Running head: Automorphisms of Mumford curves August 23, 1999 Math. Subj. Class. (1991): 14E09, 20E08, 30G06

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

by Gunther Cornelissen, Fumiharu Kato and Aristides Kontogeorgis

Introduction

All compact Riemann surfaces (vis-à-vis complex projective curves) of genus g ≥ 2 share the same universal topological covering space, and hence admit a uniformization by a discrete subgroup of P SL(2, R). This description leads to a fruitful dualism in the theory of complex curves, visible in both its theorems and their proofs. The most appropriate instance of this to be quoted here is probably the fact that Hurwitz’s upper bound 84(g − 1) on the number of automorphisms of a compact Riemann surface of genus g ≥ 2 is equivalent to Siegel’s lower bound π/21 on the volume of the fundamental domain of a Fuchsian group (cf. Lehner, [16], p. 402-405).

Although over a non-archimedean valued field k, compact rigid-analytic and projective algebraic curves can still be identified, the analogue of the dualism to discrete group theory is of quite a different nature, as not all curves admit the same universal topological covering space (e.g., curves having good reduction are analytically simply connected). But Mumford has shown that curves whose stable reduction is split multiplicative (i.e., a union of rational curves intersecting in k-rational points) are isomorphic to an analytic space of the form Γ\(P_{k}^{1} − L_{Γ}), where Γ is a discontinuous group in P GL(2, k) with L_{Γ} as set of limit points. Thus, the theory of non-archimedean discrete groups is both more restrictive than the complex analytic one (as it cannot be applied to any curve), and more powerful, as it can lead to stronger results for such so-called Mumford curves.

As an example of this, Herrlich has shown that for p-adic Mumford curves of genus g ≥ 2, Hurwitz’s bound can be strengthened to 12(g − 1) (if p ≥ 7, cf. [10]). One can wonder what happens for Mumford curves over non-archimedean fields of positive characteristic.

It ought to be mentioned that in positive characteristic, general algebraic curves can have many more automorphisms than expected from Hurwitz’s bound, e.g., there exist algebraic curves of arbitrary high genus with more than 16g^{4} automorphisms (Stichtenoth, [26]). It has been observed on many occasions that the most anomalous examples invariably have a low p-rank. One is therefore led to expect better properties for so-called ordinary curves, i.e., curves X of genus g for which

dim_{Z/p} Jac(X)[p] = g,

which are Zariski-dense in the moduli space of curves of genus g. Indeed, S. Nakajima ([19]) has shown that for such curves, |Aut(X)| ≤ 84g(g − 1). However, one knows of no infinite set of ordinary curves of arbitrary high genus all of whose elements attain this bound. Rather, the number of automorphisms in all such known collections does not exceed a cubic polynomial in \sqrt{g}, and it has been suggested that worse cannot happen.

Mumford curves are known to be ordinary (§1), whence the aforementioned pathologies are moderated by Nakajima’s result. It is our aim to show that a bound of the type suggested before indeed holds for Mumford curves:

Theorem. Let X be a Mumford curve of genus g ≥ 2 over a nonarchimedean valued field of characteristic p > 0. Then

|Aut(X)| ≤ max\{12(g − 1), 2\sqrt{g}(\sqrt{g} + 1)^{2}\}.

Actually, the proof yields more: it provides a kind of classification of those curves X for which |Aut(X)| > 12(g − 1) (§4).

Remark. A reformulation of the above theorem in the style of Siegel’s lower bound is as follows: if Γ is a Schottky group of rank g, then the μ-invariant of its normalizer N (cf. [15], [10]) is bounded from below by

μ(N) ≥ min\{\frac{1}{12}, \frac{\sqrt{g} − 1}{2\sqrt{g}(\sqrt{g} + 1)}\}.

The proof of the main theorem uses the description of the automorphism group by the normalizer N of its Schottky group Γ, whose structure can be studied by its action on the Bruhat-Tits tree 𝒯 of P GL(2, k). Although this principle of proof was also used by Herrlich in the p-adic case, the quintessence of our techniques is rather different: since the expected bound in positive characteristic is not linear in the genus, we cannot in general restrict to normalizers that are the amalgam of two finite groups. Instead, we investigate directly the link between the ramification in π : X → S := Aut(X)\X and the combinatorial geometry of the analytic reduction of S. From Hurwitz’s formula applied to π, it is immediate that the bound 12(g − 1) on the order of the automorphism group holds, unless if S = P^{1} and π is branched above m = 2 or m = 3 points. Let 𝒯_{N} (respectively 𝒯_{N}^{∗}) be the subtree of 𝒯 which is generated by the limit points of Γ, seen as ends of 𝒯 (respectively the limit points together with the fixed points of torsion elements in N). The quotients T_{N} = N\𝒯_{N} and T_{N}^{∗} = N\𝒯_{N}^{∗} are homotopic to the intersection graph of the analytic reduction of S as a rigid analytic space, but T_{N}^{∗} has finitely many ends attached to it, which are in bijection with the points above which π is branched. The advantage of using T_{N}^{∗} instead of the usual T_{N} lies in the following key proposition, which replaces the “restriction to an amalgam of two groups” in the p-adic case:

Proposition 1. If T is a subtree of T_{N}^{∗} having the same ends as T_{N}^{∗}, then T is a contraction of T_{N}^{∗}, i.e., every geodesic connecting a point from T_{N}^{∗} − T to T is a path on which the stabilizers of vertices are ordered increasingly w.r.t. inclusion in the direction of T. In particular, the amalgams associated to T and T_{N}^{∗} are the same (= N).

The proof of this result is very combinatorial and depends on the structure theorem for finite subgroups of P GL(2, k) (§3).

By applying proposition 1, it is enough to consider a simpler subtree T of 𝒯_{N}^{∗} which is a “line” if m = 2 and a “star” if m = 3. We then show that only finitely many types of such trees T (and hence, of such groups N) exist (§4). For each such type of group, we use the link between the ends of the corresponding T and the ramification in π to prove the desired bound (§5).

Remark. Proposition 1 holds in any characteristic, and can be applied to obtain a classification of Galois covers of P_{C_{p}}^{1} which are Mumford curves ramified above three points, as in [14].

We now turn to some comments on the main theorem. Note that 12(g−1) > 2\sqrt{g}(\sqrt{g} + 1)^{2} only for g ∈ \{5, 6, 7, 8\}. But there do exist curves of such genus for which the bound is attained. Let M_{g} (respectively, M_{g}) denote the moduli space of algebraic curves of genus g (respectively, of Mumford curves of genus g), considered as algebraic space (respectively, rigid analytic space). We then have the following result, which identifies a very special locus in M_{6}:

Proposition 2. If X is a Mumford curve of genus g over a field k of characteristic p > 0 such that Aut(X) is a non-solvable group of order > 2\sqrt{g}(\sqrt{g} + 1)^{2}, then p ≠ 2, 5 and X is a curve of genus 6 with an icosahedral isomorphism group. For p > 5, there is a one-dimensional open stratum M_{6}, along which the automorphism group is icosahedral. It intersects M_{6} in a one-dimensional connected rigid analytic open set (§7).

Let ∗ denote amalgamated product. The bound in the above theorem is sharp in the following sense:

Proposition 3. For fixed p, there exist Mumford curves X_{t,c} of arbitrary high genus g_{t} = (p^{t} − 1)^{2} that attain the bound in the main theorem; X_{t,c} is the projective curve with affine equation (x^{p^{t}} − x)(y^{p^{t}} − y) = c, for some c ∈ k^{∗} with |c| < 1. The normalizer of its Schottky group is isomorphic to

N_{t} = (Z_{p}^{t} ⋊ Z_{p^{t}−1}) ∗_{Z_{p^{t}−1}} D_{p^{t}−1},

where Z_{n}, D_{n} are the cyclic and dihedral group of order n and 2n respectively. The Schottky group of X_{t,c} is generated by the commutators [ε, γεγ], for all ε ∈ Z_{p}^{t}, where γ is a fixed involution in D_{p^{t}−1} − Z_{p^{t}−1}. For varying c, the X_{t,c} form the one-dimensional connected open rigid analytic stratum of curves in M_{g_{t}} that attain the bound (§9).

Another less obvious family of Mumford curves whose number of automorphisms exceeds Hurwitz’s bound is given by moduli schemes for rank two Drinfeld modules with principal level structure (e.g., Gekeler & Reversat [4]). The analytic description is as follows: Let q = p^{t}, F = F_{q}(T), and A = F_{q}[T]; let F_{∞} = F_{q}((T^{−1})) be the completion of F and C a completion of the algebraic closure of F_{∞}. On Drinfeld’s “upper half plane” Ω := P_{C}^{1} − P_{F_{∞}}^{1} (which is a rigid analytic space over C), the group GL(2, A) acts by fractional transformations. Let Z ≅ F_{q}^{∗} be its center. For 𝔫 ∈ A, the quotients of Ω by congruence subgroups Γ(𝔫) = {γ ∈ GL(2, A) : γ = 1 mod 𝔫} are open analytic curves which can be compactified to projective curves X(𝔫) by adding finitely many cusps. They turn out to be Mumford curves for the free group which splits the inclusion Γ(𝔫)_{tor} ⊲ Γ(𝔫), where Γ(𝔫)_{tor} is the subgroup generated by torsion elements (cf. Reversat [21]).

Proposition 4. Let p ≠ 2 or q ≠ 3. The automorphism group of the Drinfeld modular curves X(𝔫) is the “modular” automorphism group G(𝔫) := Γ(1)/Γ(𝔫)Z. The normalizer of its Schottky group is isomorphic to

P GL(2, p^{t}) ∗_{Z_{p}^{t}⋊Z_{p^{t}−1}} Z_{p}^{td} ⋊ Z_{p^{t}−1},

which only depends on d = deg(𝔫). For fixed g with g = g(X(𝔫)) for some n, X(𝔫) is the unique Mumford curve with G(𝔫) as automorphism group (§10).

Note that, whereas the corresponding statement for classical modular curves is easy to prove using the Hurwitz formula ([24]), the occurence of wild ramification in X(𝔫) → X(1) prevents us from giving an easy proof of the above result. Note also that the final statement of proposition 4 highlights the exceptional status (as a point of the moduli space) of such modular curves which admit simultaneous uniformization by a Schottky group and by an arithmetic group on an open part.

We want to finish this introduction by addressing a few questions raised by this work.

Question 1 (ordinary curves). Does the above bound with 12(g − 1) replaced by 84(g − 1) hold for all ordinary curves?

Questions 2 (Drinfeld modular curves). Can one be more precise about the abstract structure of the Schottky group of X(𝔫)? It should depend, not only on deg(𝔫), but also on the decomposition of 𝔫 into prime ideals of A.

