Automorphisms of Harbater-Katz-Gabber curves
aa r X i v : . [ m a t h . AG ] N ov AUTOMORPHISMS OF HARBATER–KATZ–GABBER CURVES
FRAUKE M. BLEHER*, TED CHINBURG**, BJORN POONEN † , AND PETER SYMONDS Abstract.
Let k be a perfect field of characteristic p >
0, and let G be a finite group. Weconsider the pointed G -curves over k associated by Harbater, Katz, and Gabber to faithfulactions of G on k [[ t ]] over k . We use such “HKG G -curves” to classify the automorphisms of k [[ t ]] of p -power order that can be expressed by particularly explicit formulas, namely thosemapping t to a power series lying in a Z /p Z Artin–Schreier extension of k ( t ). In addition,we give necessary and sufficient criteria to decide when an HKG G -curve with an action ofa larger finite group J is also an HKG J -curve. Introduction
Let k be a field, let k [[ t ]] be the power series ring, and let Aut( k [[ t ]]) be its automorphismgroup as a k -algebra. When the characteristic of k is positive, Aut( k [[ t ]]) contains many in-teresting finite subgroups. One way to construct such subgroups is to start with an algebraiccurve X on which a finite group G acts with a fixed point x having residue field k ; then G acts on the completion ˆ O X,x of the local ring of x at X , and ˆ O X,x is isomorphic to k [[ t ]]for any choice of uniformizing parameter t at x . In fact, results of Harbater [14, §
2] and ofKatz and Gabber [19, Main Theorem 1.4.1] show that every finite subgroup G of Aut( k [[ t ]])arises in this way. Their results connect the ´etale fundamental group of Spec( k (( t ))) to thatof P k − { , ∞} . See Section 4.C for further discussion. The value of this technique is thatone can study local questions about elements of Aut( k [[ t ]]) using global tools such as theHurwitz formula for covers of curves over k .In this paper we use the above method to study two closely related problems when k isa perfect field of characteristic p >
0, which we assume for the rest of this paper. The firstproblem, described in Section 1.A, is to find explicit formulas for p -power-order elements σ ofAut( k [[ t ]]). In particular, we study σ that are “almost rational” in the sense of Definition 1.1.Our main result in this direction, Theorem 1.2, classifies all such σ .The second problem, described in Sections 1.B and 1.C, is to study the full automorphismgroup of the so-called Harbater–Katz–Gabber G -curves (HKG G -curves), which are certain Mathematics Subject Classification.
Primary 14H37; Secondary 14G17, 20F29.*Supported by NSA Grant † Supported by NSF Grants urves X with a G -action as above. One reason for this study is that it turns out thatalmost rational automorphisms arise from HKG G -curves X for which Aut( X ) is strictlylarger than G . In fact, our Theorems 5.1(c) and 5.9 concerning such X are needed for ourproof of Theorem 1.2 on almost rational automorphisms of k [[ t ]].For some other applications of HKG G -curves, e.g., to the problem of lifting automor-phisms of k [[ t ]] to characteristic 0, see [4] and its references.1.A. Finite-order automorphisms of k [[ t ]] . Every order p element of Aut( k [[ t ]]) is conju-gate to t t (1 + ct m ) − /m for some c ∈ k × and some positive integer m prime to p (see [20,Proposition 1.2], [21, § p n for n >
1. Each such automorphism is conjugate to t σ ( t ) forsome σ ( t ) ∈ k [[ t ]] that is algebraic over k ( t ) (see Corollary 4.11). In this case, the field L := k ( t, σ ( t ) , . . . , σ p n − ( t )) ⊆ k (( t )) is algebraic over k ( t ). When n >
1, we cannot have L = k ( t ), because the group Aut k ( k ( t )) ≃ PGL ( k ) has no element of order p . The nextsimplest case from the point of view of explicit power series is the following: Definition 1.1.
Call σ ∈ Aut( k [[ t ]]) almost rational if the field L := k ( { σ ( t ) : σ ∈ G } ) is a Z /p Z Artin–Schreier extension of k ( t ); i.e., L = k ( t, β ) where β ∈ k (( t )) satisfies ℘ ( β ) = α for some α ∈ k ( t ); here ℘ is the Artin–Schreier operator defined by ℘ ( x ) := x p − x .By subtracting an element of k [ t − ] from β , we may assume that β ∈ tk [[ t ]] and hence α ∈ k ( t ) ∩ tk [[ t ]]. Then we have an explicit formula for β , namely β = − ∞ X i =0 α p i , and σ ( t ) is a rational function in t and β . This is the sense in which almost rationalautomorphisms have explicit power series.Prior to the present article, two of us found one explicit example of an almost rational σ of order p n > p (and its inverse); see [5]. Our first main theorem describes all such σ up toconjugacy. Theorem 1.2.
Suppose that σ is an almost rational automorphism of k [[ t ]] of order p n forsome n > . Then p = 2 , n = 2 , and there exists b ∈ k (unique modulo ℘ ( k ) = { ℘ ( a ) : a ∈ k } ) such that σ is conjugate to the order almost rational automorphism σ b ( t ) := b t + ( b + 1) t + βb + t , (1.3) where β is the unique solution to β − β = t + ( b + b + 1) t in tk [[ t ]] .Remark . If k is algebraically closed, then ℘ ( k ) = k , so Theorem 1.2 implies that allalmost rational automorphisms of order 4 lie in one conjugacy class in Aut( k [[ t ]]). emark . The example in [5] was σ ( t ) = t + t + ∞ X j =0 2 j − X ℓ =0 t · j +2 ℓ = t + t + ( t ) + ( t + t ) + ( t + t + t + t ) + · · · = t t + γ (1 + t ) over F , where the series γ := P ∞ i =0 ( t + t ) i satisfies γ − γ = t + t . (If β is as inTheorem 1.2, then γ = β + t .) Zieve and Scherr communicated to us that the inverse of σ has a simpler series, namely σ ( t ) = t − ∞ X i =0 ( t + t ) i = ∞ X i =0 t · i − + ∞ X j =2 t j − . In general, the inverse of σ b is σ b +1 (Remark 5.14). Remark . Let σ be any element of finite order in Aut( k [[ t ]]). Even if σ is not almostrational, we can assume after conjugation that the power series σ ( t ) = P i ≥ a i t i is algebraicover k ( t ), as mentioned above. When k is finite, this implies that the sequence ( a i ) is Turingcomputable, and even p -automatic ; i.e., there is a finite automaton that calculates a i whensupplied with the base p expansion of i [6, 7].1.B. Harbater–Katz–Gabber G -curves. An order p n element of Aut( k [[ t ]]) induces aninjective homomorphism Z /p n Z −→ Aut( k [[ t ]]). Suppose that we now replace Z /p n Z withany finite group G . Results of Harbater [14, §
2] when G is a p -group, and of Katz and Gabber[19, Main Theorem 1.4.1] in general, show that any injective α : G −→ Aut( k [[ t ]]) arises froma G -action on a curve. More precisely, α arises from a triple ( X, x, φ ) consisting of a smoothprojective curve X , a point x ∈ X ( k ), and an injective homomorphism φ : G −→ Aut( X )such that G fixes x : here α expresses the induced action of G on the completed local ring b O X,x with respect to some uniformizer t . In Section 4.B we will define a Harbater–Katz–Gabber G -curve (HKG G -curve) to be a triple ( X, x, φ ) as above with
X/G ≃ P k such thatapart from x there is at most one non-free G -orbit, which is tamely ramified if it exists. Wewill sometimes omit φ from the notation.HKG G -curves play a key role in our proof of Theorem 1.2. Our overall strategy is toreduce Theorem 1.2 to the classification of certain HKG G -curves, and then to use geometrictools such as the Hurwitz formula to complete the classification.1.C. Harbater–Katz–Gabber G -curves with extra automorphisms. In this section,(
X, x ) is an HKG G -curve and J is a finite group such that G ≤ J ≤ Aut( X ). We do notassume a priori that J fixes x . Let g X be the genus of X . Question 1.7.
Must ( X, x ) be an HKG J -curve? he answer is sometimes yes, sometimes no. Here we state our three main theorems inthis direction; we prove them in Section 7. Theorem 1.8.
We have that ( X, x ) is an HKG J -curve if and only if J fixes x . When g X >
1, Theorem 1.10 below gives a weaker hypothesis that still is sufficient toimply that (
X, x ) is an HKG J -curve. Let J x be the decomposition group Stab J ( x ). Definition 1.9.
