Some 4-point Hurwitz numbers in positive characteristic
aa r X i v : . [ m a t h . AG ] J un SOME -POINT HURWITZ NUMBERS IN POSITIVECHARACTERISTIC IRENE I. BOUW AND BRIAN OSSERMAN
Abstract.
In this paper, we compute the number of covers of curves withgiven branch behavior in characteristic p for one class of examples with fourbranch points and degree p . Our techniques involve related computations inthe case of three branch points, and allow us to conclude in many cases thatfor a particular choice of degeneration, all the covers we consider degenerateto separable (admissible) covers. Starting from a good understanding of thecomplex case, the proof is centered on the theory of stable reduction of Galoiscovers. Introduction
This paper considers the question of determining the number of covers betweengenus-0 curves with fixed ramification in positive characteristic. More concretely,we consider covers f : P → P branched at r ordered points Q , . . . , Q r of fixed ramification type ( d ; C , . . . , C r ), where d is the degree of f and C i = e ( i )- · · · - e s i ( i )is a conjugacy class in S d . This notation indicates that there are s i ramificationpoints in the fiber f − ( Q i ), with ramification indices e j ( i ). The Hurwitz number h ( d ; C , . . . , C r ) is the number of covers of fixed ramification type over C , up toisomorphism. This number does not depend on the position of the branch points.If p is a prime not dividing any of the ramification indices e j ( i ), the p -Hurwitznumber h p ( d ; C , . . . , C r ) is the number of covers of fixed ramification type whosebranch points are generic over an algebraically closed field k of characteristic p . Thegenericity hypothesis is necessary because in positive characteristic the number ofcovers often depends on the position of the branch points.The only general result on p -Hurwitz numbers is that they are always less thanor equal to the Hurwitz number, with equality when the degree of the Galois closureis prime to p . This is because every tame cover in characteristic p can be lifted tocharacteristic 0, and in the prime-to- p case, every cover in characteristic 0 specializesto a cover in characteristic p with the same ramification type (see Corollaire 2.12 ofExpos´e XIII in [9]). We say a cover has good reduction when such a specializationexists. However, in the general case, some covers in characteristic 0 specialize toinseparable covers in characteristic p ; these covers are said to have bad reduction .Thus, the difference h ( d ; C , . . . , C r ) − h p ( d ; C , . . . , C r ) is the number of covers incharacteristic 0 with generic branch points and bad reduction. In [14] and [15], thevalue h p ( d ; e , e , e ) is computed for genus 0 covers and any e i prime to p usinglinear series techniques. In this paper, we treat the considerably more difficult caseof genus-0 covers of type ( p ; e , e , e , e ). Our main result is the following. Theorem 1.1.
Given e , . . . , e all less than p , with P i e i = 2 p + 2 , we have h p ( p ; e , e , e , e ) = h ( p ; e , e , e , e ) − p. An important auxiliary result is the computation of the p -Hurwitz number h p ( p ; e - e , e , e ). Theorem 1.2.
Given odd integers e , e , e , e < p , with e + e ≤ p and P i e i =2 p + 2 , we have that h p ( p ; e - e , e , e ) = ( h ( p ; e - e , e , e ) − ( p + 1 − e − e ) : e = e ,h ( p ; e - e , e , e ) − ( p + 1 − e − e ) / e = e . Corollary 6.5 gives a more general result including the case that some of the e i areeven, but in some cases we also compute the p -Hurwitz number only up to a factor2. Note that there is an explicit formula for h ( p ; e , e , e , e ) and h ( p ; e - e , e , e );see Theorem 2.1 and Lemma 2.2 below.Our technique involves the use of “admissible covers,” which are certain coversbetween degenerate curves (see Section 2). Admissible covers provide a compacti-fication of the space of covers of smooth curves in characteristic 0, but in positivecharacteristic this is not the case, and it is an interesting question when, undera given degeneration of the base, a cover of smooth curves does in fact have anadmissible cover as a limit. In this case we say the smooth cover has good degener-ation . In [2] one finds examples of covers with generic branch points without gooddegeneration.In contrast, our technique for proving Theorem 1.1 simultaneously shows thatmany of the examples we consider have good degeneration. Theorem 1.3.