1. Automorphisms and uniformization in positive characteristic

(1.1) Let X be a complete irreducible curve of genus g ≥ 2 over a field k of characteristic p > 0. Let K denote the algebraic closure of k. Let γ denote the p-rank of X, i.e., γ = dim_{Z/p} Jac(X)(K)[p]. If γ = g, then X is called ordinary (such curves are dense in the moduli space of curves of genus g). Recall ([6], [18]) that X is called a Mumford curve if it is “uniformized over k by a Schottky group”. This means that (a) k is complete with respect to a non-archimedean valuation; (b) there exists a free group Γ of rank g in P GL(2, k), acting on P_{k}^{1} with limit set L_{Γ}; (c) X ≅ Γ\(P_{k}^{1} − L_{Γ}) as rigid analytic spaces. Mumford has shown that these conditions are equivalent to the stable reduction of X only having components of zero genus, which intersect in k-rational points. Because of the “GAGA”-correspondence for one-dimensional rigid analytic spaces, we do not have to (and will not) distinguish between analytic and algebraic curves. The following result is folklore:

(1.2) Lemma. Mumford curves are ordinary.

Proof. It is known by the works of Manin-Drinfeld ([17]) and Gerritzen ([5]) that the Jacobian of a Mumford curve X associated to a free group Γ can be uniformized as a rigid analytic abelian variety by a “multiplicative lattice”, i.e.,

Jac(X)(K) ≅ (K^{∗})^{g}/Λ

for some lattice Λ (actually, Λ ≅ Γ^{ab}) with multiplicative basis {λ_{1}, ..., λ_{g}}. Then

Jac(X)[p] ≅ ^{p} \sqrt{Λ}/Λ,

where ^{p} \sqrt{Λ} is generated by \{ ^{p} \sqrt{λ_{1}}, ..., ^{p} \sqrt{λ_{g}}\} (since the p-th roots of unity are trivial in k), which clearly has Z/p-rank = g. □

The following is the rigid analytic analogue of a well known theorem on conformal automorphisms of Riemann surfaces (e.g., [16], VI.3.L):

(1.3) Theorem (Gerritzen [6], VII.1). Let X be a Mumford curve over k uniformized by a Schottky group Γ. Let N be the normalizer of Γ in P GL(2, k). Then Aut(X) ≅ N/Γ. □

Although N is not free, it makes sense to consider the 1-dimensional analytic quotient space S := N\(P_{k}^{1} − L_{N}), which can be algebraized, since it is the quotient of X by the finite group Aut(X) = N/Γ.

(1.4) Notation. Denote the genus of S by r, and denote by m the number of points of S above which π : X → S is branced.

(1.5) Lemma. If r > 0 or m > 3, then |Aut(X)| ≤ 12(g−1). Furthermore, the case r = 0, m = 1 cannot occur.

Proof. The first part is an easy consequence of the Riemann-Hurwitz formula applied to π. (cf. Stichtenoth’s proof of Satz 3 in [26]). The second part follows from (1.2) and the same formula, since S. Nakajima has shown in [19], Theorem 2, that for ordinary X, the ramification groups of order ≥ 2 in π vanish. □

2. Structure of T_{N}^{∗}

From now on, we will assume that r = 0 and give a description of the map π by looking at its analytic reduction.

(2.1) The Bruhat-Tits tree. By taking a finite extension of k if necessary, we can assume that all fixed points of N acting on P_{k}^{1} are k-valued (cf. [6], I.3.3.2). Let O be the ring of integers of k and ϖ a uniformizer of k. Let 𝒯 denote the Bruhat-Tits tree of P GL(2, k) (i.e., its vertices are similarity classes Λ of rank two O-lattices in k^{2}, and two vertices are connected by an edge if the corresponding quotient module has length one – see Serre [23], Gerritzen & van der Put [6]). It is a regular tree in which the edges emanating from a given vertex are in one-one correspondence with P^{1}(k̄), where k̄ is the residue field of k. 𝒯 admits a left action by P GL(2, k), and hence also by any subgroup G ⊆ N; by our assumption on the fixed points, G acts without inversions on 𝒯.

(2.2) Notations. For any subtree T of 𝒯, let Ends(T) denote its set of ends (i.e., equivalence classes of half-lines different by a finite segment). There is a natural correspondence between P^{1}(k) and Ends(𝒯). Let V (T) and E(T) denote the set of vertices and edges of T respectively. For any Λ ∈ V (T), σ ∈ E(T), let σ ⊣ Λ mean that σ originates at Λ. Let G_{x} denote the stabilizer of a vertex or edge x of T for the action of G. Vertical bars |·| will denote geometric realization. For any u, v ∈ P^{1}(k), let ]u, v[ denote the apartment in 𝒯 connecting u and v (seen as ends of 𝒯). Furthermore, for a vertex Λ ∈ V (𝒯), let [Λ, u[ denote the half-line connecting Λ to u (seen as an end of 𝒯). The natural metric on |𝒯| is denoted by d.

(2.3) Action of elements of finite order. Let G be a subgroup of N. If γ ∈ G is an element of finite order, let the mirror M(γ) of γ be defined as the smallest subtree of 𝒯 generated by pointwise γ-fixed vertices of 𝒯 (this definition differs from the one given in [14]).

If γ is an elliptic element (i.e., an element in G of finite order having two distinct eigenvalues of the same valuation), then it has two fixed points in P^{1}(k), and M(γ) is just the geodesic connecting them (recall that k is of positive characteristic).

If γ is a parabolic element (i.e., an element in G of finite order having only one eigenvalue), then it has a unique fixed point z. Assume that k is discretely valued and k̄ is finite. For every w ∈ P^{1}(k) different from z, there is a unique vertex Λ_{w} ∈]w, z[ such that M(γ)∩]w, z[= [Λ_{w}, z[. For any two w, w′, Λ_{w} and Λ_{w′} lie at the same distance from [Λ_{w}, z[∩[Λ_{w′}, z[. Indeed, let {e_{0}, e_{1}} be the standard basis for k^{2}, and let the parabolic element γ be a translation γ : X ↦ X + t for some t ∈ k^{∗}, so that its fixed point z is z = ∞. Let w = 0. Vertices in the apartment connecting 0 and ∞ are similarity classes of lattices Λ_{j} = Oe_{0} +Oϖ^{j}e_{1}, and a short computation shows that Λ_{j} ∈ M(γ) if and only if j ≥ −v(t).

[figure: Picture 1. The mirror of a hyperbolic element if q = 2]

(2.4) Trees associated to subsets of P^{1}(k). We can construct a locally finite tree 𝒯(L) (possibly empty) from any compact subset L of P^{1}(k) (cf. [6]): it is generated by vertices that are the intersection of the three apartments connecting any three distinct points in L, seen as ends of 𝒯.

For any subgroup G of N we define 𝒯_{G} to be the tree associated to the subset L_{G} consisting of the limit points of G in P^{1}(k) (if G is finite, 𝒯_{G} is empty). We also define 𝒯_{G}^{∗} to be the tree associated to the set consisting of all limit points of G and all fixed points of elements of finite order in G.

𝒯_{G} and 𝒯_{G}^{∗} admit a natural action of G, and we denote the quotient graphs by T_{G} := G\𝒯_{G} and T_{G}^{∗} := G\𝒯_{G}^{∗} respectively, and both corresponding quotient maps will be written as π_{G}, by slight abuse of notation. We can consider T_{G} and T_{G}^{∗} as subtrees of 𝒯_{G} and 𝒯_{G}^{∗} by choosing any section of π_{G}.

Let S be as in (1.4). The dual graph of the analytic reduction of the curve S (with the admissible covering arising from the standard one on P_{k}^{1} − L_{N}) is exactly the quotient graph T_{N}. Since S is assumed to have genus r = 0, T_{N} has no cycles, and hence is a tree. Since S is compact, T_{N} is finite.

The tree T_{N}^{∗} has finitely many ends which are in one-to-one correspondence with the ramification points in π : X → S. Actually, the stabilizer of such an end (viz., the eventual stabilizer of edges and vertices in the end) is exactly the ramification group of the corresponding branching point in S. Note that since X is ordinary, only the first ramification groups in π : X → S are non-trivial (Nakajima, loc. cit.), and hence the stabilizer of an end in T_{N}^{∗} is a semi-direct product of a cyclic group and an elementary abelian p-group by the structure theorem for decomposition groups in global field extensions (cf. Serre [22], IV §2).

(2.5) Example. If G is a cyclic group generated by an elliptic element γ, then T_{G}^{∗} = M(γ).

(2.6) Graphs of groups. We turn both of T_{N} and T_{N}^{∗} into graphs of groups by labeling every vertex v ∈ V (T_{N}) and edge e = [vw] ∈ E(T_{N}) with the stabilizer of any of its lifts Λ ∈ V (𝒯_{N}) and σ ∈ E(𝒯_{N}) for the action of N, denoted by N_{v} and N_{e} = N_{vw} = N_{v} ∩ N_{w} respectively. Note that these groups are finite since N is discrete. Then N equals the tree product of T_{N}, i.e., N = ∗_{N_{vw}}N_{v}. The following observation is easy, but it allows us to deduce properties about the local structure of our tree from information about finite groups acting on the projective line:

(2.7) Lemma. Let v be any vertex of T_{N} corresponding to Λ in 𝒯_{N} and N_{v} its stabilizer. The canonical bijection {σ ∈ 𝒯 : σ ⊣ Λ} ⟶^{∼} P^{1}(k̄) induces a representation ρ : N_{v} → P GL(2, k̄). Then for every subgroup G ⊆ N_{v} there is a bijection between {σ ∈ E(𝒯_{N}^{∗}) : σ ⊣ Λ and N_{σ} ⊇ G} and points on P^{1}(k̄) fixed by ρ(G). Taking quotients by N, this leads to a bijection

\left\{{e ∈ E(T_{N}^{∗}) with e ⊣ v and e is \atop fixed by an N_{v}-conjugate of G}\right\} ⟶^{∼} \{x ∈ ρ(N_{v})\P^{1}(k̄) such that points above x in P^{1}(k̄) are fixed by a ρ(N_{v})-conjugate of ρ(G)\} .□

We now turn to a description of the possible N_{v}.

(2.8) Notation. We let Z_{n} denote the cyclic group of order n, D_{n} the dihedral group of order 2n and E_{n} the elementary abelian p-group Z_{p}^{n} of order p^{n}. Let T = A_{4}, O = S_{4}, I = A_{5} be the tetrahedral, octahedral and icosahedral groups respectively. We also introduce the following short hand notation: B(t, n) := E_{t} ⋊ Z_{n} for n|p^{t} − 1; if n = p^{t} − 1, this is a Borel subgroup of P GL(2, p^{t}). Let ⟨γ⟩ denote the cyclic group generated by γ. Groups not containing a p-group will be called classical.

We will write P?L(2, q) to denote either P GL(2, q) or P SL(2, q) by slight abuse of notation, with the convention that any related numerical quantities that appear between set delimiters {} are only to be considered for P SL(2, q).

From our point of view, the following theorem describes the structure of T_{G}^{∗} for finite subgroups G of P GL(2, k).