We call the action of J mixed if there exists σ ∈ J such that σ ( x ) = x and σ ( x ) is nontrivially but tamely ramified with respect to the action of J x , and unmixed otherwise. Theorem 1.10. If g X > and the action of J is unmixed, then ( X, x ) is an HKG J -curve. We will also answer Question 1.7 in an explicit way when g X ≤
1, whether or not theaction of J is mixed.Finally, if J is solvable, the answer to Question 1.7 is almost always yes, as the nexttheorem shows. For the rest of the paper, k denotes an algebraic closure of k . Theorem 1.11. If J is solvable and ( X, x ) is not an HKG J -curve, then one of the followingholds: • X ≃ P ; • p is or , and X is an elliptic curve of j -invariant ; • p = 3 , and X is isomorphic over k to the genus curve z = t u − tu in P ; or • p = 2 , and X is isomorphic over k to the smooth projective model of the genus affine curve z = ( u + u )( u + u + 1) . Each case in Theorem 1.11 actually arises. We prove a stronger version of Theorem 1.11in Theorem 7.13 using the examples discussed in Section 6.2.
Automorphisms of k [[ t ]]The purpose of this section is to recall some basic results about Aut( k [[ t ]]).2.A. Groups that are cyclic mod p . A p ′ -group is a finite group of order prime to p .A finite group G is called cyclic mod p if it has a normal Sylow p -subgroup such that thequotient is cyclic. Equivalently, G is cyclic mod p if G is a semidirect product P ⋊ C with P a p -group and C a cyclic p ′ -group. In this case, P is the unique Sylow p -subgroup of G , and the Schur–Zassenhaus theorem [18, Theorem 3.12] implies that every subgroup of G isomorphic to C is conjugate to C . .B. The Nottingham group.
Any k -algebra automorphism σ of k [[ t ]] preserves the max-imal ideal and its powers, and hence is t -adically continuous, so σ is uniquely determined byspecifying the power series σ ( t ) = P n ≥ a n t n (with a ∈ k × ). The map Aut( k [[ t ]]) −→ k × sending σ to a is a surjective homomorphism. The Nottingham group N ( k ) is the kernel ofthis homomorphism; it consists of the power series t + P n ≥ a n t n under composition. ThenAut( k [[ t ]]) is a semidirect product N ( k ) ⋊ k × . For background on N ( k ), see, e.g., [3].If k is finite, then N ( k ) is a pro- p group. In general, N ( k ) is pro-solvable with a filtrationwhose quotients are isomorphic to k under addition; thus every finite subgroup of N ( k ) is a p -group. Conversely, Leedham-Green and Weiss, using techniques of Witt, showed that anyfinite p -group can be embedded in N ( F p ); indeed, so can any countably based pro- p group [2].The embeddability of finite p -groups follows alternatively from the fact that the maximalpro- p quotient of the absolute Galois group of k (( t − )) is a free pro- p group of infinite rank[19, (1.4.4)].On the other hand, any finite subgroup of k × is a cyclic p ′ -group. Thus any finite subgroupof Aut( k [[ t ]]) is cyclic mod p , and any finite p -group in Aut( k [[ t ]]) is contained in N ( k ).2.C. Algebraic automorphisms of k [[ t ]] . Call σ ∈ Aut( k [[ t ]]) algebraic if σ ( t ) is algebraicover k ( t ). Proposition 2.1.
The set
Aut alg ( k [[ t ]]) of all algebraic automorphisms of k [[ t ]] over k is asubgroup of Aut( k [[ t ]]) .Proof. Suppose that σ ∈ Aut alg ( k [[ t ]]), so σ ( t ) is algebraic over k ( t ). Applying anotherautomorphism τ ∈ Aut( k [[ t ]]) to the algebraic relation shows that σ ( τ ( t )) is algebraic over k ( τ ( t )). So if τ is algebraic, so is σ ◦ τ . On the other hand, taking τ = σ − shows that t isalgebraic over k ( σ − ( t )). Since t is not algebraic over k , this implies that σ − ( t ) is algebraicover k ( t ). (cid:3) Automorphisms of order p . The following theorem was proved by Klopsch [20,Proposition 1.2] and reproved by Lubin [21, §
4] (they assumed that k was finite, but thisis not crucial). Over algebraically closed fields it was shown in [1, p. 211] by Bertin andM´ezard, who mention related work of Oort, Sekiguchi and Suwa in [22]. For completeness,we give here a short proof, similar to the proofs in [20, Appendix] and [1, p. 211]; it worksover any perfect field k of characteristic p > Theorem 2.2.
Every σ ∈ N ( k ) of order p is conjugate in N ( k ) to t t (1 + ct m ) − /m fora unique positive integer m prime to p and a unique c ∈ k × . The automorphisms given by ( m, c ) and ( m ′ , c ′ ) are conjugate in Aut( k [[ t ]]) if and only if m = m ′ and c/c ′ ∈ k × m .Proof. Extend σ to the fraction field k (( t )). By Artin–Schreier theory, there exists y ∈ k (( t ))such that σ ( y ) = y + 1. This y is unique modulo k (( t )) σ . Since σ acts trivially on theresidue field of k [[ t ]], we have y / ∈ k [[ t ]]. Thus y = ct − m + · · · for some m ∈ Z > and c ∈ k × . hoose y so that m is minimal. If the ramification index p divided m , then we could subtractfrom y an element of k (( t )) σ with the same leading term, contradicting the minimality of m . Thus p ∤ m . By Hensel’s lemma, y = c ( t ′ ) − m for some t ′ = t + · · · . Conjugating bythe automorphism t t ′ lets us assume instead that y = ct − m . Substituting this into σ ( y ) = y + 1 yields c σ ( t ) − m = ct − m + 1. Equivalently, σ ( t ) = t (1 + c − t m ) − /m . Rename c − as c .Although y is determined only modulo ℘ ( k (( t ))), the leading term of a minimal y isdetermined. Conjugating σ in Aut( k [[ t ]]) amounts to expressing σ with respect to a newuniformizer u = u t + u t + · · · . This does not change m , but it multiplies c by u m .Conjugating σ in N ( k ) has the same effect, except that u = 1, so c is unchanged too. (cid:3) Remark . For each positive integer m prime to p , let Disp m : N ( k ) −→ N ( k ) be the mapsending t f ( t ) to t f ( t m ) /m (we take the m th root of the form t + · · · ). This is aninjective endomorphism of the group N ( k ), called m -dispersal in [21]. It would be conjugationby t t m , except that t t m is not in Aut( k [[ t ]]) (for m > t t (1 + t ) − by conjugating by t ct and thendispersing. 3. Ramification and the Hurwitz Formula
Here we review the Hurwitz formula and related facts we need later.3.A.
Notation.
By a curve over k we mean a 1-dimensional smooth projective geometricallyintegral scheme X of finite type over k . For a curve X , let k ( X ) denote its function field,and let g X or g k ( X ) denote its genus. If G is a finite group acting on a curve X , then X/G denotes the curve whose function field is the invariant subfield k ( X ) G .3.B. The local different.
Let G be a finite subgroup of Aut( k [[ t ]]). For i ≥
0, definethe ramification subgroup G i := { g ∈ G | g acts trivially on k [[ t ]] / ( t i +1 ) } as usual. Let d ( G ) := P ∞ i =0 ( | G i |− ∈ Z ≥ ; this is the exponent of the local different [24, IV, Proposition 4].3.C. The Hurwitz formula.
In this paragraph we assume that k is an algebraically closedfield of characteristic p >
0. Let H be a finite group acting faithfully on a curve X over k .For each s ∈ X ( k ), let H s ≤ H be the inertia group. We may identify b O X,s with k [[ t ]] and H s with a finite subgroup G ≤ Aut( k [[ t ]]); then define d s = d s ( H ) := d ( H s ). We have d s > s is ramified. If s is tamely ramified, meaning that H s is a p ′ -group, then d s = | H s | −
1. The Hurwitz formula [15, IV, 2.4] is2 g X − | H | (2 g X/H −
2) + X s ∈ X ( k ) d s . Remark . When we apply the Hurwitz formula to a curve over a perfect field that is notalgebraically closed, it is understood that we first extend scalars to an algebraic closure. .D. Lower bound on the different.