Given odd integers < e ≤ e ≤ e ≤ e < p with P i e i = 2 p + 2 ,every cover of type ( p ; e , e , e , e ) with generic branch points (0 , , λ, ∞ ) has gooddegeneration under the degeneration sending λ to ∞ . As with Theorem 1.2, our methods do not give a complete answer in some caseswith even e i , but we do prove a more general result in Theorem 8.2.Building on the work of Raynaud [16], Wewers uses the theory of stable reduc-tion in [20] to give formulas for the number of covers with three branch points andhaving Galois closure of degree strictly divisible by p which have bad reduction tocharacteristic p . In [5], some p -Hurwitz numbers are calculated using the existenceportion of Wewers’ theorems, but these are in cases which are rigid (meaning theclassical Hurwitz number is 1) or very close to rigid, so one does not have to carryout calculations with Wewers’ formulas. In [17], Selander uses the full statement ofWewers’ formulas to compute some examples in small degree. Our result in Theo-rem 1.2 is the first explicit calculation of an infinite family of p -Hurwitz numberswhich fully uses Wewers’ formulas, and its proof occupies the bulk of the presentpaper.We begin in Sections 2 and 3 by reviewing the situation in characteristic 0 andsome group-theoretic background. We then recall the theory of stable reduction inSection 4. In order to apply Wewers’ formulas, in Section 5 we analyze the possi-ble structures of the stable reductions which arise, and then in Section 6 we applyWewers’ formulas to compute the number of smooth covers with a given stablereduction. Here we are forced to use a trick comparing the number of covers inthe case of interest to the number in a related case where we know all covers havebad reduction. In Section 7 we then apply Corollary 6.5 as well as the formulasfor h p ( d ; e , e , e ) of [14] and the classical Hurwitz number calculations in [12] to OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 3 estimate the number of admissible covers in characteristic p . This provides a suffi-cient lower bound on h p ( p ; e , e , e , e ). Finally, we use the techniques of [3], againbased on stable reduction, to directly prove in Section 8 that h p ( p ; e , e , e , e ) isbounded above by h ( p ; e , e , e , e ) − p . We thus conclude Theorems 1.1 and 1.3.We would like to thank Peter M¨uller, Bj¨orn Selander and Robert Guralnick forhelpful discussions. 2. The characteristic- situation In this paper, we consider covers f : P → P branched at r ordered points Q , . . . , Q r of fixed ramification type ( d ; C , . . . , C r ), where d is the degree of f and C i = e ( i )- · · · - e s i ( i ) is a conjugacy class in S d . This means that there are s i ramification points in the fiber f − ( Q i ), with ramification indices e j ( i ). The Hurwitz number h ( d ; C , . . . , C r ) is the number of covers of fixed ramification typeover C , up to isomorphism. This number does not depend on the position of thebranch points.Riemann’s Existence Theorem implies that the Hurwitz number h ( d ; C , . . . , C r )is the cardinality of the set of Hurwitz factorizations defined as { ( g , · · · , g r ) ∈ C × · · · × C r | h g i i ⊂ S d transitive , Y i g i = 1 } / ∼ , where ∼ denotes uniform conjugacy by S d .The group h g i i is called the monodromy group of the corresponding cover. Fora fixed monodromy group G , a variant equivalence relation is given by G -Galoiscovers , where we work with Galois covers together with a fixed isomorphism of theGalois group to G . The group-theoretic interpretation is then that the g i are in G (with the action on a fiber recovered by considering G as a subgroup of S | G | ),and the equivalence relation ∼ G is uniform conjugacy by G . To contrast with the G -Galois case, we sometimes emphasize that we are working up to S d -conjugacy byreferring to the corresponding covers as mere covers .In this paper, we are mainly interested in the pure-cycle case, where every C i isthe conjugacy class in S d of a single cycle. In this case, we write C i = e i , where e i is the length of the cycle. A cover f : Y → P over C of ramification type( d ; e , e , · · · , e r ) has genus g ( Y ) = 0 if and only if P ri =1 e i = 2 d − r .Giving closed formulae for Hurwitz numbers may get very complicated, even incharacteristic zero. The following result from [12] illustrates that the genus-0 pure-cycle case is more tractable than the general case, as one may give closed formulaefor the Hurwitz numbers, at least if the number r of branch points is at most 4. Theorem 2.1.