(2.9) Theorem (Valentini-Madan [27]). Any finite subgroup of P GL(2, k) is isomorphic to a finite subgroup of P GL(2, p^{m}) for some m > 0. The group P GL(2, p^{m}) has the following finite subgroups G, such that π_{G} is branched over d points with ramification groups isomorphic to G_{1}, ..., G_{d}:

(i) G = Z_{n} for (n; p) = 1, d = 2, G_{1} = G_{2} = Z_{n};

(ii) G = D_{n} with p ≠ 2, n|p^{m} ± 1, d = 3, G_{1} = G_{2} = Z_{2}, G_{3} = Z_{n} or also, p = 2, (n; 2) = 1, d = 2 and G_{1} = Z_{2}, G_{2} = Z_{n};

(iii) G = B(t, n) for t ≤ m and n|p^{m} − 1, n|p^{t} − 1 with d = 2 and G_{1} = G, G_{2} = Z_{n} if n > 1 and d = 1, G_{1} = G otherwise;

(iv) G = P?L(2, p^{t}) with d = 2 and G_{1} = B(t, \{\frac{1}{2}\}(p^{t} −1)), G_{2} = Z_{\{ \frac{1}{2} \}(p^{t}+1)};

(v) T of p ≠ 2, 3, d = 3, G_{1} = Z_{2}, G_{2} = G_{3} = Z_{3};

(vi) O if p ≠ 2, 3, d = 3, G_{1} = Z_{2}, G_{2} = G_{3} = Z_{4};

(vii) I if 5|p^{2m}−1 and p ≠ 2, 3, 5 with d = 3 and G_{1} = Z_{2}, G_{2} = Z_{3}, G_{3} = Z_{5} or p = 3, d = 2 and G_{1} = B(1, 2), G_{2} = Z_{5}. □

(2.10) Lemma. Let v ∈ V (T_{N}^{∗}) with preimage Λ ∈ V (𝒯_{N}^{∗}). The group representation ρ : N_{v} → P GL(2, k) arising from the action of N_{Λ} on its neighbouring vertices in 𝒯 is faithful, unless possibly when N_{v} = B(t, n), and if so, ker(ρ) is a p-group.

Proof. An element γ ∈ ker(ρ) fixes all edges σ ⊣ Λ, hence all neighbouring vertices of Λ are in its mirror M(γ). But if γ is elliptic, then its mirror consists of only one apartment, whence cannot stabilize more than two non-collinear vertices in 𝒯, whereas the valency of 𝒯 is q + 1 > 2. So γ has to be parabolic, and ker(ρ) is a normal p-subgroup of N_{Λ}. Since P SL(2, q) is simple, it follows from (2.9) that the only finite subgroups of P GL(2, k) which have such a nontrivial normal p-subgroup are of the form B(t, n). □

3. Contraction of T_{N}^{∗} – proof of proposition 1

(3.1) Definition. Let T be a tree of groups in 𝒯. A subtree of groups T′ of T is called a contraction of T if every geodesic connecting a point from T −T′ to T′ is a path on which the stabilizers of vertices are ordered increasingly w.r.t. inclusion upon approaching T′. Obviously, the tree products of T and T′ are isomorphic.

(3.2) Example. The mirror of a parabolic element can be contracted to a half-line.

(3.3) Definition. Let T and T′ be subtrees of 𝒯_{N}^{∗}. We say that T′ is perpendicular to T (notation: T ⊥T′) if for every half-line ℓ = [Λ, ∗[⊂ T′ and every edge σ ∈ E(T) with σ ⊣ Λ, π_{N}(σ) ≠ π_{N}(σ′), where σ′ is the unique edge σ′ ∈ ℓ with σ′ ⊣ Λ.

Since σ and σ′ emanate from the same vertex, the last condition is equivalent to σ ∉ N_{Λ} · σ′.

(3.4) Lemma. Let Λ ∈ V (𝒯_{N}^{∗}). If there exists γ ∈ N_{Λ} and σ ⊣ Λ such that |⟨γ⟩ · σ| > 1 (i.e., σ is not fixed by all powers of γ), then there exists a non-trivial γ′ ∈ N_{Λ} whose mirror is perpendicular to the γ-orbit of σ, i.e., such that M(γ′)⊥⟨γ⟩ · σ.

Proof. Let ν = |⟨γ⟩ · σ|, and fix a numbering ⟨γ⟩ · σ = {σ_{i}}_{i=1}^{ν}. By the identification in (2.7), (letting H = ρ(G), G = N_{v}), it suffices to prove the following: if p_{1}, ..., p_{ν+1} := p_{1} is a sequence of points on P^{1} which are cyclically permuted by an element γ ∈ H (identified with σ_{1}, ..., σ_{ν+1} = σ_{1}), then there exists a non-trivial γ′ ∈ H whose fixed points on P^{1} are not mapped to π_{H}(p_{1})(= π_{H}(p_{i})).

Notice that H ≠ 1 since γ acts non-trivially and that π_{H}(p_{1}) is not totally ramified since ν > 1. To prove the above statement, observe that if H is cyclic, we can let γ′ = γ since π_{H} is totally ramified (hence, not above p_{0}).

If H is not cyclic nor of type B(t, n), then by (2.9), the map π_{H} is branched above at least two points, and since (again, by (2.9)) there exist at least two points on P^{1} with disjoint decomposition groups, one can choose γ′ in the decomposition group of a ramified point that does not fix any point above π_{H}(p_{1}).

On the other hand, if H = B(t, n), it suffices to take γ′ ∈ B(t, 1) ⊆ B(t, n), since then, γ′ fixes a unique (totally ramified) point, which hence does not map to π_{H}(p_{1}). □

(3.5) Lemma. Let 𝒯′ be a subtree of 𝒯_{N}^{∗} such that Ends(T_{N}^{∗}) = Ends(T), where T = π_{N}(𝒯′). Then there is no half-line ℓ in 𝒯_{N}^{∗} emanating from a vertex Λ ∈ 𝒯′ which is perpendicular to 𝒯′.

Proof. Suppose that such a half-line ℓ starting at Λ ∈ 𝒯′ and pointing to x ∈ P^{1}(k) is perpendicular to 𝒯′. Then by our assumption on the ends, the image π_{N}(ℓ) is not a half-line, and hence there exists a vertex Λ_{0} on ℓ which is not contained in 𝒯′, and γ ∈ N_{Λ_{0}} which maps [Λ_{0}, x[ to a half-line containing [Λ_{0}, Λ] (say we choose Λ_{0} nearest to Λ). Set 𝒯_{0} := 𝒯′ ∪ ℓ and v_{0} = π_{N}(Λ_{0}). Let T_{0} = π_{N}(𝒯_{0}), which is the union of T and the finite path [v, v_{0}] (where v = π_{N}(Λ)). Then:

(3.5.1) There exist infinite sequences {ℓ_{n}}_{n=0}^{∞} and {ς_{n}}_{n=0}^{∞} of half-lines in 𝒯_{N}^{∗} pointing to x_{i} ∈ P^{1}(k) and finite paths of length > 0 in T_{N}^{∗}, respectively, such that

(a) Λ_{0} ⊢ ℓ_{0} and T_{0} ∩ ς_{0} = {v_{0}},

(b) ς_{i} ⊆ π_{N}(ℓ_{i}), (c) ℓ_{i} ∩ ℓ_{j} = ∅ and ς_{i} ∩ ς_{j} = ∅ unless |i − j| ≤ 1,

(d) ℓ_{i} ∩ ℓ_{i+1} is the unique vertex (say, Λ_{i+1}) at which ℓ_{i+1} starts,

(e) ς_{i} ∩ ς_{i+1} is a unique vertex, namely v_{i+1} := π_{N}(Λ_{i+1}); moreover, ς_{i} = [v_{i}, v_{i+1}],

(f) there exists an element γ_{i} in Γ_{Λ_{i}} which maps [Λ_{i}, x_{i−1}[ to a half-line containing [Λ_{i}, Λ_{i−1}].

[figure: ^{𝒯′}; ^{𝒯{∗ \atop N}}; ℓ_{0}; ℓ; ℓ_{1}; γ; γ_{1}; Λ_{0}; γ_{2}; Λ_{1}; Λ_{2}; Picture 2. The infinite sequence \{ℓ_{i}\}]

The proof of (3.5.1) is inductive. Since γ does not fix all edges emanating from Λ_{0}, the conditions of (3.4) are satisfied. Hence there exists ℓ_{0} ∈ 𝒯_{N}^{∗} starting at Λ_{0} (say, pointing to x_{0} ∈ P^{1}(k)) which is perpendicular to 𝒯_{0}. Since the ends of π_{N}(𝒯_{0} ∪ ℓ_{0}) coincide with those of T by assumption, there has to be a vertex Λ_{1} not in 𝒯_{0} and an element γ_{1} ∈ N_{Λ_{1}} which maps [Λ_{1}, x_{0}[ to a half-line containing [Λ_{1}, Λ_{0}]. Take such a vertex Λ_{1} nearest to Λ_{0}, and set v_{1} = π_{N}(Λ_{1}). Let ς_{0} = [v_{0}, v_{1}]. Then (a) follows from the fact that ℓ_{0} is perpendicular to 𝒯_{0},

(b) is clear by definition, and we also have (f).

Assume that we have constructed all ℓ_{j} and ς_{j} for j < n. By lemma (3.4), there exists a half-line ℓ_{n} with

ℓ_{n}⊥𝒯_{n}, where 𝒯_{n} := 𝒯_{0} ∪ ( ⋃_{j<n}[Λ_{j}, Λ_{j+1}]).

Replacing 𝒯_{0} by 𝒯_{n} in the argumentation above, we obtain Λ_{n+1}, v_{n+1} and ς_{n}. Since a tree does not contain a cycle, all the properties (b)∼(f) are clear. This finishes the proof of the (3.5.1).

Finally,

ℓ = [v, v_{0}] ∪ ( ⋃_{n=0}^{∞} ς_{n})

is a half-line in T_{N}^{∗} such that ℓ ∩ T = {Λ}. But this means that T_{N}^{∗} has an end which is not of T, contradicting the assumption. □

(3.6) Proof of Proposition 1. Fix a lift 𝒯′ of T to 𝒯_{N}^{∗}. Let Λ ∈ 𝒯_{N}^{∗} − 𝒯′. Since T_{N}^{∗} is connected, there exist a vertex Λ′ ∈ 𝒯′ closest to Λ. The proof proceeds by induction on d(Λ, Λ′).