We continue to assume that k is an algebraicallyclosed field of characteristic p >
0. The following material is taken from [24, IV], as inter-preted by Lubin in [21]. Let G and the G i be as in Section 3.B. An integer i ≥ break in the lower numbering of the ramification groups of G if G i = G i +1 . Let b , b , . . . be thebreaks in increasing order; they are all congruent modulo p . The group G /G embeds into k × , while G i /G i +1 embeds in the additive group of k if i ≥ G is a cyclic group of order p n with generator σ . Then G = G and each quotient G i /G i +1 is killed by p . Thus there must be exactly n breaks b , . . . , b n − .If 0 ≤ i ≤ b , then G i = G ; if 1 ≤ j ≤ n − b j − < i ≤ b j , then | G i | = p n − j ; and if b n − < i , then G i = { e } . According to the Hasse–Arf theorem, there exist positive integers i , . . . , i n − such that b j = i + pi + · · · + p j i j for 0 ≤ j ≤ n −
1. Then d ( G ) = ( i + 1)( p n −
1) + i ( p n − p ) + · · · + i n − ( p n − p n − ) . (3.2)The upper breaks b ( j ) we do not need to define here, but they have the property that in thecyclic case, b ( j ) = i + · · · + i j for 0 ≤ j ≤ n − p ∤ b (0) , that b ( j ) ≥ pb ( j − for 1 ≤ j ≤ n −
1, andthat if this inequality is strict then p ∤ b ( j ) ; this is proved in [24, XV, § b (0) , . . . , b ( n − that satisfies these threeconditions is realized by some element of order p n in Aut( k [[ t ]]) [21, Observation 5].Thus i ≥
1, and i j ≥ ( p − p j − for 1 ≤ j ≤ n −
1. Substituting into (3.2) yields thefollowing result.
Lemma 3.3. If G is cyclic of order p n , then d ( G ) ≥ p n + p n +1 + p n − p − p + 1 and this bound is sharp.Remark . Lemma 3.3 is valid over any perfect field k of characteristic p , because extendingscalars to k does not change d ( G ).4. Harbater–Katz–Gabber G -curves Let k be a perfect field of characteristic p > Pointed G -curves.Definition 4.1. A pointed G -curve over k is a triple ( X, x, φ ) consisting of a curve X , apoint x ∈ X ( k ), and an injective homomorphism φ : G −→ Aut( X ) such that G fixes x . (Wewill sometimes omit φ from the notation.) uppose that ( X, x, φ ) is a pointed G -curve. The faithful action of G on X induces afaithful action on k ( X ). Since G fixes x , the latter action induces a G -action on the k -algebras O X,x and b O X,x . Since Frac( O X,x ) = k ( X ) and O X,x ⊆ b O X,x , the G -action on b O X,x is faithfultoo. Since x ∈ X ( k ), a choice of uniformizer t at x gives a k -isomorphism b O X,x ≃ k [[ t ]]. Thuswe obtain an embedding ρ X,x,φ : G ֒ → Aut( k [[ t ]]). Changing the isomorphism b O X,x ≃ k [[ t ]]conjugates ρ X,x,φ by an element of Aut( k [[ t ]]), so we obtain a map { pointed G -curves } −→ { conjugacy classes of embeddings G ֒ → Aut( k [[ t ]]) } (4.2)( X, x, φ ) [ ρ X,x,φ ] . Also, G is the inertia group of X −→ X/G at x . Lemma 4.3. If ( X, x, φ ) is a pointed G -curve, then G is cyclic mod p .Proof. The group G is embedded as a finite subgroup of Aut( k [[ t ]]). (cid:3) Harbater–Katz–Gabber G -curves.Definition 4.4. A pointed G -curve ( X, x, φ ) over k is called a Harbater–Katz–Gabber G -curve(HKG G -curve) if both of the following conditions hold:(i) The quotient X/G is of genus 0. (This is equivalent to
X/G ≃ P k , since x maps to a k -point of X/G .)(ii) The action of G on X − { x } is either unramified everywhere, or tamely and nontriviallyramified at one G -orbit in X ( k ) − { x } and unramified everywhere else. Remark . Katz in [19, Main Theorem 1.4.1] focused on the base curve
X/G as startingcurve. He fixed an isomorphism of
X/G with P k identifying the image of x with ∞ and theimage of a tamely and nontrivially ramified point of X ( k ) − { x } (if such exists) with 0. Hethen considered Galois covers X −→ X/G = P k satisfying properties as above; these werecalled Katz–Gabber covers in [4]. For our applications, however, it is more natural to focuson the upper curve X .HKG curves have some good functoriality properties that follow directly from the defini-tion: • Base change:
Let X be a curve over k , let x ∈ X ( k ), and let φ : G −→ Aut( X ) be ahomomorphism. Let k ′ ⊇ k be a field extension. Then ( X, x, φ ) is an HKG G -curveover k if and only if its base change to k ′ is an HKG G -curve over k ′ . • Quotient:
If (
X, x, φ ) is an HKG G -curve, and H is a normal subgroup of G , then X/H equipped with the image of x and the induced G/H -action is an HKG
G/H -curve.
Example 4.6.
Let P be a finite subgroup of the additive group of k , so P is an elementaryabelian p -group. Then the addition action of P on A k extends to an action φ : P −→ Aut( P k )totally ramified at ∞ and unramified elsewhere, so ( P k , ∞ , φ ) is an HKG P -curve. xample 4.7. Suppose that C is a p ′ -group and that ( X, x, φ ) is an HKG C -curve. ByLemma 4.3, C is cyclic. By the Hurwitz formula, X must have genus 0 since there are atmost two C -orbits of ramified points and all the ramification is tame. Moreover, X hasa k -point (namely, x ), so X ≃ P k , and C is a p ′ -subgroup of the stabilizer of x insideAut( X ) ≃ Aut( P k ) ≃ PGL ( k ). It follows that after applying an automorphism of X = P k ,we can assume that C fixes the points 0 and ∞ and corresponds to the multiplication actionof a finite subgroup of k × on A k . Conversely, such an action gives rise to an HKG C -curve( P k , ∞ , φ ).The following gives alternative criteria for testing whether a pointed G -curve is an HKG G -curve. Proposition 4.8.
Let ( X, x, φ ) be a pointed G -curve. Let P be the Sylow p -subgroup of G .Then the following are equivalent: (i) ( X, x, φ ) is an HKG G -curve. (ii) ( X, x, φ | P ) is an HKG P -curve. (iii) The quotient
X/P is of genus , and the action of P on X − { x } is unramified. (iv) Equality holds in the inequality g X ≥ − | P | + d x ( P ) / .Proof. Let C = G/P .(iii) ⇒ (ii): Trivial.(i) ⇒ (iii): By the quotient property of HKG curves, X/P is an HKG C -curve, so X/P ≃ P k by Example 4.7. At each y ∈ X ( k ) − { x } , the ramification index e y for the P -action divides | P | but is prime to p , so e y = 1. Thus the action of P on X − { x } is unramified.(ii) ⇒ (i): Applying the result (i) ⇒ (iii) to P shows that X −→ X/P is unramified outside x . There is a covering P k ≃ X/P −→ X/G , so
X/G ≃ P k . We may assume that C = { } .By Example 4.7, the cover X/P −→ X/G is totally tamely ramified above two k -points, andunramified elsewhere. One of the two points must be the image of x ; the other is the imageof the unique tamely ramified G -orbit in X ( k ), since X −→ X/P is unramified outside x .(iii) ⇔ (iv): The Hurwitz formula (see Remark 3.1) for the action of P simplifies to theinequality in (iv) if we use g X/P ≥ X − { x } . Thus equality holds in (iv) if and only if g X/P = 0 and the action of P on X − { x } is unramified. (cid:3) The Harbater–Katz–Gabber theorem.
The following is a consequence of work ofHarbater [14, §
2] when G is a p -group and of Katz and Gabber [19, Main Theorem 1.4.1]when G is arbitrary. Theorem 4.9 (Harbater, Katz–Gabber) . The assignment ( X, x, φ ) ρ X,x,φ induces a sur-jection from the set of HKG G -curves over k up to equivariant isomorphism to the set ofconjugacy classes of embeddings of G into Aut( k [[ t ]]) . orollary 4.10. Any finite subgroup of
Aut k ( k [[ t ]]) can be conjugated into Aut k ′ ( k ′ [[ t ]]) forsome finite extension k ′ of k in k .Proof. The subgroup is realized by some HKG curve over k . Any such curve is defined oversome finite extension k ′ of k . (cid:3) Corollary 4.11.
Any finite subgroup of
Aut( k [[ t ]]) can be conjugated into Aut alg ( k [[ t ]]) .Proof. The subgroup is realized by some HKG curve X . By conjugating, we may assumethat the uniformizer t is a rational function on X . Then each power series σ ( t ) representsanother rational function on X , so σ ( t ) is algebraic over k ( t ). (cid:3) Almost rational automorphisms
The field generated by a group of algebraic automorphisms.
Let G be a finitesubgroup of Aut alg ( k [[ t ]]). Let L := k ( { σ ( t ) : σ ∈ G } ) ⊆ k (( t )). Then L is a finite extensionof k ( t ), so L ≃ k ( X ) for some curve X . The t -adic valuation on k (( t )) restricts to a valuationon L associated to a point x ∈ X ( k ). The G -action on k (( t )) preserves L . This induces anembedding φ : G −→ Aut( X ) such that G fixes x , so ( X, x, φ ) is a pointed G -curve over k . Theorem 5.1.