Under the hypothesis P ri =1 e i = 2 d − r , we have the following. (a) h ( d ; e , e , e ) = 1 . (b) h ( d ; e , e , e , e ) = min i ( e i ( d + 1 − e i )) . (c) Let f : P C → P C be a cover of ramification type ( d ; e , e , . . . , e r ) with r ≥ . The Galois group of the Galois closure of f is either S d or A d unless ( d ; e , e , . . . , e r ) = (6; 4 , , in which case the Galois group is S acting transitively on letters. These statements are Lemma 2.1, Theorem 4.2, and Theorem 5.3 of [12]. Wemention that Boccara ([1]) proves a partial generalization of Theorem 2.1.(a). Hegives a necessary and sufficient condition for h ( d ; C , C , ℓ ) to be nonzero in the IRENE I. BOUW AND BRIAN OSSERMAN case that C , C are arbitrary conjugacy classes of S d and only C = ℓ is assumedto be the conjugacy class of a single cycle.Later in our analysis we will be required to study covers of type ( d ; e - e , e , e ),so we mention a result which is not stated explicitly in [12], but which follows easilyfrom the arguments therein. We will only use the case that e = d , but we statethe result in general since the argument is the same. Lemma 2.2.
Given e , e , e , e and d with d + 2 = P i e i and e + e ≤ d , if e = e we have h ( d ; e - e , e , e ) = ( d + 1 − e − e ) min( e , e , d + 1 − e , d + 1 − e ) , and if e = e we have h ( d ; e - e , e , e ) = ⌈
12 ( d + 1 − e − e ) min( d + 1 − e , d + 1 − e ) ⌉ . Note that this number is always positive. In particular, when e = d we have h ( d ; e - e , e , d ) = d + 1 − e − e if e = e , ( d + 2 − e − e ) / if e = e , d even , ( d + 1 − e − e ) / if e = e , d odd . Proof.
Without loss of generality, we may assume that e ≤ e and e ≤ e . Thus,we want to prove that h ( d ; e - e , e , e ) is given by the smaller of e ( d + 1 − e − e )and ( d + 1 − e )( d + 1 − e − e ) when e = e , by (( d + 1 − e )( d + 1 − e − e ) + 1) / e = e and all of d, e , e are even, and by ( d + 1 − e )( d + 1 − e − e ) / e ≤ e , this formula still follows fromthe argument of Theorem 4.2.(ii) of [12]. The first observations to make are thatsince e + e ≤ d , we have e + e ≥ d + 2, and it follows that although we may nothave e ≤ e , we have e < e . Moreover, we have e + e ≤ d +1 and e + e ≥ d +1.We are then able to check that the Hurwitz factorizations ( σ , σ , σ , σ ) describedin case (ii) of loc. cit. still give valid Hurwitz factorizations ( g , g , g ) by setting g = σ σ , just as in Proposition 4.7 of loc. cit. Moreover, just as in Proposition 4.9of loc. cit. we find that every Hurwitz factorization must be one of the enumeratedones.It remains to consider when two of the described possibilities yield equivalentHurwitz factorizations. If e = e , we can extract σ and σ as the disjoint cycles(of distinct orders) in g , so we easily see that the proof of Proposition 4.8 of loc.cit. is still valid. Thus the Hurwitz number is simply the number of possibilitiesenumerated in Theorem 4.2 (ii) of [12], which is the minimum of e ( d + 1 − e − e )and ( d + 1 − e )( d + 1 − e − e ), as desired.Now suppose e = e . We then check easily that e + e ≥ d + 1, so that thenumber of enumerated possibilities is ( d + 1 − e )( d + 1 − e − e ). Here, we see thatwe potentially have a given Hurwitz factorization ( g , g , g ) being simultaneouslyequivalent to two of the enumerated possibilities, since σ and σ can be switched.Indeed, the argument of Proposition 4.8 of loc. cit. describing how to intrinsicallyrecover the parameters m, k of Theorem 4.2 (ii) of loc. cit. lets us compute how m, k change under switching σ and σ , and we find that the pair ( m, k ) is sent to( e + 2 e − d − m, e + e − d − k ). We thus find that each Hurwitz factorization isequivalent to two distinct enumerated possibilities, with the exception that if d and e (and therefore necessarily e ) are even, the Hurwitz factorization corresponding OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 5 to ( m, k ) = (( e + 2 e − d ) / , ( e + e − d ) /