Assume d(Λ, Λ′) = 1. If N_{Λ} = 1, then contraction is clear. Otherwise, take any non-trivial element γ ∈ N_{Λ}, and consider a half-line ℓ which starts at Λ and converges to an end of M(γ). We can then apply (3.5) to the tree 𝒯′′ := 𝒯′ ∪ [Λ, Λ′] to find that ℓ is not perpendicular to 𝒯′′, and hence there exists γ′ ∈ N_{Λ} such that γ′ · ℓ passes through the edge σ = [Λ, Λ′]. Since ℓ is pointwise fixed by γ, γ′ · ℓ is pointwise fixed by γ′γγ′^{−1}. Hence

N_{Λ} = ⋃_{γ′∈N_{Λ}} γ′N_{σ}γ′^{−1},

i.e., N_{Λ} is the union of the conjugates of its subgroup N_{σ}. Hence ([13], I.2, Aufgabe 4) N_{Λ} = N_{σ}, so that 𝒯′′ can be contracted to 𝒯′. It is clear how to proceed by induction. □

4. Mumford covers of P^{1} branched above at most three points

We will now describe the abstract group structure of N if π is only branched above m = 2 or m = 3 points.

The possible structure of stabilizers of edges in T_{N} turns out to be further restricted by the following lemma:

(4.1) Lemma. (i) If p > 0, then stabilizers of edges in T_{N} are of the form E_{t} ⋊ Z_{n}.

Proof. As is shown in Proposition (4.5.1) of [6], the stabilizer group of an edge contains a normal p-group with cyclic quotient. The statement follows from (2.9). □

We also have some information on the “relative position” of the p-prime part of the stabilizer of an edge in the stabilizers of the corresponding vertices. We say a subgroup of a group is maximally cyclic if it is not contained in any bigger cyclic subgroup.

(4.2) Lemma (Herrlich [10], lemma 1). If it is non-trivial, then the prime-to-p part of the stabilizer of an edge is a maximal cyclic subgroup of the stabilizer of the corresponding vertices. □

(4.3) Lemma. Let v ∈ V (T_{N}), q = p^{t}. If N_{v} = P?L(2, q), then for all e ∈ E(T_{N}) with e ⊣ v, either N_{e} = B(t, \{\frac{1}{2}\}(q − 1)) or N_{e} = Z_{\{ \frac{1}{2} \}(q+1)} (i.e., the p-part is also maximally cyclic in N_{v}).

Proof. The action of N_{v} on the edges emanating from a lift Λ of v to 𝒯_{N} induces a representation ρ : N_{v} → P GL(2, k̄), which is faithful by (2.10). It suffices to invoke (2.7). □

(4.4) Lemma. (i) The stabilizer of a vertex on the interior of a mirror of a parabolic element is a Borel group (interior meaning not an end point).

(ii) If 𝒯_{N}^{∗} contains a half-line ℓ which is pointwise fixed by a parabolic element of N, then π_{N}(ℓ) ⊆ T_{N}^{∗} is not finite (whence, a half-line).

Proof. (i) Let Λ be such a vertex on M(γ) for a parabolic element γ of N. By the description of M(γ) given in (2.3), one sees that all edges emanating from Λ are fixed by γ; hence the group representation ρ : N_{Λ} → P GL(2, k̄) is not faithful, so the result follows from (2.10).

(ii) Let ℓ be fixed by γ and let Λ be any interior vertex of ℓ∩M(γ). It suffices to show that any two edges emanating from Λ are mapped to different edges by π_{N}, viz., they are not identified by an element from N_{Λ}, which we know is a Borel group by (i). Let e be such an edge on ℓ pointing in the direction of the fixed point (say, ∞) of γ. Since all elements of N_{Λ} fix the same point (they are upper-triagonal), in particular, e is fixed by N_{Λ}, and hence cannot be identified with any other edge emanating from Λ. □

(4.5) Remark. If the mirror of a parabolic element is contracted to a half-line [v_{0}, v_{1}, ...[ as in (3.2), its structure as a tree of groups is as follows: the stabilizer N_{v_{0}} can be a larger group (of type P?L(2, q)); all interior points are stabilized by a group N_{v_{i}} = B(t_{i}, n) for some fixed n by (4.2); the sequence t_{1}, t_{2}, ... is increasing, and N_{v_{i}} = N_{[v_{i},v_{i+1}]} ⊆ N_{v_{i+1}}.

(4.6) Proposition. If π : X → S is branched above two points, then N is isomorphic to one of the following:

(A1) P?L(2, q) ∗_{B(t,n_{−})} B(t_{1}, n_{−});

(A2) B(t_{2}, n_{+}) ∗_{Z_{n_{+}}} P?L(2, q) ∗_{B(t,n_{−})} B(t_{1}, n_{−});

(A3) B(t_{3}, n_{−}) ∗_{B(t,n_{−})} P?L(2, q) ∗_{Z_{n_{+}}} P?L(2, q) ∗_{B(t,n_{−})} B(t_{1}, n_{−}); (A4) B(t_{4}, n_{+}) ∗_{Z_{n_{+}}} P?L(2, q); (A5) B(t_{5}, n_{−}) ∗_{B(t,n_{−})} P?L(2, q) ∗_{Z_{n_{+}}} P?L(2, q); (B) B(t′_{1}, n) ∗_{Z_{n}} B(t′_{2}, n); (C) E_{t′_{3}} ∗ E_{t′_{4}},

for q = p^{t}, t|t_{i}(i = 1, 3, 5), 2t|t_{2j}(j = 1, 2), n|p^{t′_{1}}−1, n|p^{t′_{2}}−1 and n_{±} = \{\frac{1}{2}\}(q±1).

If p = 3, a list (Ai′)_{i=1}^{5} similar to (Ai)_{i=1}^{5} can occur with P?L(2, q), t, n_{+}, n_{−} replaced by I, 1, 5, 2 respectively. If p = 2, there are the following extra possibilities:

(A1′′) E_{t_{2}} ∗_{E_{1}} D_{n} ∗_{E_{1}} B(t_{1}, n);

(A2′′) E_{t_{2}} ∗_{E_{1}} D_{n} ∗_{Z_{n}} D_{n} ∗_{E_{1}} E_{t_{1}};

(A3′′) E_{t_{2}} ∗_{E_{1}} D_{n_{+}} ∗_{Z_{n_{+}}} P GL(2, q) ∗_{B(t,n_{−})} B(t_{1}, n_{−}), where n is odd, q is a power of 2 and n_{±} = q ± 1.

Proof. Let T_{0} be a straight line in T_{N}^{∗} such that Ends(T_{0}) = Ends(T_{N}^{∗}). By Proposition 1, we can contract T_{N}^{∗} to T_{0} without affecting the structure of N.

There exists at least one vertex in T_{0} whose stabilizer is not classical, since there is wild ramification in π : X → S. Let v be such a vertex. Then there are the following possibilities:

(A) There is a vertex v such that N_{v} = P?L(2, q = p^{t}). By (4.4), the end emanating from a representative of v in 𝒯_{N}^{∗} which is fixed by the Borel subgroup B := B(t, n_{−}) is mapped to a half-line of T_{0} by π_{N}. Let {v_{i}} be a numbering of the vertices on this half-line in T_{0} such that v_{i} is at distance i from v = v_{0}. Since all v_{i} are fixed by B, the stabilizer N_{v_{i}} can only be a group of the form B(t_{i}, n_{i}) by (2.9). Indeed, would N_{v_{i}} = P?L(2, p^{t_{i}}), then v_{i+1} would have to be stabilized by both Z_{n_{+}} and the Borel group N_{[v_{i−1}v_{i}]} = B(t_{i}, n_{−}) (using (4.3) and (4.4)), and hence by a group containing B(2t_{i}, n_{+}), contradicting (4.3).

Actually, n_{i} = n_{0} = n_{−} by (4.2), and {t_{i}} is increasing and eventually terminates since v_{i} is stabilized by the N_{v_{j}} for j ≤ i.

The edge emanating from v_{0} in the other direction is stabilized by Z_{n_{+}}. For the stabilizer of the first neighbouring vertex v′ of v on T_{0} in the other direction for which N_{v′} ≠ N_{v}, by (2.7) and (2.9) there are three possibilities:

(a) N_{v′} = Z_{n_{+}}; (b) N_{v′} = B(t′, n_{+}) for q+1|p^{t′} −1, t′ > 0 or (c) N_{v′} = P?L(2, q). In case (b) and (c), the subsequent neighbouring vertices of v′ in T_{0} can be dealt with as above, i.e., their stabilizers form an increasing stabilizing chain of Borel groups. This leads in the end to the prescribed form of N.

(A′) Suppose we are not in case (A), but p = 3 and N_{v} = A_{5} for some vertex v ∈ T_{0}. By (2.9), one end of T_{N_{v}}^{∗} is stabilized by the Borel group B(1, 2) and one end by the cyclic group Z_{5}. A similar reasoning as in (A) can be performed.

(B) Suppose we are not in case (A) or (A′), but there is a vertex with N_{v} = B(t, n), n > 1. One of the half-lines emanating from v is stabilized by an increasing sequence of Borel groups as in case (A). We can assume that the edge emanating from v in the other direction is not fixed by parabolic elements (since all parabolic elements of N_{v} have the same unique fixed point), whence it is Z_{n} by (4.2). Let v′ be the vertex closest to v in this direction, such that N_{v′} ≠ Z_{n} (which exists, since otherwise, N would be finite). Then N_{v′} can only be a Borel group (since it cannot be P?L(2, q) by assumption, and it is not a classical group by (2.10)). We find in the end that N = B(t, n) ∗_{Z_{n}} B(t′, n).

(C) Suppose that all N_{v}, v ∈ V (T_{0}) are p-groups, say, N_{v} = E_{t_{v}}. There exist at least two vertices v, v′ the mirrors of whose stabilizers map to different ends of T_{0} under π_{N} because of lemma (4.4). If we take such v and v′ at minimal distance of each other, then all vertices between them are stabilized only by the trivial group, since the mirrors of elements in N_{v} and in N_{v′} have to map to disjoint half-lines in T_{0}. On edges outside of such a minimal segment, the p-groups are ordered increasingly and eventually stabilize. Hence, N = E_{t} ∗E_{t′}.

The proofs if p = 2 and there is a vertex v on T_{0} with N_{v} = D_{n} are entirely similar. □

(4.7) Proposition. If π : X → S is branched above three points, then T_{N}^{∗} can be contracted to a tree T_{0} consisting of three half-lines meeting at one common vertex v_{0}. Then either:

(D) There is an edge e emanating from v_{0} with N_{e} = {1};

(E) The stabilizer of v_{0} is classical non-dihedral, i.e., N_{v_{0}} ∈ {T, O, I}; (F) p ≠ 2 and N is isomorphic to one of the following:

Q(t_{1}) ∗_{Z_{2}} B(t_{3}, n_{−}) ∗_{B(t,n_{−})} P?L(2, q) ∗_{Z_{n_{+}}} D_{n_{+}}  ; ∗_{Z_{2}}Q(t^{2}) (F1)

Q(t_{1}) ∗_{Z_{2}} B(t, n) ∗_{Z_{n}} D_{n}  , ∗_{Z_{2}}Q(t_{2}) (F2)

where q = p^{t}, t|t_{3}, Q(t_{i}) = B(t_{i}, 2) (i = 1, 2) and two cyclic subgroups Z_{2} occuring in (F) are the same in the dihedral group D_{n}.