Let G be a finite subgroup of Aut alg ( k [[ t ]]) . Let L and ( X, x, φ ) be as above.Let d := [ L : k ( t )] . (a) We have g X ≤ ( d − . (b) If G is cyclic of order p n , then g X ≥ p ( p n − p n − − p + 1) . Moreover, if equality holds,then ( X, x, φ ) is an HKG G -curve. (c) Suppose that G is cyclic of order p n . Then d ≥ s p ( p n − p n − − p + 1) . (5.2) In particular, if d ≤ p and n ≥ , then d = p = n = 2 and ( X, x, φ ) is an HKG Z / Z -curve of genus .Proof. (a) In [23, § F ⊆ L is called d -controlled if there exists e ∈ Z > such that[ L : F ] ≤ d/e and g F ≤ ( e − . In our setting, the G -action on k (( t )) preserves L , so[ L : k ( σ ( t ))] = d for every σ ∈ G . By [23, Corollary 2.2], L ⊆ L is d -controlled. Here d/e = 1, so g L ≤ ( e − = ( d − .(b) In the inequality g X ≥ − | G | + d x ( G ) / | G | = p n and the bound of Lemma 3.3. If equality holds, then Proposition 4.8(iv) ⇒ (i) shows that( X, x, φ ) is an HKG G -curve. c) Combine the upper and lower bounds on g X in (a) and (b). If d ≤ p and n ≥
2, then p ≥ d ≥ s p ( p − p − p + 1) = 1 + ( p − r p ≥ p −
1) = p, so equality holds everywhere. In particular, p = d , n = 2, and p/ d = p = n = 2.Also, (b) shows that ( X, x, φ ) is an HKG G -curve, and g X = ( d − = 1. (cid:3) Remark . Part (c) of Theorem 5.1 implies the first statement in Theorem 1.2, namelythat if σ is an almost rational automorphism of order p n > p , then p = n = 2. To completethe proof of Theorem 1.2 we will classify in Section 5.B the σ when p = n = 2.5.B. Almost rational automorphisms of order . In this section, k is a perfect field ofcharacteristic 2, and G = Z / Z . Definition 5.4.
For a, b ∈ k , let E a,b be the projective closure of z − z = w + ( b + b + 1) w + a. Let O ∈ E a,b ( k ) be the point at infinity, and let φ : Z / Z −→ Aut( E a,b ) send 1 to the order 4automorphism σ : ( w, z ) ( w + 1 , z + w + b ) . Proposition 5.5.
Each ( E a,b , O, φ ) in Definition 5.4 is an HKG Z / Z -curve over k .Proof. The automorphism σ fixes O . Also, σ maps ( w, z ) to ( w, z + 1), so σ fixes only O ;hence the G -action on E a,b − { O } is unramified. Since E a,b −→ E a,b /G is ramified, the genusof E a,b /G is 0. (cid:3) Proposition 5.6.
Let k be a perfect field of characteristic . Let G = Z / Z . For an HKG G -curve ( X, x, φ ′ ) over k , the following are equivalent: (i) The genus of X is . (ii) The lower ramification groups for X −→ X/G at x satisfy | G | = | G | = 4 , | G | = | G | = 2 , and | G i | = 1 for i ≥ . (iii) The ramification group G equals { } . (iv) There exist a, b ∈ k such that ( X, x, φ ′ ) is isomorphic to the HKG G -curve ( E a,b , O, φ ) of Definition 5.4.Proof. Let g be the genus of X . Since G is a 2-group, | G | = | G | = 4.(ii) ⇒ (i): This follows from the Hurwitz formula (see Remark 3.1)2 g − −
2) + X i ≥ ( | G i | − . (i) ⇒ (ii): If g = 1, then the Hurwitz formula yields 0 = − P i ≥ ( | G i | − | G i | form a decreasing sequence of powers of 2 and include all the numbers 4, 2, and 1(see Section 3.D), the only possibility is as in (ii). ii) ⇒ (iii): Trivial.(iii) ⇒ (ii): The lower breaks (see Section 3.D) satisfy 1 ≤ b < b <
4. Since b ≡ b (mod 2), (ii) follows.(iv) ⇒ (i): The formulas in [25, III. §
1] show that E a,b is an elliptic curve, hence of genus 1.(i) ⇒ (iv): By [25, A.1.2(c)], an elliptic curve with an order 4 automorphism has j -invariant1728 = 0 ∈ k . By [25, A.1.1(c)], it has an equation y + a y = x + a x + a . Substituting y y + a − a x leads to an alternative form y + a y = x + a x + a . Let u ∈ k × besuch that σ ∗ acts on H ( X, Ω ) by multiplication by u − . Then u = 1, so u = 1. By [25,p. 49], σ has the form ( x, y ) ( x + r, y + sx + t ) for some r, s, t ∈ k . Since σ = 1, we have s = 0. Conjugating by a change of variable ( x, y ) ( ǫ x, ǫ y ) lets us assume that s = 1.The condition that ( x, y ) ( x + r, y + x + t ) preserves y + a y = x + a x + a implies that a = r = 1 and a = t + t + 1. Rename t, x, y as b, w, z . (cid:3) Corollary 5.7.
The HKG Z / Z -curves that are minimally ramified in the sense of havingthe smallest value of inf { i : G i = { }} are those satisfying the equivalent conditions inProposition 5.6. Let ℘ ( x ) := x − x be the Artin–Schreier operator in characteristic 2. The following lemmais clear. Lemma 5.8.
Let
L/K be a Z / Z Artin–Schreier extension, so there exist a ∈ K and b ∈ L − K such that ℘ ( b ) = a . If x ∈ L − K satisfies ℘ ( x ) ∈ K , then x ∈ b + K . Theorem 5.9.
Let k be a perfect field of characteristic . Let G = Z / Z . Let X be the setof HKG G -curves satisfying the equivalent conditions in Proposition 5.6. Then (a) The map (4.2) restricts to a surjection from X to the set of conjugacy classes in Aut( k [[ t ]]) containing an almost rational automorphism of order . (b) Explicitly, E a,b (made into an HKG G -curve as in Proposition 5.5) maps to the conjugacyclass of σ b ( t ) := b t + ( b + 1) t + βb + t , (5.10) where β := P ∞ i =0 ( t + ( b + b + 1) t ) i is the unique solution to β − β = t + ( b + b + 1) t in tk [[ t ]] . (c) For b, b ′ ∈ k , the automorphisms σ b , σ b ′ ∈ Aut( k [[ t ]]) are conjugate if and only if b ≡ b ′ (mod ℘ ( k )) .Proof. (a) First we show that each E ,b maps to a conjugacy class containing an almost rationalautomorphism; the same will follow for E a,b for a = 0 once we show in the proof of (c) that E a,b gives rise to the same conjugacy class as E ,b . Let P := (0 , ∈ E ,b ( k ). Composing w with translation-by- P yields a new rational function w P = z/w on E ,b ; define z P similarly,so z P = 1 − z /w . Since w has a simple zero at P , the function t := w P has a simple ero at O . Also, σ j ( t ) ∈ k ( E ,b ) = k ( t, z P ), which shows that σ is almost rational since z P − z P = w P + ( b + b + 1) w P .Now suppose that σ is any almost rational automorphism of order 4. Theorem 5.1(c) showsthat σ arises from an HKG Z / Z -curve of genus 1, i.e., a curve as in Proposition 5.6(i).(b) Again by referring to the proof of (c), we may assume a = 0. Follow the first half ofthe proof of (a) for E ,b . In terms of the translated coordinates ( w P , z P ) on E ,b , the order 4automorphism of the elliptic curve is( t, β ) σ (( t, β ) − P ) + P. It is a straightforward but lengthy exercise to show that the first coordinate equals theexpression σ b ( t ) in (5.10). One uses t = w P = z/w , β = z P = 1 − z /w , and theformulas σ ( w ) = w + 1 and σ ( z ) = z + w + b . In verifying equalities in the field k ( t, β ),one can use the fact that k ( t, β ) is the quadratic Artin–Schreier extension of k ( t ) defined by β − β = t + ( b + b + 1) t .(c) Let v := w − w . Let b O be the completion of the local ring of E a,b at the point O atinfinity, and let b K := Frac( b O ) = k (( w − ))( z − ). With respect to the discrete valuation on b K ,the valuations of w , z and v are − − −
4, respectively. With respect to the discretevaluation on k (( w − )), the valuation of w is − v is −