2) is not equivalent to any other. Wetherefore conclude the desired statement. (cid:3)
We now explain how Theorem 4.2 of [12] can be understood in terms of degener-ations. Harris and Mumford [10] developed the theory of admissible covers , givinga description of the behavior of branched covers under degeneration. Admissiblecovers in the case we are interested in may be described geometrically as follows:let X be the reducible curve consisting of two smooth rational components X and X joined at a single node Q . We suppose we have points Q , Q on X , and Q , Q on X . An admissible cover of type ( d ; C , C , ∗ , C , C ) is then a connected, finiteflat cover f : Y → X which is ´etale away from the preimage of Q and the Q i ,and if we denote by Y → X and Y → X the (possibly disconnected) coversof X and X induced by f , we require also that Y → X has ramification type( d ; C , C , C ) for Q , Q , Q and Y → X has ramification type ( d ; C, C , C ) for Q, Q , Q , for some conjugacy class C in S d , and furthermore that for P ∈ f − ( Q ),the ramification index of f at P is the same on Y and Y . In characteristic p ,we further have to require that ramification above the node is tame. We refer to Y → X and Y → X as the component covers determining f . When we wishto specify the class C , we say the admissible cover is of type ( d, C , C , ∗ C , C , C ).The two basic theorems on admissible covers concern degeneration and smooth-ing. First, in characteristic 0, or when the monodromy group has order primeto p , if a family of smooth covers of type ( d ; C , C , C , C ) with branch points( Q , Q , Q , Q ) is degenerated by sending Q to Q , the limit is an admissiblecover of type ( d ; C , C , ∗ , C , C ). On the other hand, given an admissible cover oftype ( d ; C , C , ∗ , C , C ), irrespective of characteristic there is a deformation to acover of smooth curves, which then has type ( d ; C , C , C , C ). Such a deformationis not unique in general; we call the number of smooth covers arising as smoothingsof a given admissible cover (for a fixed smoothing of the base) the multiplicity ofthe admissible cover.Suppose we have a family of covers f : X → Y , with smooth generic fiber f : X → Y , and admissible special fiber f : X → Y . If we choose local mon-odromy generators for π tame1 ( Y r { Q , Q , Q , Q } ) which are compatible with thedegeneration to Y , we then find that if we have a branched cover of Y correspond-ing to a Hurwitz factorization ( g , g , g , g ), the induced admissible cover of Y will have monodromy given by ( g , g , ρ ) over Y and ( ρ − , g , g ) over Y , where ρ = g g . The multiplicity of the admissible cover arises because it may be possibleto apply different simultaneous conjugations to ( g , g , ρ ) and to ( ρ − , g , g ) whilemaintaining the relationship between ρ and ρ − . It is well-known that when ρ is apure-cycle of order m , the admissible cover has multiplicity m , although we recoverthis fact independently in our situation as part of the Hurwitz number calculationof [12].To calculate more generally the multiplicity of an admissible cover of the abovetype, we define the action of the braid operator Q on the set of Hurwitz factoriza-tions as Q · ( g , g , g , g ) = ( g g g g − g − , g g g − , g , g ) . One easily checks that Q · ¯ g is again a Hurwitz factorization of the same ramificationtype as ¯ g . The multiplicity of a given admissible cover is the length of the orbit of Q acting on the corresponding Hurwitz factorization. IRENE I. BOUW AND BRIAN OSSERMAN
In this context, we can give the following sharper statement of Theorem 2.1 (b),phrased in somewhat different language in [12].