If p = 3, a case (F 1′) similar to (F 1) can occur, where Q(t_{i}) is either B(t_{i}, 2) or P SL(2, 3) ∗_{B(1,2)} B(t_{i}, 2), and P?L(2, q), t, n_{+}, n_{−} is replaced by I, 1, 5, 2, respectively.

Proof. Let T_{0} be the subtree of T_{N}^{∗} which is the union of three half-lines ℓ_{1}, ℓ_{2} and ℓ_{3} such that ℓ_{i} ⋂ ℓ_{j} = {v_{0}} for i ≠ j for some (uniquely determined) v_{0}. By Proposition 1, we can contract T_{N}^{∗} to T_{0}.

Suppose G = N_{v_{0}} is such that π_{G} is branched over at most 2 points (cf. (2.9)). Then it follows from (2.7) that at least one of the edges emanating from v_{0} has trivial stabilizer; hence (D) occurs.

Next, we suppose π_{G} is branched over 3 points. If we are not in case (D), then p ≠ 2 and the group N_{v_{0}} is a dihedral group D_{n}. Let v_{1} be the neighbouring vertex of v_{0} such that the edge connecting v_{0} and v_{0} is stabilized by Z_{n}. For N_{v_{1}}, by (2.7) and (2.9) there are three possibilities: (a) N_{v_{1}} = Z_{n};

(b) N_{v_{1}} = B(t, n) for t > 0 or (c) N_{v_{1}} = P?L(2, q). Then an argument similar to that in the proof of (4.6) can be applied to classify all the possible groups appearing in the amalgam product in this direction. The other directions and the case (F 1′) can be treated similarly. □

5. The classical bound

In this section, we will prove that |Aut(X)| is bounded “classically” by 12(g − 1) in case (Ai′)_{i=1}^{5}, (D) and (E). The proof is very similar to Herrlich’s computations in the p-adic case ([10]).

(5.1) Definition. If (T, G_{∗}) is a finite tree of groups, we define its μinvariant μ(T) to be

(∗) μ(T) = ∑_{[vw]∈E(T)} \frac{1}{|G_{vw}|} − ∑_{v∈V (T)} \frac{1}{|G_{v}|}.

The following group theoretical result provides the direct link between T_{N} as a tree of groups and the automorphism group of X.

(5.2) Theorem (Karrass, Pietrowski, Solitar [15]). With notations as above, |N/Γ| · μ(T_{N}) = (g − 1). □

This theorem, together with (1.3), implies that in order to bound |Aut(X)| from above, one has to bound μ(T_{N}) from below.

(5.3) Lemma ([6], (4.7.1)). If in the above situation, B is a subtree of T_{N}, then μ(T) ≥ μ(B). □

(5.4) Proposition. The bound |Aut(X)| ≤ 12(g − 1) holds in case (D) and (E) and (Ai′)_{i=1}^{5}.

Proof. Suppose that, in the notations of the proof of (4.7), N_{v_{0}} is non-trivial such that N_{e} = 1 for some edge emanating from v_{0} in T_{0}. Suppose that e ∈ ℓ_{1} for some half-line ℓ_{1} of T_{0}. Let v_{1} be the vertex closest to v_{0} on ℓ_{1} such that N_{v_{1}} ≠ 1 (which exists, since eventually the stabilizer of ℓ_{1} is a (non-trivial) ramification group of π), and let B be the subtree of T_{0} consisting of the path from v_{0} to v_{1}. Then

μ(B) = 1 − \frac{1}{|N_{v_{0}}|} − \frac{1}{|N_{v_{1}}|} ≥ 1 − \frac{1}{2} − \frac{1}{3} = \frac{1}{6},

unless if N_{v_{0}} and N_{v_{1}} are both isomorphic to Z_{2} (whence, p ≠ 2). Let Λ_{1} be a lift of v_{1} to 𝒯_{N}^{∗}. If N_{v_{1}} = Z_{2} is generated by γ, then the mirror of γ is an apartment ]x, y[ in 𝒯_{N}, and since [v_{0}v_{1}] is not stabilized by γ, by (3.4) there has to exist an element γ′ ≠ γ ∈ N_{v_{1}} which maps ]x, Λ_{1}] to [Λ_{1}, y[; a contradiction.

In case (E), let e be any edge emanating from v_{0}. We can assume that N_{e} ≠ 1, since otherwise, case (D) occurs. Let v_{1} be the vertex closest to v_{0} on T_{0} in the direction of e such that N_{v_{1}} ≠ N_{e}. Let B be the segment connecting v_{0} and v_{1}. We will now prove that μ(B) ≥ \frac{1}{12} for all such possible B. Recall that N_{e} is maximally cyclic in both N_{v_{i}} (i = 0, 1).

(5.4.1) The following table provides the necessary data in case N_{v_{1}} is also a classical group:

N_{v_{1}}, N_{e}, N_{v_{0}}  μ_{B}  N_{v_{1}}, N_{e}, N_{v_{0}}  μ_{B}
D_{n_{1}}, Z_{2}, T  ≥ \frac{1}{6}  D_{3}, Z_{3}, T  = \frac{1}{12}
D_{n_{1}}, Z_{2}, O  ≥ \frac{5}{24}  D_{3}, Z_{3}, O  = \frac{1}{8}
D_{4}, Z_{4}, O  = \frac{1}{12}  D_{n_{1}}, Z_{2}, I  ≥ \frac{7}{30}
D_{3}, Z_{3}, I  = \frac{3}{20}  D_{5}, Z_{5}, I  = \frac{1}{12}
T, Z_{2,3}, T  = \frac{1}{3} , \frac{1}{6}  T, Z_{2,3}, O  = \frac{3}{8} , \frac{5}{24}
T, Z_{2,3}, I  = \frac{2}{5} , \frac{7}{30}  O, Z_{2,3,4}, O = \frac{5}{12} , \frac{1}{4} , \frac{1}{6}
O, Z_{2,3}, I  = \frac{53}{120} , \frac{11}{40}  I, Z_{2,3,5}, I = \frac{7}{15} , \frac{3}{10} , \frac{1}{6}

(5.4.2) Let N_{v_{1}} contain a p-group. The classification theorem (2.9) shows that, except if p = 3, we can assume that N_{e} is cyclic of order n prime to p; and it is maximally cyclic in both N_{v_{0}}, N_{v_{1}}. This implies that n = 2, 3 if N_{v_{0}} = T, n = 2, 3, 4 if N_{v_{0}} = O and n = 2, 3, 5 if N_{v_{0}} = I. If N_{v_{1}} = P GL(2, q), then n = q + 1 by (4.3). This leads to q = 3, 4 for N_{V_{0}} = T and q = 2, 3, 4, 5 for N_{v_{0}} = O and N_{v_{0}} = I. We can argue similarly using 2n = q + 1 for N_{v_{1}} = P SL(2, q). After we eliminate the Consulting (2.9), the following possibilities remain:

N_{v_{1}}, N_{e}, N_{v_{0}}  μ_{B}
P SL(2, 5), Z_{3}, T  = \frac{7}{30}
P SL(2, 7), Z_{4}, O  = \frac{17}{84}
P SL(2, 9), Z_{5}, I  = \frac{13}{72}
E_{t} ⋊ Z_{2,3}, Z_{2,3}, T  ≥ \frac{19}{60} , \frac{1}{8}
E_{t} ⋊ Z_{2,3,4}, Z_{2,3,4}, O ≥ \frac{43}{120} , \frac{41}{168} , \frac{19}{120}
E_{t} ⋊ Z_{2,3,5}, Z_{2,3,5}, I ≥ \frac{23}{60} , \frac{113}{420} , \frac{109}{660}

(5.4.3) We are left to consider p = 3 with N_{e} = D_{3} = E_{1} ⋊ Z_{2} and N_{v_{0}} = I. The following cases remain: (i) if N_{v_{1}} = P GL(2, 3) = O, then μ(B) = \frac{13}{120}; (ii) if N_{v_{1}} = E_{t_{1}} ⋊ Z_{2}, then μ(B) ≥ \frac{17}{180}; (iii) if N_{v_{1}} = I, then μ(B) = \frac{2}{15}.

(5.4.4) In each of the cases (Ai′)_{i=1}^{5}, one easily computes that μ(T_{0}) ≥ \frac{1}{12}. □

6. The extreme cases with 2 points ramified

We will now prove the bound in all remaining cases. Since the bound is not linear in the genus, one cannot simply restrict to a suitably chosen segment of T_{0} and apply the techniques of the previous section. Instead, we use the following lemma to further eliminate cases.

(6.1) Notation. Let F be the real function

F : R → R : g ↦ F (g) := 2\sqrt{g}(\sqrt{g} + 1)^{2}.

Let p_{Γ} : N → N/Γ be the projection map.

Recall that for any finite group G ⊂ N, p_{Γ}(G) ≅ G since Γ is free.

(6.2) Lemma. Let X be a Mumford curve of genus g. Let λ be the greatest common divisor of g−1 and |Aut(X)|, and write g−1 = λ·a, |Aut(X)| = λ·b. If we can find a lower bound λ_{0} for λ such that

λ_{0}b ≤ F (λ_{0}a + 1),

then the bound |Aut(X)| ≤ F (g) holds.

Proof. We have

|Aut(X)| = (g − 1)\frac{λ_{0}b}{λ_{0}a} ≤ (g − 1)\frac{F (λ_{0}a + 1)}{λ_{0}a}.

Since the function x ↦ F (x)/(x − 1) is increasing for x ≥ 2 and λ_{0}a + 1 ≤ λa + 1 = g, the result follows. □

(6.3) Proposition. The bound |Aut(X)| ≤ F (g) holds for all cases (Ai)_{i=1}^{5}, except possibly (A1) with t_{1} = 2t and (A5) with t_{5} = t.

Proof. We know that the ramification groups of π : X → S are the stabilizers of the ends of T_{0}. The Riemann-Hurwitz formula applied to π allows us to compute the ratio of g − 1 to |Aut(X)|, and one finds the following values of a and b in each of the cases (Ai)_{i=1}^{5}:

a  b  where
(A1) q^{n} − q − 1  \{ \frac{1}{2} \}q^{n}(q^{2} − 1) nt = t_{1}
(A2) q^{n+1} − q^{n−m+1} − q^{n−m} − q + 1 \{ \frac{1}{2} \}q^{n}(q^{2} − 1) nt = 2t_{2} ≥ mt = t_{1}
q^{m+1} − q^{m−n+1} + q^{m−n} − q − 1 \{ \frac{1}{2} \}q^{m}(q^{2} − 1) nt = 2t_{2} ≤ mt = t_{1}
(A3) q^{m} − q^{m−n} − 1  \{ \frac{1}{2} \}q^{m}(q − 1) nt = t_{1} ≤ mt = t_{3}
(A4) q^{n+1} − q^{n} − q^{n−1} − q + 1  \{ \frac{1}{2} \}q^{n}(q^{2} − 1) nt = t_{4}
(A5) q^{n} − q^{n−1} − 1  \{ \frac{1}{2} \}q^{n}(q − 1)  nt = t_{5}

One computes that the above lemma can be applied with λ_{0} = 1 in all but the above mentioned cases. □

(6.4) Proposition. The bound |Aut(X)| ≤ F (g) holds in case (A1) with t_{1} = 2t, i.e., n = 2 in the above table.