2. We have b K G = k (( v − )). Define w ′ , z ′ , v ′ , σ ′ , b O ′ , and b K ′ = k (( w ′− ))( z ′− ) similarly for E a ′ ,b ′ . Bydefinition of the map (4.2), E a,b and E a ′ ,b ′ give rise to the same conjugacy class if and only ifthere exists a G -equivariant continuous isomorphism b O ∼ −→ b O ′ or equivalently α : b K ∼ −→ b K ′ .It remains to prove that α exists if and only if b ≡ b ′ (mod ℘ ( k )).= ⇒ : Suppose that α exists. Lemma 5.8 shows that α ( w ) = w ′ + f for some f ∈ k (( v ′− )).Since α preserves valuations, f ∈ k [[ v ′− ]]. Since v ′ has valuation − k (( w ′− )), thevaluation of v ′− in this field is 2. Therefore f ∈ k [[ v ′− ]] implies f = c + P i ≥ f i w ′− i forsome c, f i ∈ k . Similarly, α ( z ) = z ′ + h for some h = P i ≥− h i w ′− i ∈ w ′ k [[ w ′− ]]. Subtractingthe equations α ( z ) − α ( z ) = α ( w ) + ( b + b + 1) α ( w ) + az ′ − z ′ = w ′ + ( b ′ + b ′ + 1) w ′ + a ′ yields h − h = ( w ′ + f ) − w ′ + ( b + b + 1)( w ′ + f ) − ( b ′ + b ′ + 1) w ′ + a − a ′ (5.11)= w ′ f + w ′ f + f + ℘ ( b − b ′ ) w ′ + ( b + b + 1) f + a − a ′ h − h ≡ ( c + ℘ ( b − b ′ )) w ′ + c w ′ + ( f + c + ( b + b + 1) c + a − a ′ ) (mod w ′− k [[ w ′− ]]) . (5.12) quating coefficients of w ′ yields h − = c . The G -equivariance of α implies α ( σ ( z )) = σ ′ ( α ( z ))( z ′ + h ) + ( w ′ + f ) + b = ( z ′ + w ′ + b ′ ) + σ ′ ( h ) h + f + b = b ′ + σ ′ ( h ) h − w ′ + h + c + b ≡ b ′ + h − ( w ′ + 1) + h (mod w ′− k [[ w ′− ]]) (5.13) b − b ′ = h − − c = c − c = ℘ ( c ) . ⇐ = : Conversely, suppose that b − b ′ = ℘ ( c ) for some c ∈ k . We must build a G -equivariant continuous isomorphism α : b K ∼ −→ b K ′ . Choose f := c + P i ≥ f i w ′− i in k [[ v ′− ]]so that the value of f makes the coefficient of w ′ in (5.12), namely the constant term, equalto 0. The coefficient of w ′ in (5.12) is c + ℘ ( ℘ ( c )) = c . So (5.12) simplifies to h − h ≡ c w ′ + c w ′ (mod w ′− k [[ w ′− ]]) . Thus we may choose h := c w ′ + P i ≥ h i w ′− i so that (5.11) holds. Define α : k (( w − )) −→ k (( w ′− )) by α ( w ) := w ′ + f . Equation (5.11) implies that α extends to α : b K −→ b K ′ bysetting α ( z ) := z ′ + h . Then α | k (( w − )) is G -equivariant since ( w ′ + 1) + f = ( w ′ + f ) + 1. Inother words, σ − α − σ ′ α ∈ Gal( b K/k (( w − ))) = { , σ } . If σ − α − σ ′ α = σ , then ασ = σ ′ αα ( σ ( z )) = σ ′ ( α ( z )) α ( z + w + b + 1) = σ ′ ( z ′ + h )( z ′ + h ) + ( w ′ + f ) + b + 1 = ( z ′ + w ′ + b ′ ) + σ ′ ( h );by the calculation leading to (5.13), this is off by 1 modulo w ′− k [[ w ′− ]]. Thus σ − α − σ ′ α = 1instead. In other words, α is G -equivariant. (cid:3) Remark . Changing b to b + 1 does not change the curve E a,b , but it changes σ to σ − .Thus σ and σ − are conjugate in Aut( k [[ t ]]) if and only if 1 ∈ ℘ ( k ), i.e., if and only if k contains a primitive cube root of unity.Combining Theorems 5.1(c) and 5.9 proves Theorem 1.2 (and a little more).6. Constructions of Harbater–Katz–Gabber curves
In this section we construct some examples needed for the proofs of Theorems 1.11 and7.13. Let k be an algebraically closed field of characteristic p >
0. Let (
Y, y ) be an HKG H -curve over k . If the H -action on Y − { y } has a tamely ramified orbit, let S be that orbit;otherwise let S be any H -orbit in Y − { y } . Let S ′ = S ∪ { y } . Let m, n ∈ Z ≥ . Suppose that p ∤ n , that mn divides | S ′ | , that the divisor P s ∈ S ′ ( s − y ) is principal, and that for all s ∈ S ′ ,the divisor m ( s − y ) is principal. hoose f ∈ k ( Y ) × with divisor P s ∈ S ′ ( s − y ). Let π : X −→ Y be the cover with k ( X ) = k ( Y )( z ), where z satisfies z n = f . Let C := Aut( X/Y ), so C is cyclic of order n . Let x bethe point of X ( k ) such that π ( x ) = y . Let G := { γ ∈ Aut( X ) : γ | k ( Y ) ∈ H } . Proposition 6.1.
Let k, Y, H, S ′ , n, X, C, G be as above. (a) Every automorphism of Y preserving S ′ lifts to an automorphism of X (in n ways). (b) The sequence −→ C −→ G −→ H −→ is exact. (c) We have that ( X, x ) is an HKG G -curve.Proof. (a) Suppose that α ∈ Aut( Y ) preserves S ′ . Then div( α f /f ) = ( | S | + 1)( α y − y ), which is n times an integer multiple of the principal divisor m ( α y − y ), so α f /f = g n for some g ∈ k ( Y ) × . Extend α to an automorphism of k ( X ) by defining α z := gz ; this is well-defined since the relation z n = f is preserved. Given one lift, all others are obtained bycomposing with elements of C .(b) Only the surjectivity of G −→ H is nontrivial, and that follows from (a).(c) The quotient X/G is isomorphic to (
X/C ) / ( G/C ) =
Y /H , which is of genus 0. In thecovers X −→ X/C ≃ Y −→ X/G ≃ Y /H , all the ramification occurs above and below S ′ . The valuation of f at each point of S ′ is 1 mod n , so X −→ Y is totally ramifiedabove S ′ . Hence each ramified G -orbit in X maps bijectively to an H -orbit in Y , andeach nontrivial inertia group in G is an extension of a nontrivial inertia group of H by C . Thus, outside the totally ramified G -orbit { x } , there is at most one ramified G -orbitand it is tamely ramified. (cid:3) Example 6.2.
Let (
Y, y ) = ( P , ∞ ), with coordinate function t ∈ k ( P ). Let H ≤ PGL ( F q )be a group fixing ∞ and acting transitively on A ( F q ). (One example is H := F q ! .)Let n be a positive divisor of q + 1. Then the curve z n = t q − t equipped with the pointabove ∞ is an HKG G -curve, where G is the set of automorphisms lifting those in H . (Here S ′ = P ( F q ), m = 1, and f = t q − t ∈ k ( P ). Degree 0 divisors on P are automaticallyprincipal.) Example 6.3.
Let p = 2. Let ( Y, y ) be the j -invariant 0 elliptic curve u + u = t with itsidentity, so Y, y ) = 24 [16, Chapter 3, § H be Aut( Y, y ) or its Sylow 2-subgroup.Then k ( Y )( √ t + t ) is the function field of an HKG G -curve X , for an extension G of H by a cyclic group of order 3. (Here S ′ = Y ( F ), which is also the set of 3-torsion points on Y , and m = n = 3, and f = t + t .) Eliminating t by cubing z = t + t and substituting t = u + u leads to the equation z = ( u + u )( u + u + 1) for X . Example 6.4.
Let p = 3. Let ( Y, y ) be the j -invariant 0 elliptic curve u = t − t withits identity, so Y, y ) = 12 [16, Chapter 3, § H be a group between Aut( Y, y ) nd its Sylow 3-subgroup. Then k ( Y )( √ u ) is the function field of an HKG G -curve X , foran extension G of H by a cyclic group of order 2. (Here S ′ is the set of 2-torsion points on Y , and m = n = 2, and f = u .) Thus X has affine equation z = t − t . (This curve isisomorphic to the curve in Example 6.2 for q = 3, but | C | here is 2 instead of 4.)7. Harbater–Katz–Gabber curves with extra automorphisms
We return to assuming only that k is perfect of characteristic p . Throughout this section,( X, x ) is an HKG G -curve over k , and J is a finite group such that G ≤ J ≤ Aut( X ). Let J x be the decomposition group of x in J . Choose Sylow p -subgroups P ≤ P x ≤ P J of G ≤ J x ≤ J , respectively. In fact, P ≤ G is uniquely determined since G is cyclic mod p byLemma 4.3; similarly P x ≤ J x is uniquely determined.7.A. General results.