Theorem 2.3.
Given a genus- ramification type ( d ; e , e , e , e ) , with e ≤ e ≤ e ≤ e the only possibilities for an admissible cover of type ( d ; e , e , ∗ , e , e ) aretype ( d ; e , e , ∗ m , e , e ) or type ( d ; e , e , ∗ e - e , e , e ) . (a) Fix m ≥ . There is at most one admissible cover of type ( d ; e , e , ∗ m , e , e ) ,and if such a cover exists, it has multiplicity m . (i) Suppose that d + 1 ≤ e + e . There exists an admissible cover of type ( d ; e , e , ∗ m , e , e ) if and only if e − e + 1 ≤ m ≤ d + 1 − e − e , m ≡ e − e + 1 (mod 2) . (ii) Suppose that d + 1 ≥ e + e . There exists an admissible cover of type ( d ; e , e , ∗ m , e , e ) if and only if e − e + 1 ≤ m ≤ d + 1 − e − e , m ≡ e − e + 1 (mod 2) . (b) Admissible covers of type ( d ; e , e , ∗ e - e , e , e ) have multiplicity . Thecomponent cover of type ( d ; e , e , e - e ) is uniquely determined, so the ad-missible cover is determined by its second component cover and the gluingover the node. Moreover, the gluing over the node is unique when e = e .When e = e , there are always two possibilities for gluing except for asingle admissible cover in the case that e , e , and d are all even.The number of admissible covers of this type is ( e ( d + 1 − e − e ) if d + 1 ≤ e + e , ( e + e − d − d + 1 − e ) if d + 1 ≥ e + e . Proof.
We briefly explain how this follows from Theorem 4.2 of [12]. As statedabove, the possibilities for admissible covers are determined by pairs ( g , g , ρ ),( ρ − , g , g ) where ( g , g , g , g ) is a Hurwitz factorization of type ( d ; e , e , e , e ). Loc. cit. immediately implies that ρ is always either a single cycle of length m ≥ loc. cit. that the ranges for m (which is e + e − k is the notation of loc. cit. ) are as asserted, and that for a given m , the numberof possibilities with ρ an m -cycle is precisely m , when counted with multiplicity.On the other hand, in this case both component covers are three-point pure-cyclecovers, and thus uniquely determined (see Theorem 2.1 (a)). Thus the admissiblecover is unique in this case, with multiplicity m .For (b), we see by inspection of the description of part (ii) of loc. cit. that g is disjoint from g . It immediately follows that the braid action is trivial, so themultiplicity is always 1, and the asserted count of covers follows immediately fromthe proof of Proposition 4.10 of loc. cit. Moreover, the component cover of type( d ; e , e , e - e ) is a disjoint union of covers of type ( e ; e , e ) and ( e ; e , e ) (aswell as d − e − e copies of the trivial cover), so it is uniquely determined, asasserted. Furthermore, we see that the second component cover is always a singleconnected cover of degree d , and g , g are recovered as the disjoint cycles of ρ − , sothe gluing is unique when e = e . When e = e , it is possible to swap g and g ,so we see that there are two possibilities for gluing. The argument of Lemma 2.2shows that we do in fact obtain two distinct admissible covers in this way, exceptfor a single cover occurring when e , e and d are all even. (cid:3) OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 7 Group theory
In several contexts, we will have to calculate monodromy groups other than thosetreated by Theorem 2.1 (c). We will also have to pass between counting mere coversand counting G -Galois covers. In this section, we give basic group-theoretic resultsto address these topics.Since we restrict our attention to covers of prime degree, the following propositionwill be helpful. Proposition 3.1.