Proof. Paradoxically(?), we will show that |Aut(X)| is bounded from below non-trivially. Let N_{1} = P?L(2, q), N_{2} = B(2t, n_{−}), B = B(t, n_{−}) be the subgroups of P GL(2, k) occuring in T_{0}, i.e., N = N_{1} ∗_{B} N_{2}. We can assume that N_{1} is the “standard” copy P?L(2, F_{q}) of P?L(2, q) in P GL(2, k) induced by the inclusion F_{q} ⊂ k. Since the p-part of N_{1} equals B, hence is a subgroup of N_{2}, the p-parts of N_{1} and N_{2} have to commute in N; so they can be simultaneously put into upper triagonal form, i.e., we can suppose that they are of the form

τ(α_{i}) ∈ N_{1} ∩ N_{2}; τ(β_{j}) ∈ N_{2} − N_{1}

for some α_{i}(i = 1, ..., t), β_{j}(j = 1, ..., t) ∈ k. where we set

τ(x) := ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )

Now observe that the images of these matrices under the map p_{Γ} : N → N/Γ (where Γ is the Schottky group of X), stay distinct. Indeed, if p_{Γ}(τ(x)) = p_{Γ}(τ(y)), then τ(x − y) ∈ Γ; but τ(x − y) is of finite order, so x = y since Γ is free.

On the other hand, N_{1} contains the “non-split” element

γ = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ,

so also the lower triagonal elements τ^{t}(x) = γτ(x)γ^{−1} are contained in N for all x ∈ {α_{i}, β_{j}}. For the same reason as above, the elements τ^{t}(x) remain mutually distinct modulo Γ.

It is also impossible that p_{Γ}(τ(x)) = p_{Γ}(τ^{t}(y)) for some x, y ∈ {α_{i}, β_{j}}. This is clear if x, y are both in {α_{i}}, since then τ(x) ≠ τ^{t}(y) in N_{1} and p_{Γ}(N_{1}) ≅ N_{1}.

Now suppose that p_{Γ}(τ(α_{i})) = p_{Γ}(τ^{t}(β_{j})) for some i, j. Since the commutator [α_{i}, β_{j}] = 1 in N, we have

[p_{Γ}(τ^{t}(β_{i}), p_{Γ}(τ^{t}(α_{i}))] = 1.

We conclude that

[p_{Γ}(τ(α_{i})), p_{Γ}(τ^{t}(α_{i}))] = 1.

But since τ(α_{i}) and τ^{t}(α_{i}) are both in p_{Γ}(N_{1}) ≅ N_{1}, this gives a contradiction, since τ(α_{i}) and τ^{t}(α_{i}) are not commuting in N_{1} = P GL(2, q).

Finally, if p_{Γ}(τ(β_{i})) = p_{Γ}(τ^{t}(β_{j})), then since the first element commutes with p_{Γ}(τ(α_{1})) and the second with p_{Γ}(τ^{t}(α_{1})), we would find that p_{Γ}(τ(α_{1})) and p_{Γ}(τ^{t}(α_{1})) commute, leading to the same contradiction.

We find in the end that the order of A = N/Γ is at least divisible by q^{4} (viz., the number of distinct p-order elements constructed above), which means that we can choose λ_{0} = q^{2} in lemma (6.2) and this leads to the desired bound, except if q = 2, but then, P GL(2, 2) = D_{3} and another case applies. □

(6.5) Proposition. The bound |Aut(X)| ≤ F (g) holds in case (A5) with t_{5} = t.

Proof. We see that the group p_{Γ}(Z_{n_{+}}) = Z_{n_{+}} which stabilizes the central edge of T_{0} acts without fixed point on X (by the correspondence between fixed points and ends). By Hurwitz’s formula applied to this unramified action, n_{+} divides g − 1. Hence we can set λ_{0} = n_{+} to see that the criterion of lemma (6.2) is satisfied. □

(6.6) Lemma. ([23], I.1., Prop. 4) If G_{1} and G_{2} are finite groups, then we have a natural exact sequence

1 → [G_{1}, G_{2}] → G_{1} ∗ G_{2} → G_{1} × G_{2} → 1,

where the group [G_{1}, G_{2}] generated by the commutators [g_{1}, g_{2}] := g_{1}g_{2}g_{1}^{−1}g_{2}^{−1} for g_{1} ∈ G_{1} − {1} and g_{2} ∈ G_{2} − {1} is a maximal free subgroup of G_{1} ∗ G_{2} of rank (|G_{1}| − 1)(|G_{2}| − 1). □

(6.7) Lemma. Let X be an ordinary curve, admitting a Galois cover X → P^{1} branched above two points. If its Galois group is of the form G = B(t, n), then n = 2.

Proof. Let Ē be a subgroup of index p in a p-Sylow group of G and consider the curve Y := Ē\X, which is a Galois cover of Aut(X)\X = P^{1}, totally ramified above two points with Galois group Z_{p} ⋊ Z_{n}. It follows that the intermediate curve Y′ := Z_{p}\Y is a cyclic p-prime Galois cover of P^{1} ramified above two points, and hence equal to P^{1}. Let x be a coordinate on Y′; then the equation of Y is of the form

y^{p} − y = B := c\frac{x − α}{x} for some α ≠ 0

and c ∈ k. Indeed, the ramification points are the poles of B, and since X is ordinary, they are simple (Subrao [25]). This curve has to admit an automorphism of order n. It is easy to see that such an automorphism has to be of the form B ↦ ζB + B_{0}^{p} − B_{0}, y ↦ ζy + B_{0} for some n-th root of unity ζ and some B_{0}(x) ∈ k(x) (cf. Hasse [7], p. 38). On the other hand, the automorphism should be induced from an automorphism of k(x) of order n, i.e., of the form x ↦ ζ · x. This leads to the identity

B(ζx) − ζB(x) = B_{0}(x)^{p} − B_{0}(x),

which is only satisfied if ζ = ±1 with B_{0} a (constant) root of c(1−ζ) = B_{0}^{p}−B_{0}. Hence n = 2 as claimed. □

(6.8) Proposition. The bound |Aut(X)| ≤ F (g) holds in case (B) and (C).

Proof. Recall that in this case, N = N_{1}∗_{B}N_{2} with N_{1} = B(t, n), N_{2} = B(t′, n) and B = Z_{n} for n|(p^{t} − 1; p^{t′} − 1). The criterion in lemma (6.2) fails to hold with λ_{0} = 1 only if t = t′, which we will assume from now on. Observe that then, N = (E_{1} ∗ E_{2}) ⋊ Z_{n}, where E_{1} and E_{2} are the respective p-parts of N_{1} and N_{2}.

We will distinguish two cases. First of all, assume that E_{1} = E_{2} mod Γ. It follows from this that [E_{1}, E_{2}] ⊆ Γ. In particular, we find that Aut(X) = N/Γ is a quotient of (E_{1} ∗ E_{2}) ⋊ Z_{n}/[E_{1}, E_{2}] = (E_{1} × E_{2}) ⋊ Z_{n} by (6.6). Since E_{1} = E_{2} mod Γ, Aut(X) is even a quotient of E_{1} ⋊ Z_{n}. On the other hand, since E_{1} ⋊ Z_{n} ≅ N_{1} ⊆ N/Γ, we find in the end that

Aut(X) = E_{1} ⋊ Z_{n}.

The previous lemma implies that n = 2, and knowing this, the bound μ(T_{0}) ≥ \frac{1}{12} is easy to check.

Let E′_{1} denote the part of E_{1} which is identified with a part (called E′_{2}) of E_{2} modulo Γ. Denote by E′′_{i} the respective complements of E′_{i} in E_{i}, and let p^{τ} be the order of E′′_{i}. By the result of the previous paragraph, we can assume that τ > 0. We claim that E′′_{i} are Z_{n}-modules (for the action of conjugation). Indeed, it suffices to show this for E′_{i}. Let ε be an element of E′_{1}, and choose an element ε̄ ∈ E′_{2} with p_{Γ}(ε) = p_{Γ}(ε̄). We have to show that for every σ ∈ Z_{n}, σεσ^{−1} ∈ E′_{1}, viz., p_{Γ}(σεσ^{−1}) = p_{Γ}(x) for some x ∈ E_{2}, and it suffices to let x = σε̄σ^{−1}.

Since P^{1} does not admit unramified extensions, the automorphism group of X is generated by its decomposition groups, and in particular, it contains the group generated by E_{1} ⋊ Z_{n} and E_{2} ⋊ Z_{n}. By assumption, E_{1} and E_{2} intersect modulo Γ in a group of order p^{t−τ}, so that

|Aut(X)| ≥ \frac{|E_{1}||E_{2}||Z_{n}|}{|E_{1} ∩ E_{2}|} = p^{t+τ} n,

hence we can let λ_{0} = p^{τ} in (6.2). If we then use that n ≤ p^{τ} − 1 (since E′′_{1} is a Z_{n}-module, n divides p^{τ} − 1), we find that the inequality in (6.2) is fulfilled, so that the desired bound holds (note that if p^{t} = 2, we have g = 1). □

(6.9) Proposition. The bound |Aut(X)| ≤ F (g) holds in case (F 1), (F 1′) and (F 2).

Proof. In all such cases, one can compute a corresponding table of (a, b) depending on the values of t, t_{1}, t_{2}. We will skip the detailed computation; let it suffice to say that lemma (6.2) cannot be applied with λ_{0} = 1 only if t_{1} = t_{2} = 0, in which case the order of the ramification groups is (2, 2, nq) (with q = p^{t}) and a = q − 2, b = 2nq (respectively, a = \frac{1}{2}(q − 2), b = nq if p = 2).

For the case (F 1), observe that, as in the proof of (6.5), the group Z_{n_{+}} acts freely on X, leading to λ_{0} ≥ n_{+}, and with this value, the bound follows from (6.2).

The only case that remains to be settled is (F 2), viz., N = N_{1} ∗_{B} N_{2}, with N_{1} = B(t, n), B = Z_{n} and N_{2} = D_{n}. Let E be a p-Sylow of N_{1}, let γ be an involution in D_{n} − Z_{n} and set

E′ = {ε ∈ E : p_{Γ}(γεγ) ∈ p_{Γ}(E)}.

By lemma (6.7), if E = E′, then n = 2 and the bound follows easily.

Let E′′ be the complementary Z_{n}-module of E′ in E, and let |E′′| = p^{τ}; then we can assume that τ > 0. We find as in (6.8) that Aut(X) ≥ 2np^{t+τ}, and the desired bound follows from (6.2) taking λ_{0} = 2p^{τ}. □

(6.10) Remark. Stichtenoth has shown in [26] that Aut(X) ≤ 84(g − 1) if π is ramified above three points and at least two ramification indices are > 2.