Proof of Theorem 1.8.
If (
X, x ) is an HKG J -curve, then J fixes x , by definition.Now suppose that J fixes x . By Lemma 4.3, J is cyclic mod p . By Proposition 4.8(i) ⇒ (ii),( X, x ) is an HKG P -curve. Identify X/P with P k so that x maps to ∞ ∈ X/P ≃ P k . Case 1: J normalizes G . Then J normalizes also the unique Sylow p -subgroup P of G . In particular, P is normal in P J . If a p -group acts on P k fixing ∞ , it must act bytranslations on A k ; applying this to the action of P J /P on X/P shows that
X/P −→ X/P J is unramified outside ∞ . Also, X −→ X/P is unramified outside x . Thus the composition X −→ X/P −→ X/P J is unramified outside x . On the other hand, X/P J is dominated by X/P , so g X/P J = 0. By Proposition 4.8(iii) ⇒ (i), ( X, x ) is an HKG J -curve. Case 2: J is arbitrary. There exists a chain of subgroups beginning at P and endingat P J , each normal in the next. Ascending the chain, applying Case 1 at each step, showsthat ( X, x ) is an HKG curve for each group in this chain, and in particular for P J . ByProposition 4.8(ii) ⇒ (i), ( X, x ) is also an HKG J -curve. (cid:3) Corollary 7.1.
We have that ( X, x ) is an HKG J x -curve and an HKG P x -curve.Proof. Apply Theorem 1.8 with J x in place of J . Then apply Proposition 4.8(i) ⇒ (ii). (cid:3) Lemma 7.2.
Among p ′ -subgroups of J x that are normal in J , there is a unique maximalone; call it C . Then C is cyclic, and central in J x .Proof. Let C be the group generated by all p ′ -subgroups of J x that are normal in J . Then C is another group of the same type, so it is the unique maximal one. By Lemma 4.3, J x iscyclic mod p , so J x /P x is cyclic. Since C is a p ′ -group, C −→ J x /P x is injective. Thus C iscyclic. The injective homomorphism C −→ J x /P x respects the conjugation action of J x oneach group. Since J x /P x is abelian, the action on J x /P x is trivial. Thus the action on C istrivial too; i.e., C is central in J x . (cid:3) .B. Low genus cases.
Define A := Aut( X, x ), so G ≤ A . By Theorem 1.8, ( X, x ) is anHKG J -curve if and only if J ≤ A . When g X ≤
1, we can describe A very explicitly. Example 7.3.
Suppose that g X = 0. Then ( X, x ) ≃ ( P k , ∞ ). Thus Aut( X ) ≃ PGL ( k ),and A is identified with the image in PGL ( k ) of the group of upper triangular matrices inGL ( k ). Example 7.4.
Suppose that g X = 1. Then ( X, x ) is an elliptic curve, and Aut( X ) ≃ X ( k ) ⋊ A . Let A := Aut( X k , x ) be the automorphism group of the elliptic curve over k . Now p divides | G | , since otherwise it follows from Example 4.7 that g X = 0. Thus G contains anorder p element, which by the HKG property has a unique fixed point. Since G ≤ A ≤ A ,the group A also contains such an element. By the computation of A (in [16, Chapter 3],for instance), p is 2 or 3, and X is supersingular, so X has j -invariant 0. Explicitly: • If p = 2, then A ≃ SL ( F ) ≃ Q ⋊ Z / Z (order 24), and G is Z / Z , Z / Z , Q , orSL ( F ). • If p = 3, then A ≃ Z / Z ⋊ Z / Z (order 12), and G is Z / Z , Z / Z , or Z / Z ⋊ Z / Z .Because of Corollary 7.1, the statement about G is valid also for J x .7.C. Cases in which p divides | G | . If p divides | G | , then we can strengthen Theorem 1.8:see Theorem 7.6 and Corollary 7.7 below. Lemma 7.5. If p divides | G | and G is normal in J , then J fixes x .Proof. Ramification outside x is tame, so if p divides | G | , then x is the unique point fixedby G . If, in addition, J normalizes G , then J must fix this point. (cid:3) Theorem 7.6. If p divides | G | , then the following are equivalent: (i) ( X, x ) is an HKG J -curve. (ii) J fixes x . (iii) J is cyclic mod p .Proof. (i) ⇔ (ii): This is Theorem 1.8.(ii) ⇒ (iii): This is Lemma 4.3.(iii) ⇒ (i): By Proposition 4.8(i) ⇒ (ii), ( X, x ) is an HKG P -curve. Again choose a chainof subgroups beginning at P and ending at P J , each normal in the next. Since J is cyclicmod p , we may append J to the end of this chain. Applying Lemma 7.5 and Theorem 1.8to each step of this chain shows that for each group K in this chain, K fixes x and ( X, x ) isan HKG K -curve. (cid:3) Corollary 7.7. If p divides | G | , then (a) P x = P J . b) The prime p does not divide the index ( J : J x ) . (c) If j ∈ J x , then j P x = P x . (d) If j / ∈ J x , then j P x ∩ P x = 1 . (e) If J contains a nontrivial normal p -subgroup A , then ( X, x ) is an HKG J -curve.Proof. (a) Since p divides | P x | and P J is cyclic mod p , Corollary 7.1 and Theorem 7.6(iii) ⇒ (ii)imply that P J fixes x . Thus P J ≤ P x , so P x = P J .(b) The exponent of p in each of | J x | , | P x | , | P J | , | J | is the same.(c) By Lemma 4.3, J x is cyclic mod p , so P x is normal in J x .(d) A nontrivial element of P x ∩ j P x would be an element of p -power order fixing both x and jx , contradicting the definition of HKG J x -curve.(e) The group A is contained in every Sylow p -subgroup of J ; in particular, A ≤ P J = P x .This contradicts (d) unless J x = J . By Theorem 7.6(ii) ⇒ (i), ( X, x ) is an HKG J -curve. (cid:3) Lemma 7.8.
Suppose that g X > . Let A ≤ J be an elementary abelian ℓ -subgroup for someprime ℓ . Suppose that P x normalizes A . Then A ≤ J x .Proof. It follows from Example 4.7 that p divides | G | . If ℓ = p , then P x A is a p -subgroup of J , but P x is a Sylow p -subgroup of J by Corollary 7.7(a), so A ≤ P x ≤ J x .Now suppose that ℓ = p . The conjugation action of P x on A leaves the group A x = J x ∩ A invariant. By Maschke’s theorem, A = A x × C for some other subgroup C normalized by P x . Then C x = 1. By Corollary 7.1, ( X, x ) is an HKG P x -curve. Since P x normalizes C , thequotient X/C equipped with the image y of x and the induced P x -action is another HKG P x -curve. Since C x = 1, we have d x ( P x ) = d y ( P x ); thus Proposition 4.8(i) ⇒ (iv) implies that g X = g X/C . Since g X >
1, this implies that C = 1. So A = A x ≤ J x . (cid:3) Unmixed actions.
Proof of Theorem 1.10.
By the base change property mentioned after Remark 4.5, we mayassume that k is algebraically closed. By Corollary 7.1, we may enlarge G to assume that G = J x .First suppose that the action of G has a nontrivially and tamely ramified orbit, say Gy ,where y ∈ X ( k ). The Hurwitz formula applied to ( X, G ) gives2 g X − − | G | + d x ( G ) + | G/G y | ( | G y | − . (7.9)Since the action of J is unmixed, J x and
J y are disjoint. The Hurwitz formula for (
X, J )therefore gives 2 g X − ≥ − | J | + | J/G | d x ( G ) + | J/J y | ( | J y | − . (7.10) alculating | J/G | times the equation (7.9) minus the inequality (7.10) yields( | J/G | − g X − ≤ | J/J y | − | J/G y | ≤ , because G y ≤ J y . Since g X >
1, this forces J = G .If a nontrivially and tamely ramified orbit does not exist, we repeat the proof whileomitting the terms involving y . (cid:3) Mixed actions.
Here is an example, mentioned to us by Rachel Pries, that shows thatTheorem 1.10 need not hold if the action of J is mixed. Example 7.11.
