j ) is mapped to E( dj ) if and only if d j has order 4 in Cl(A).
We recall from Proposition 4.12 that when δ is odd Quinn(G) also acts on { v ∈ vert(G\T ) : G v ∼ = F * q 2 }.Our final result in this section concerns the action of N G (G) on T .it is known [Se,Corollary,p.75]that G acts without inversion (on the edges) of T .
Proposition 7.11.Suppose that δ is odd.Then every ι g acts without inversion on T and hence on every quotient graph H\T .
Proof.As in Theorem 2.2 we can represent ι g with a matrix M in G and we can assume that all its entries lie in A. Let ∆ = det(M).Then the A-ideal generated by ∆ is the square of an ideal in A, again by Theorem 2.2.It follows that, for all places v = v ∞ , v(∆) is even.By the product formula then δv ∞ (∆) and hence v ∞ (∆) 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 δ(K) = 2.We recall that there exists a quadratic polynomial π ∈ F q [t], irreducible over F q , such that In this case it is known that Cl(A) 2 = Cl(A) ∼ = Quinn(G) ∼ = Z/2Z.It is wellknown that G\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\T lifts to a doubly infinite line D in T which we now describe in detail.For some Then D maps onto (and is isomorphic to) G\T which has the following structure.
The action of the (essentially only) non-trivial quasi-inner automorphism of G\T (represented by g 0 ) is given by v i ↔ v * i (i ≥ 0).We note two features of D which are of interest relevant to this section.(i) From the structure of 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 δ in Proposition 7.11 is necessary.(ii) For this case there is only one stabilizer invariant involution.However the graph G\T has many automorphisms.Infinitely many examples include translations (which have infinite order) and reflections in any vertex (which are involutions).

Two instructive examples
We conclude with two examples which demonstrate how our results apply to the structure of the quotient graph G\T .Both are elliptic function fields K/F q .We record some of their basic properties.
where x, y satisfy a (smooth) Weierstass equation F (x, y) = 0 with Here the group operation is point addition ⊕ according to the chord-tangent law.
Here a rational point (a, b) ∈ E(F q ) corresponds to the ideal class of A(x−a)+A(y−b).
We also require some "elliptic" properties of K = K.F q 2 (which is a constant field extension of K).
Corollary 8.3.Suppose that K/F q is elliptic.Then K/F q 2 is also elliptic and defined by the same Weierstrass equation.
With our choice of infinite place we have where x and y satisfy the Weierstrass equation F (x, y) = 0.In an analagous way We recall that the image of any α ∈ F q 2 under the Galois automorphism of F q 2 /F q is denoted by α.For each rational point P = (α, β) ∈ E(F q 2 ) we put P = (α, β).
Corollary 8.4.Suppose that K/F q is elliptic.Under the identifications of Cl 0 ( K) (resp.Cl 0 (K)) with E(F q 2 ) (resp.E(F q )) the norm map N : Cl 0 ( K) → Cl 0 (K) translates to a map N E : E(F q 2 ) → E(F q ) defined by N E (P ) = P ⊕ P , so that P ∈ ker N E ⇔ 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\T is a tree.Since δ = 1, for the case of a finite field of constants, the isolated vertices of G\T are precisely those whose stabilizer is isomorphic to GL 2 (F q ) or F * q 2 by [MS3, Theorem 5.1].For each cusp c ∈ Cl(A) 2 the cuspidal ray R(c) in G\T has attached to its terminal vertex (appearing as a "spike") an isolated vertex with stabilizer isomorphic to GL 2 (F q ).The remaining cuspidal rays consist of 1 2 | Cl(A)\ Cl(A) 2 | inverse pairs {R(c), R(c −1 )} which share a terminal vertex (appearing in G\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 where a, b ∈ F q and f (x) has no repeated roots.

For
each subgroup H of G we recall that the elements of H\T are vert(H\T ) = {Hv : v ∈ vert(T )} and edge(H\T ) = {He : e ∈ edge(T )} .Definition 7.5.Let H, H * be isomorphic subgroups of G.An isomorphism of graphs φ : H\T → H * \T , is said to be stabilizer invariant if the following condition holds.For any w ∈ vert(T ) ∪ edge(T ) let φ(Hw) = H * w * , (where w * ∈ vert(T ) if and only if w ∈ vert(T )).Then, for all u ∈ H w and u * ∈ H * w * H u ∼ = 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 κ = ι g , where g ∈ N G (G) and let H be a subgroup of G. Then the map κ H : H\T → κ(H)\T , defined by κ H (Hw) = H ′ w ′ , (iii) Quinn(G) acts on {E(d j ), E( dj )} : 1 ≤ j ≤ t ′ .