(6.11) Proposition. The bound |Aut(X)| ≤ F (g) holds if p = 2 in case (Ai′′)_{i=1}^{3}.

Proof. Unconditionally in case (A2′′) and in case (A1′′) and (A3′′) if t_{2} ≠ 1, it is easy to compute that μ(T_{0}) ≥ \frac{1}{12}.

In case (A1′′) with t_{2} = 1, one finds that a = 2^{t_{1}−1} − 1, b = 2^{t_{1}}n, and π has ramification type (2, 2^{t}n). As in (6.9), let E be the p-part of B(t_{1}, n) and E′ the image of the conjugate of E by an involution of D_{n}. If E = E′, then again, n = 2 and the bound holds. If we let τ be the order of a complement of E′ in E, then we can set a = 2^{t_{1}} − 1, b = 2^{t_{1}}n, λ_{0} = 2^{τ+1} and using n ≤ 2^{τ} − 1, prove that (6.2) is satisfied (noting that t ≥ τ).

In case (A3′′) with t_{2} = 1, we define n by tn = t_{1}; then a = \frac{1}{2}q^{n} − 1, b = q^{n}(q − 1), and since Z_{n_{+}} acts freely on X, we can let λ_{0} = q + 1 as in (6.5). □

(6.12) Remark. A careful inspection of the estimates shows that only case (F 2) with t_{1} = t_{2} = 0, n = p^{t} − 1 exactly attains the bound of the main theorem (if a free subgroup of N of rank g = (q − 1)^{2} with normalizer N exists, cf. infra).

7. The icosahedral group – proof of proposition 2

(7.1) We have already remarked that 12(g−1) > F (g) only if g ∈ {5, 6, 7, 8}, and hence there is only a finite number of groups A (namely, 134) for which 12(g − 1) < |A| ≤ F (g), which one can easily write down, e.g., using GAP ([28]), and the only non-solvable such group is I, whose order is 60. It follows that g = 6. Note that I is a subgroup of P GL(2, k) only if p ≠ 2, 5, and then, a geometric construction of a Mumford curve X of genus 6 with 60 automorphisms was given by Herrlich in [9], pp. 50-51. The normalizer of its Schottky group is of the form

N = I ∗_{Z_{5}} D_{5},

and X is a cover of P^{1} ramified above 4 points with ramification indices (2, 2, 2, 3) if p ≠ 3 and (2, 2, 6) if p = 3.

(7.2) We will now compute the dimension of the stratum of the moduli space M_{6} (which we consider as an algebraic space) of curves of genus 6 containing this particular example. Observe that the dimension of the infinitesimal deformation space of X consisting of curves whose automorphism group contains I is given by h^{1}(X, T_{X})^{I}, where T_{X} is the tangent sheaf of X and the superscript I means taking group invariants. By Serre duality, h^{1}(X, T_{X})^{I} = h^{0}(X, Ω_{X}^{⊗2})^{I}, where Ω_{X} is the sheaf of regular differentials on X. Finally, this space of invariant 2-differentials is computed for (p, |I|) = 1 to be of dimension 3ḡ − 3 + n, where ḡ is the genus of I\X and n is the number of branch points of X → I\X ([2], pp. 254-255). Hence in our case, the closed stratum S̄_{I} of curves in M_{6} whose automorphism group contains I is one-dimensional. We do not know whether S̄_{I} is connected, and whether or not it equals the open stratum S_{I}.

(7.3) Let us now switch to M_{g}, the moduli space of Mumford curves in of genus g over k (which can conveniently be described by non-archimedean Teichmüller theory, cf. Herrlich [12]). Since this is a rigid-analytic open subspace of M_{g} over k, it follows immediately from the computation in (7.2) that the stratum S̄_{I} intersects M_{6} in a one-dimensional space. Since we know that |Aut(X)| ≤ 60 on M_{6}, we have S_{I} ∩ M_{6} = S̄_{I} ∩ M_{6}, hence all curves in this one-dimensional rigid analytic stratum have automorphism group exactly equal to I.

However, there is a more direct way to the computation of the dimension of closed strata in the space of Mumford curves that avoids the use of invariant theory (hence, of the assumption that the characteristic is coprime to the order of the automorphism group):

(7.4) Theorem (Herrlich [12]). With the above notations, let A = N/Γ. If g ≥ 4, then the closed stratum S̄_{A} of curves whose automorphism group contains A intersects the Mumford locus M_{g} in a space of dimension

dim(S̄_{A} ∩ M_{g}) = 3(f + d_{v} − d_{e} − 1) + 2(c_{v} − c_{e}),

where c_{v} (resp. d_{v}) is the number of non-trivial cyclic (resp. non-cyclic) vertex groups of T_{N}, c_{e}, d_{e} are similarly defined for edges of T_{N}, and f is the number of free generators of N.

Furthermore, the open stratum S_{A} ∩ M_{g} is connected.

Using (7.4), one computes that in our case, dim(S̄_{I} ∩ M_{6}) = 1 as expected, and the same result as in (7.3) follows. Note, however, that (7.4) does not say anything about the dimension of the corresponding stratum in M_{g}.

(7.5) Remark. We do not know any equations for the one-parameter family of curves S_{I} ∩ M_{6}.

8. Discreteness – attaining the bound

(8.1) It has not yet been proven that the groups on the lists in (4.6) and (4.7) actually occur as discrete subgroups of P GL(2, k). This is the case exactly when there is an action of N on 𝒯 whose quotient equals T_{N}. For this, it suffices that the action of N is well-defined, and has a finite stabilizer at at least one point of 𝒯. Checking this is not always so obvious, but there is a direct criterion to check whether a free product exists as a discrete subgroup of P GL(2, k).

(8.2) Definition. The isometric circle of a non-trivial element γ of finite order in P GL(2, k) is defined by Ī_{γ} = {z ∈ P^{1}(k) : |cz + d| ≤ 1}, where (c, d) is the second row of γ.

(8.3) Lemma ([10]). If G, H are finite subgroups of P GL(2, k), then G ∗ H exists as a discrete subgroup of P GL(2, k) if and only if Ī_{γ} ∩ Ī_{δ} = ∅ for all non-trivial γ ∈ G and δ ∈ H.

(8.4) We will use the above criterion to prove that the segment that attains the bound of the main theorem ((F_{1}) with t_{1} = t_{2} = 0, n = p^{t} − 1) does correspond to a discontinuous group, and hence a corresponding Mumford curve exists. Let us consider the corresponding segment B with N_{1} = B(t, n) and N_{2} = D_{n} = Z_{n} ⋊⟨γ⟩ for some element γ of order 2. We will study this case as a kind of “cover” of case (B). Let B′ be the segment defined by N_{1} = B(t, n) = E_{t} ⋊ Z_{n}, N_{2} = E′_{t} ⋊ Z_{n} and H = Z_{n}, where E_{t}, E′_{t} are two disjoint p-groups generated by {ε_{i}}_{i=1}^{t_{1}} and {ε′_{i}}_{i=1}^{t_{1}} respectively, and on which, by definition, Z_{n} acts componentwise by conjugation. Looking at explicit presentations, one sees that

N_{B} = N_{B′} ∗ ⟨γ⟩/⟨γ^{2} = 1, γσγ = σ^{−1}, γε_{i}γ = ε′_{i}, i = 1...t⟩,

with obvious notations.

Remark. Geometrically, the link between the segments B and B′ is as follows: one subdivides the segment B like

(E_{t} ⋊ Z_{n}) ∗_{Z_{n}} Z_{n} ∗_{Z_{n}} (E′_{t} ⋊ Z_{n}),

and then lets an element γ of order two act on this like a vertical “mirror” along the central vertex; then it identifies E_{t} and E′_{t} by conjugation and inverts the elements on the stabilizers of edges.

Rearranging words shows that N_{B} = ⟨γ⟩ · N_{B′}, so that N_{B′} is a subgroup of (finite) index 2 in N_{B} (of course, we actually mean the image of N_{B′} in N_{B}). To show that N_{B} is discontinuous, it therefore suffices to show this for N_{B′}. One sees that N_{B′} = (E_{t} ∗ E′_{t}) ⋊ Z_{n}. We can let E′_{t} = QE_{t}Q^{−1} for some Q ∈ P GL(2, k). It suffices to show that the free product E_{t} ∗ E′_{t} is contained in P GL(2, k). Now the isometric circles of elements in E′_{t} are translates by Q of the corresponding circles in E_{t}, which we can assume to be unit circles with centers at finitely many elements of F̄_{p} (if we let E_{t} be generated by suitable lower triagonal matrices). Call their union I. It then suffices to choose Q such that Q·I ∩I = ∅, which is clearly possible since k contains elements of non-unit valuation, so that we can let Q be translation over an element of high enough valuation.

The corresponding Schottky group Γ_{t} = [E_{t}, γE_{t}γ] is free of rank g = (p^{t} − 1), and normal in N. By our theorem, N is then exactly equal to the normalizer of Γ.

9. Artin-Schreier-Mumford curves – proof of proposition 3

(9.1) Let q = p^{t}. To avoid problems with the singularities of the particular plane model given, we think of the curve X_{t,c} : (y^{q} −y)(x^{q} −x) = c as embedded in P^{1} × P^{1}. Since |c| < 1, its analytic reduction is given by the “chess board” of lines (x^{q} − x)(y^{q} − y) = 0, and hence it is a Mumford curve whose genus (the number of squares of the board) is given by g_{t} = (q − 1)^{2}. It seems appropriate to call these curves Artin-Schreier-Mumford curves.

[figure: q; ; q {; Picture 3. the analytic reduction of X_{t,c}]

(9.2) One can see from the equation that the curve admits the following automorphisms: (x ↦ αx, y ↦ α^{−1}y) for α ∈ F_{q}^{∗}, (x ↦ x + β, y ↦ y + γ) for β, γ ∈ F_{q} and (x ↦ y, y ↦ x). This accounts for 2q^{2}(q − 1) = F (g_{t}) automorphisms, and hence by the main theorem, there are no more, and N_{t} is as expected (note that this gives a less intricate proof of the fact that N_{t} is discrete). It is also clear that the automorphism group is given by Aut(X_{t}) = Z_{p}^{2t} ⋊D_{p^{t}−1}, and hence the Schottky group (which is the kernel of N_{t} → Aut(X_{t})) is generated by the commutators [Z_{p}^{t}, γZ_{p}^{t}γ] for a fixed involution γ ∈ D_{p^{t}−1} − Z_{p^{t}−1}.

(9.3) The dimension of the stratum of curves X in M_{g_{t}} whose automorphism group contains Aut(X_{t,c}) can not be determined in a straightforward way because of wild ramification in π ([2] does not apply immediately), but Herrlich’s formula (7.4) does imply that the intersection of the closed stratum S̄_{Aut(X_{t})} with the Mumford locus is a one-dimensional connected subspace. By the main theorem, this is also the intersection with the open stratum. Note that the family X_{t,c} is not constant as c varies in A_{k}^{1}, since it is flat and has a singular fibre at zero. As c varies through {|c| < 1}, we move through the open stratum in M_{g_{t}} (and the embedding of Z_{p}^{t} in P GL(2, k) changes).