Let n be a power of p ; assume that n >
2. Let k = F n . Let X be thecurve over k constructed by Giulietti and Korchm´aros in [11]; it is denoted C in [13]. Let J = Aut( X ). Let G be a Sylow p -subgroup of J ; by [11, Theorem 7], | G | = n . Then X isan HKG G -curve by [13, Lemma 2.5 and proof of Proposition 3.12], and g X > σ in Definition 1.9 to be the automorphism denoted ˜ W on [11, p. 238] shows thatthe action of J on X is mixed. In fact, [11, Theorem 7] shows that J fixes no k -point of X ,so the conclusion of Theorem 1.10 does not hold.7.F. Solvable groups.
Here we prove Theorem 1.11. If p does not divide | G | , then Exam-ple 4.7 shows that X ≃ P k , so the conclusion of Theorem 1.11 holds. For the remainder ofthis section, we assume that p divides | G | . In this case we prove Theorem 1.11 in the strongerform of Theorem 7.13, which assumes a hypothesis weaker than solvability of J . We retainthe notation set at the beginning of Section 7, and let C denote the maximal p ′ -subgroup of J x that is normal in J , as in Lemma 7.2. Lemma 7.12.
Suppose that g X > and that ( X, x ) is not an HKG J -curve. If J containsa nontrivial normal abelian subgroup, then C = 1 .Proof. The last hypothesis implies that J contains a nontrivial normal elementary abelian ℓ -subgroup A for some prime ℓ . By Corollary 7.7(e), ℓ = p . By Lemma 7.8, A ≤ J x . Thus1 = A ≤ C . (cid:3) Theorem 7.13.
Suppose that p divides | G | and ( X, x ) is not an HKG J -curve. (a) Suppose that g X = 0 , so Aut( X ) ≃ Aut( P k ) ≃ PGL ( k ) . Then J is conjugate in PGL ( k ) to precisely one of the following groups: • PSL ( F q ) or PGL ( F q ) for some finite subfield F q ≤ k (these groups are the same if p = 2 ); note that PSL ( F q ) is simple when q > . • If p = 2 and m is an odd integer at least such that a primitive m th root of unity ζ ∈ k satisfies ζ + ζ − ∈ k , the dihedral group of order m generated by ζ ζ − ! and ! if ζ ∈ k , and generated by ζ + ζ − + 1 11 1 ! and ! if ζ / ∈ k .(The case m = 3 is listed already, as PSL ( F ) .) If p = 3 and F ≤ k , a particular copy of the alternating group A in PSL ( F ) (allsuch copies are conjugate in PGL ( F ) ); the group A is simple.Suppose, in addition, that J contains a nontrivial normal abelian subgroup; then p ∈{ , } and | P J | = p , and if J is conjugate to PSL ( F q ) or PGL ( F q ) , then q = p . (b) Suppose that g X = 1 . Then p is or , and the limited possibilities for X and J x aredescribed in Example 7.4. The group J is a semidirect product of J x with a finite abeliansubgroup T ≤ X ( k ) . (c) Suppose that g X > . Let C ≤ J be as in Lemma 7.2. Let Y = X/C , let y be the imageof x under X −→ Y , and let U = Stab J/C ( y ) . If J/C contains a nontrivial normalabelian subgroup (automatic if J is solvable), then one of the following holds: i. p = 3 , g X = 3 , g Y = 0 , C ≃ Z / Z , P x ≃ Z / Z , ( J : J x ) = 4 , and ( X, x ) isisomorphic over k to the curve z = t u − tu in P equipped with ( t : u : z ) =(1 : 0 : 0) , which is the curve in Example 6.2 with q = 3 . Moreover, PSL ( F ) ≤ J/C ≤ PGL ( F ) . ii. p = 2 , g X = 10 , g Y = 1 , C ≃ Z / Z , P x ≃ Q , ( J : J x ) = 9 , and ( X, x ) is isomorphicover k to the curve in Example 6.3. The homomorphism J −→ J/C sends thesubgroups J x ⊃ P x to subgroups J x /C ⊃ P x C/C of U . Also, P x C/C ≃ P x ≃ Q and U ≃ SL ( Z / Z ) , and U acts faithfully on the -torsion subgroup Y [3] ≃ ( Z / Z ) ofthe elliptic curve ( Y, y ) . The group J/C satisfies Y [3] ⋊ Q ≃ ( Z / Z ) ⋊ Q ≤ J/C ≤ ( Z / Z ) ⋊ SL ( Z / Z ) ≃ Y [3] ⋊ U. iii. p = 3 , g X = 3 , g Y = 1 , C ≃ Z / Z , P x ≃ Z / Z , ( J : J x ) = 4 , and ( X, x ) isisomorphic over k to the curve z = t u − tu in P equipped with ( t : u : z ) =(1 : 0 : 0) as in Example 6.4. The homomorphism J −→ J/C sends the subgroups J x ⊃ P x to subgroups J x /C ⊃ P x C/C of U . Also P x C/C ≃ P x ≃ Z / Z and U ≃ Z / Z ⋊ Z / Z , and U/Z ( U ) acts faithfully on the group Y [2] ≃ ( Z / Z ) . Thegroup J/C satisfies Y [2] ⋊ Z / Z = ( Z / Z ) ⋊ Z / Z ≤ J/C ≤ ( Z / Z ) ⋊ ( Z / Z ⋊ Z / Z ) = Y [2] ⋊ U. In each of i., ii., and iii., if ( X, x ) is the curve over k specified, from Examples 6.2–6.4, then any group satisfying the displayed upper and lower bounds for J/C is actuallyrealized as
J/C for some subgroup J ≤ Aut( X ) satisfying all the hypotheses.Proof. (a) The groups listed in the statement of (a) are pairwise non-isomorphic, hence notconjugate. Thus it remains to prove that J is conjugate to one of them. By Corollary 7.7(e), J has no normal Sylow p -subgroup. We will show that every finite subgroup J ≤ PGL ( k )with no normal Sylow p -subgroup is conjugate to a group listed in (a). This would follow mmediately from [9, Theorem B], but [9] has not yet been published, so we now give a proofnot relying on it. We will use the exact sequence1 −→ PSL ( k ) −→ PGL ( k ) det −→ k × /k × −→ . Case 1: k is finite and J ≤ PSL ( k ) . For finite k , the subgroups of PSL ( k ) up toconjugacy were calculated by Dickson [8, § § § p -subgroup are among those listed in (a). (Dickson sometimeslists two PSL ( k )-conjugacy classes of subgroups of certain types, but his proof shows thatthey map to a single PGL ( k )-conjugacy class.) Case 2: k is infinite and J ≤ PSL ( k ) . Let e J be the inverse image of J under the finiteextension SL ( k ) ։ PSL ( k ). So e J is finite. The representation of e J on k is absolutelyirreducible, since otherwise e J would inject into the group ∗ ∗ ∗ ! of 2 × k , and e J would have a normal Sylow p -subgroup e J ∩ ∗ ! , and J would have one too, contrary to assumption. By [10, Theorem 19.3], this representation isdefinable over the field k generated by the traces of the elements of e J . Each trace is a sum ofroots of unity, so k is finite. Thus J is conjugate in PGL ( k ) to a subgroup J ≤ PGL ( k ).Conjugation does not change the determinant, so J ≤ PSL ( k ). By Case 1, J is conjugateto a group in our list, so J is too. Case 3: k is finite or infinite, and J ≤ PGL ( k ) , but J (cid:2) PSL ( k ) . If p = 2, then, since k is perfect, k × = k × , so PGL ( k ) = PSL ( k ). Thus p >
2. Let J ′ := J ∩ PSL ( k ). Then J/J ′ injects into k × /k × , so p ∤ ( J : J ′ ). The Sylow p -subgroups of J ′ are the same as those of J ,so J ′ has exactly one if and only if J has exactly one; i.e., J ′ has a normal Sylow p -subgroupif and only if J has one. Since J does not have one, neither does J ′ . By Case 1, we mayassume that J ′ appears in our list.The group J is contained in the normalizer N PGL ( k ) ( J ′ ). We now break into cases ac-cording to J ′ . If J ′ is PSL ( F q ) or PGL ( F q ) for some subfield F q ≤ k , then N PGL ( k ) ( J ′ ) =PGL ( F q ) by [8, § k is infinite), so J = PGL ( F q ), whichis in our list. Recall that p >
2, so J ′ is not dihedral. Thus the only remaining possibilityis that J ′ ≃ A ≤ PSL ( F ) ≤ PGL ( k ). Let { , a } be a subgroup of order 2 in the imageof J in k × /k × and let J ′′ be its inverse image in J . Then J ′′ < PSL ( k ( √ a )), so J ′′ shouldappear in our list, but | J ′′ | = 120 and there is no group of order 120 there for p = 3.(b) In the notation of Example 7.4, let ψ : J −→ A be the projection. Let T := ker ψ ≤ X ( k ). Since X is supersingular, T is a p ′ -group. Let J := ψ ( J ) ≤ A . Since G ≤ J ≤ A ,the group J is in the list of possibilities in Example 7.4 for G given p . Checking each caseshows that its Sylow p -subgroup P J := ψ ( P J ) is normal in J . The action of Aut( X ) on X ( k )restricts to the conjugation action of J on the abelian group T , which factors through J , so H ( P J , T ) = T P J = T P J = 0, since P J has a unique fixed point on X . Also, H i ( P J , T ) = 0 or all i ≥
1, since | P J | and | T | are coprime. Thus, by the Lyndon–Hochschild–Serre spectralsequence applied to P J ⊳ J , we have H i ( J , T ) = 0 for all i ≥
1. Therefore the short exactsequence 0 −→ T −→ J −→ J −→ K be theimage of a splitting J −→ J . Then K contains a Sylow p -subgroup of J . Equivalently, someconjugate K ′ of K contains P J . Since K ′ ≃ J and P J is normal in J , the group P J is normalin K ′ . Since x is the unique fixed point of P J , this implies that K ′ fixes x ; i.e., K ′ ≤ J x . Onthe other hand, | K ′ | = | J | ≥ | J x | since J x ∩ T = { e } . Hence K ′ = J x and J = T ⋊ J x .(c) We may assume that k is algebraically closed. By Theorem 1.8, ( X, x ) is an HKG J x -curve. Then ( Y, y ) is an HKG J x /C -curve, but not an HKG J/C -curve since
J/C does notfix y . If g Y >
1, then Lemma 7.12 applied to Y yields a nontrivial p ′ -subgroup C ≤ J x /C that is normal in J/C , and the inverse image of C in J is a p ′ -subgroup C ≤ J x normal in J with C (cid:13) C , contradicting the maximality of C . Thus g Y ≤
1. Since g X >
1, we have C = 1. Let n = | C | . Let ζ be a primitive n th root of unity in k . Let c be a generator of C .By Lemma 7.2, C is central in J x , so P x C is a direct product. By Corollary 7.1, X isan HKG P x -curve. Thus X/P x ≃ P , and the P x -action on X is totally ramified at x andunramified elsewhere. The action of C on X/P x fixes the image of x , so by Example 4.7, thecurves in the covering X/P x −→ X/P x C have function fields k ( z ) ⊇ k ( f ), where z n = f and c z = ζ z . Powers of z form a k ( X/P x C )-basis of eigenvectors for the action of c on k ( X/P x ).We may assume that the (totally ramified) image of x in X/P x is the point z = ∞ . Weobtain a diagram of curves X C w w ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ P x totally ramified above z = ∞ , unramified elsewhere ( ( PPPPPPPPPPPPPP Y ≃ X/C P x ' ' ❖❖❖❖❖❖❖❖❖❖❖❖ X/P x ≃ P zC totally ramified above f = ∞ , f = 0 w w ♦♦♦♦♦♦♦♦♦♦♦♦ X/P x C ≃ P f where the subscript on each P indicates the generator of its function field, and the grouplabeling each morphism is the Galois group. The field k ( X ) is the compositum of its subfields k ( Y ) and k ( X/P x ).Let S be the preimage of the point f = 0 under Y −→ X/P x C , and let S ′ := S ∪ { y } .Comparing the p -power and prime-to- p ramification on both sides of the diagram showsthat the point f = ∞ totally ramifies in X −→ Y −→ X/P x C , while the point f = 0 splitscompletely into a set S of | P x | points of Y , each of which is totally ramified in X −→ Y . Thusthe extension k ( X ) ⊇ k ( Y ) is Kummer and generated by the same z as above, and powersof z form a k ( Y )-basis of eigenvectors for the action of c on k ( X ). This extension is totallyramified above S ′ and unramified elsewhere. The divisor of f on Y is S − | S | y = S ′ − | S ′ | y ,where S here denotes the divisor P s ∈ S s , and so on. et j ∈ J . Since C ⊳ J , the element j acts on Y and preserves the branch locus S ′ of X −→ Y . Since X −→ Y is totally ramified above S ′ , the automorphism j fixes x if andonly if it fixes y . Since P x acts transitively on S , and J does not fix x or y , the set S ′ is the J -orbit of y . Thus ( J : J x ) = | J x | = | J y | = | S ′ | = | P x | + 1 . Suppose that j ∈ J − J x , so j y = y . Then the divisor of j f /f on Y is (cid:0) S ′ − | S ′ | j y (cid:1) − ( S ′ − | S ′ | y ) = | S ′ | ( y − j y ) , which is nonzero. Since C is cyclic and normal, j − cj = c r for some r , and hence c ( j z/z ) = jc r z/ c z = ζ r − ( j z/z ). Thus j z/z is a ζ r − -eigenvector, so j z/z = z r − g for some g ∈ k ( Y ) × .Taking n th powers yields j f /f = f r − g n . The corresponding equation on divisors is | S ′ | ( y − j y ) = ( r − S ′ − | S ′ | y ) + n div( g ) . (7.14)Considering the coefficient of a point of S ′ − { y, j y } shows that r − ≡ n ). Then,considering the coefficient of y shows that n divides | S ′ | , and dividing equation (7.14) throughby n shows that ( | S ′ | /n )( y − j y ) is div( f ( r − /n g ), a principal divisor. If, moreover, g Y > Y cannot be a principal divisor, so n = | S ′ | . Case 1: g Y = 0 . Applying (a) to Y shows that p ∈ { , } and any Sylow p -subgroup of J/C has order p . Since C is a p ′ -group, | P J | = p too. By Corollary 7.7(a), P x = P J , so | P x | = p , and n divides | S ′ | = p + 1. Thus ( p, n ) is (2 , , , X −→ Y yields2 g X − n (2 · −
2) + X s ∈ S ′ ( n −
1) = − n + ( p + 1)( n − . Only the case ( p, n ) = (3 ,
4) yields g X >
1. By (a), we may choose an isomorphism Y ≃ P t mapping y to ∞ such that the J/C -action on Y becomes the standard action of PSL ( F )or PGL ( F ) on P t . Then S ′ = J y = P ( F ). Then f has divisor S ′ − y = A ( F ) − · ∞ on P , so f = t − t up to an irrelevant scalar. Since k ( X ) = k ( Y )( n √ f ), the curve X hasaffine equation z = t − t . This is the same as the q = 3 case of Example 6.2. Case 2: g Y = 1 . Applying (b) (i.e., Example 7.4) to Y shows that either p is 2 and | P x | divides 8, or p = 3 and | P x | = 3; also, Y has j -invariant 0. Also, n divides | S ′ | = | P x | + 1,but n is not 1 or | S ′ | . Thus ( p, n, | P x | , | S ′ | ) is (2 , , ,
9) or (3 , , , g X = 10 or g X = 3, respectively. Let m = | S ′ | /n . Since m ( y − j y ) is principalfor all j ∈ J , if y is chosen as the identity of the elliptic curve, then the J -orbit S ′ of y is contained in the group Y [ m ] of m -torsion points. But in both cases, these sets have thesame size | S ′ | = m . Thus S ′ = Y [ m ].If p = 2, the j -invariant 0 curve Y has equation u + u = t , and Y [3] − { y } is the setof points with t ∈ F , so f = t + t up to an irrelevant scalar, and k ( X ) = k ( Y )( √ t + t ).Thus X is the curve of Example 6.3. f p = 3, the j -invariant 0 curve Y has equation u = t − t , and Y [2] − { y } is the set ofpoints with u = 0, so f = u up to an irrelevant scalar, and k ( X ) = k ( Y )( √ u ) = k ( t )( √ t − t ).Thus X is the curve of Example 6.4.Finally, Proposition 6.1 implies that in each of i., ii., and iii., any group satisfying thedisplayed upper and lower bounds, viewed as a subgroup of Aut( Y ), can be lifted to asuitable group J of Aut( X ). (cid:3) Remark . Suppose that (
X, x ) is not an HKG J -curve, g X >
1, and P J is not cyclic orgeneralized quaternion. Then [13, Theorem 3.16] shows that J/C is almost simple with soclefrom a certain list of finite simple groups.
Acknowledgement
We thank the referee for suggestions on the exposition.
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E-mail address : [email protected] Ted Chinburg, Department of Mathematics, University of Pennsylvania, Philadelphia, PA19104, U.S.A.
E-mail address : [email protected] Bjorn Poonen, Department of Mathematics, Massachusetts Institute of Technology,Cambridge, MA 02139, U.S.A.
E-mail address : [email protected] Peter Symonds, School of Mathematics, University of Manchester, Oxford Road, Manch-ester M13 9PL, Manchester M13 9PL United Kingdom