10. Drinfeld modular curves – proof of proposition 4

(10.1) Lemma (Gekeler [3], VII.5). The cover X(𝔫) → X(1) = P^{1} of Drinfeld modular curves is ramified above two points with respective indices q + 1 and q^{d}(q − 1), where d = deg(𝔫). The genus of X(𝔫) satifies

g(X(𝔫)) − 1 = |G(𝔫)|\frac{q^{d} − q − 1}{q^{d}(q^{2} − 1)}, where G(𝔫) = Γ(1)/Γ(𝔫)Z

is of order

|G(𝔫)| = q^{3d} ∏_{p|n}(1 − \frac{1}{q^{2 deg(p)}} ).

(10.2) Notation. If G is a group of automorphisms acting on a curve X, we denote by G_{i,P} for a point P ∈ G\X and i ∈ Z_{≥0} the order of the i-th ramification group of X → G\X at P. We will also write G to denote the order of a group G if there is no possibility for confusion. As usual, let π_{G} denote the corresponding covering morphism.

The following lemma characterizes when a group of automorphisms of a curve is the full group of automorphisms in terms of ramification data.

(10.3) Lemma. Assume p ≠ 2. Let A be a group of automorphisms acting on an ordinary curve X with quotient A\X = P^{1} wildly branched above two points P, Q, and suppose that there exists a subgroup G of A such that X → G\X = P^{1} is branched above two points x, y, with x tamely ramified and y wildly ramified, such that x and y map to P and Q respectively on A\X. Then A_{0,P} = A_{1,P} (A_{1,P} − 1).

The same holds if π_{A} is wildly branched above one point P and branched of order 2 above two more points, where x and y map to P on A\X.

Proof. Since X is ordinary, the second ramification groups in both covers are trivial. Let g be the genus of X. The Riemann-Hurwitz formula applied to π_{G} gives that

M := \frac{2(g − 1)}{G} = \frac{G_{1,y} − 2}{G_{0,y}} − \frac{1}{G_{0,x}} ,

so that 0 < M < 1. Applying similarly Hurwitz’s formula to π_{A} and dividing M by the result gives that

[A : G] = M \frac{A_{0,P} A_{0,Q}}{A_{0,Q}(A_{1,P} − 2) + A_{0,P} (A_{1,Q} − 2)}.

Since both terms in the denominator are positive (as P and Q are wildly ramified and p > 2),

[A : G] ≤ M \frac{A_{0,P}}{A_{1,P} − 2}.

Henn has proved ([8], Lemma 1) that \frac{A_{0,P}}{A_{1,P}} divides A_{1,P} − 1, so unless if A_{0,P} = A_{1,P} (A_{1,P} − 1), we have [A : G] ≤ MA_{1,P}. Since ramification above x in π_{G} is only tame, the p-part A_{1,P} is bounded by [A : G], so that finally, [A : G] ≤ M[A : G], a contradiction to M < 1.

A similar computation works for the second case. □

(10.4) Proof of proposition 4. We see from (10.1) that, except for q = 2, 3, X(𝔫) has more than 12(g(X(𝔫)) − 1) automorphisms.

Let Γ be the Schottky group of X(𝔫) and N its normalizer. We observe that G(𝔫) ⊆ A := Aut(X(𝔫)), and the cover X(𝔫) → X(1) is a cover of P^{1} by a Mumford curve ramified above exactly two points P, Q with ramification indices q + 1 and q^{d}(q − 1) (by (10.1)). But such covers where classified in (4.6), and the only case in which the ramification behaviour is compatible with the one of X(𝔫) → X(1) is case (A1) with P?L = P GL, t_{1} = dt. Since we have a tower of coverings of the form X(𝔫) → X(1) → A\X(𝔫), this implies that N contains N′ = P GL(2, q) ∗_{B(t,n_{−})} B(td, n_{−}). Again, the cover π_{A} has to belong to one of the cases in (4.6) and (4.7), since X(𝔫) has more than 12(g − 1) automorphisms (for q > 3). The only possible such N containing N′ are (A1) and (F 1) with t_{1} = t_{2} = 0 (the latter since the bound 12(g(X(n)) − 1) is exceeded). By the previous lemma (since p ≠ 2), we conclude that A_{0,P} = A_{1,P} (A_{1,P} − 1). In case (A1), we have A_{0,P} = p^{t_{2}}n_{+}, and the above identity leads to t = t_{2}, whereas 2t|t_{2}, a contradiction. Similarly, in case (F 1) with t_{1} = t_{2} = 0, we have A_{0,P} = p^{t_{3}}n_{−}, leading to t = t_{3}, whereas dt|t_{3} for d > 1 since N contains N′; a contradiction. We conclude that N = N′ and Aut(X(𝔫)) = G(𝔫).

Application of (7.4) to this situation now leads to the fact that closed the stratum of Mumfurd curves of genus g(X(𝔫)) having an automorphism group containing G(𝔫) (hence, N as normalizer of their Schottky group) has dimension zero. Since it is also connected, it has to be a point. This point then xcorresponds to the Drinfeld modular curve above, this proves the final claim of proposition 4. □

(10.5) Remark. The curve X(𝔫) is defined over a finite extension F_{𝔫} of F, generated by the n-torsion of the Carlitz module over F. A standard argument in model theory implies that for all but a finite number of primes 𝔭 of F_{𝔫}, the curve X(𝔫) ×_{F_{𝔫}} F_{𝔫}/𝔭 has the same automorphism group. It would be interesting to know what happens at such special primes (cf. Adler [1], Rajan [20]).

Acknowledgments. The first author is post-doctoral fellow of the Fund for Scientific Research - Flanders (FWO -Vlaanderen). The third author is supported by the TMR-programme of the EU. We thank F. Herrlich for sending us a copy of his thesis. This work was done at the MPIM; the authors are delighted to notice that the emblem of this institution is an icosahedron that goes particularly well with the proposition 2.

References

[1] A. Adler, The Mathieu group M_{11} and the modular curve X(11), Proc. London Math. Soc. (3) 74 (1997), no. 1, 1–28.

[2] H.M. Farkas and I. Kra, Riemann surfaces, Graduate Texts in Mathematics 71, Springer-Verlag, Berlin - Heidelberg - New York, 1980.

[3] E.-U. Gekeler, Drinfeld modular curves, Lecture Notes in Math., vol. 1231, Springer, Berlin - Heidelberg - New York, 1986.

[4] E.-U. Gekeler and M. Reversat, Jacobians of Drinfeld modular curves, J. Reine Angew. Math. 476 (1996), 27–93.

[5] L. Gerritzen, On non-Archimedean representations of abelian varieties, Math. Ann. 196 (1972), 323–346.

[6] L. Gerritzen and M. van der Put, Schottky groups and Mumford curves, Lecture Notes in Math., vol. 817, Springer, Berlin - Heidelberg - New York, 1980.

[7] H. Hasse, Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper, J. Reine Angew. Math. 172 (1935), 37–54.

[8] H.-W. Henn, Funktionenkörper mit großer Automorphismengruppe, J. Reine Angew. Math. 302 (1978), 96–115.

[9] F. Herrlich, Über Automorphismen p-adischer Schottkykurven, Dissertation, Ruhr-Universität Bochum, 1978.

[10] F. Herrlich, Die Ordnung der Automorphismengruppe einer p-adischen Schottkykurve, Math. Ann. 246 (1979/80), no. 2, 125–130.

[11] F. Herrlich, Endlich erzeugbare p-adische diskontinuierliche Gruppen, Arch. Math. (Basel) 35 (1981), no. 6, 505–515.

[12] F. Herrlich, On the stratification of the moduli space of Mumford curves, Groupe d’étude d’Analyse ultramétrique (Y. Amice, G. Christol and P. Robba, eds.), 11e année (1983/84), Exp. nr. 18, 9 pp., Secrétariat Math., Paris, 1985.

[13] B. Huppert, Endliche Gruppen I, Grundl. d. math. Wiss. 134, Springer-Verlag, Berlin - Heidelberg - New York, 1979.

[14] F. Kato, p-adic Schwarzian triangle groups of Mumford type, preprint, 1999.

[15] A. Karrass, A. Pietrowski, and D. Solitar, Finite and infinite cyclic extensions of free groups, J. Austral. Math. Soc. 16 (1973), 458–466.

[16] J. Lehner, Discontinuous groups and automorphic functions, Mathematical Surveys, No. VIII, American Mathematical Society, Providence, R.I., 1964 (reprinted with corrections, 1982).

[17] Yu. Manin and V. Drinfeld, Periods of p-adic Schottky groups, J. Reine Angew. Math. 262/263 (1973), 239–247.

[18] D. Mumford, An analytic construction of degenerating curves over complete local rings, Compositio Math. 24 (1972), 129–174.

[19] S. Nakajima, p-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc. 303 (1987), no. 2, 595–607.

[20] C.S. Rajan, Automorphisms of X(11) over characteristic 3 and the Mathieu group M_{11}, J. Ramanujan Math. Soc. 13 (1998), 63–72.

[21] M. Reversat, Sur les revêtements de Schottky des courbes modulaires de Drinfeld, Arch. Math. (Basel) 66 (1996), no. 5, 378–387.

[22] J.-P. Serre, Corps Locaux, Actualités scientifiques et industrielles 1296, Hermann, Paris, 1968.

[23] J.-P. Serre, Trees, Springer, Berlin - Heidelberg - New York, 1980.

[24] J.-P. Serre, The automorphism group of X(p) (copied from a letter dated June 26, 1996), appendix to: B. Mazur, Open problems regarding rational points on curves and varieties, in: Galois representations in arithmetic algebraic geometry (A.J. Scholl and R.L. Taylor, eds.), London Math. Soc. Lecture Note Series 254, Cambridge University Press, Cambridge, 1998.

[25] D. Subrao, The p-rank of Artin-Schreier curves, manus. math. 16 (1975), 169–193.

[26] H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe, Arch. Math. (Basel) 24 (1973), 527–544.

[27] R.C. Valentini and M.L. Madan, A Hauptsatz of L. E. Dickson and Artin-Schreier extensions, J. Reine Angew. Math. 318 (1980), 156–177.

[28] Martin Schönert et al., GAP – Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, RWTH-Aachen, Germany, fifth edition, 1995.

Max-Planck-Institut für Mathematik, P.O. Box 7280, D-53072 Bonn (gc^{1}, fk & ak)

University of Gent, Dept. of Pure Mathematics, Galglaan 2, B-9000 Gent (gc)

Graduate School of Mathematics, Kyushu University 33, Fukuoka 812-8581, Japan (fk)

e-mail: gc@cage.rug.ac.be, fkato@math.kyushu-u.ac.jp, kontogeo@mpim-bonn.mpg.de

1correspondence should be sent to this author at this address