Let p be a prime number and G a transitive group on p letters.Suppose that G contains a pure cycle of length < e < p − . Then G is either A p or S p .Moreover, if e = p − , and G is neither A p nor S p , then p = 2 r + 1 for some r , and G contains a unique minimal normal subgroup isomorphic to PSL (2 r ) , andis contained in PΓL (2 r ) ≃ PSL (2 r ) ⋊ Z /r Z . If e = p − , and G is not S p , then G = F p ⋊ F ∗ p . Note that this does not contradict the exceptional case d = 6 and G = S inTheorem 2.1 (c), since we assume that the degree d is prime. Proof.
Since p is prime, G is necessarily primitive, and a theorem usually attributedto Marggraff ([11]) then states that G is at least ( p − e + 1)-transitive. When e ≤ p −
2, we have that p − e + 1 ≥
3. The 2-transitive permutation groups havebeen classified by Cameron (Section 5 of [7]). Specifically, G has a unique minimalnormal subgroup which is either elementary abelian or one of several possible simplegroups. Since G is at least 3-transitive, one easily checks that the elementary abeliancase is not possible: indeed, one checks directly that if a subgroup of a 3-transitivegroup inside S p contains an element of prime order ℓ , then it is not possible forall its conjugates to commute with one another. Similarly, most possibilities inthe simple case cannot be 3-transitive. If G is not S p or A p , then G must havea unique minimal normal subgroup N which is isomorphic to a Mathieu group M , M , or to N = PSL (2 r ). We then have that G is a subgroup of Aut( N ).For N = M , M , we have N = G = Aut( N ), and it is easy to check that theMathieu groups M and M do not contain any single cycles of order less than p , for example with the computer algebra package GAP. Therefore these cases donot occur. The group PSL (2 r ) can only occur if p = 2 r + 1. In this case, we havethat G is a subgroup of Aut(PSL (2 r )) = PΓL (2 r ) and G is at most 3-transitive,so we have e = p −
2, as desired.Finally, if e = p −
1, M¨uller has classified transitive permutation groups contain-ing ( p − S p is F p ⋊ F ∗ p , as asserted. (cid:3) We illustrate the utility of the proposition with:
Corollary 3.2.
Fix e , e , e , e with ≤ e i ≤ p for each i , and e + e ≤ p . For p > , any genus- cover of type ( p ; e - e , e , e ) has monodromy group S p or A p ,with the latter case occurring precisely when e and e are odd, and e + e is even.For p = 5 , the only exceptional case is type (5; 2 - , , , where the monodromygroup is F ⋊ F ∗ .Proof. Without loss of generality, we assume e ≤ e and e ≤ e . ApplyingProposition 3.1, it is clear that the only possible exception occurs for types with IRENE I. BOUW AND BRIAN OSSERMAN e , e ≥ p −
2. We thus have to treat types ( p ; 3-3 , p − , p − p ; 2-4 , p − , p − p ; 2-2 , p − , p ), ( p ; 2-3 , p − , p − p ; 2-2 , p − , p − G contains both a ( p − p − p > F p ⋊ F ∗ p does not contain a 2-2-cycle.For the first three cases, we must have that p = 2 r + 1 for some r and G is a subgroup of Γ := PΓL (2 r ). Since p = 2 r + 1 is a Fermat prime number,we have that r is a power of 2. Moreover, since PSL (4) = A as permutationgroups in S , we may assume r ≥
4. Since r is even, any element of order 3 inΓ ∼ = PSL (2 r ) ⋊ Z /r Z must lie inside PSL (2 r ), and a non-trivial element of thisgroup can fix at most 2 letters. Thus, in order to contain a 3-3-cycle, we would haveto have 6 ≤ p = 2 r + 1 ≤
8, which contradicts the hypothesis r ≥
4. This rules outthe first case. In the second case, if we square the 2-4-cycle we obtain a 2-2-cycle.To complete the argument for both the second and third cases it is thus enough tocheck directly that if r >
4, an element of order 2 cannot fix precisely p − (16)does not contain a 2-2-cycle, which one can do directly with GAP. (cid:3) Because the theory of stable reduction is developed in the G -Galois context, itis convenient to be able to pass back and forth between the context of mere coversand of G -Galois covers. The following easy result relates the number of mere coversto the number of G -Galois covers in the case we are interested in. Lemma 3.3.
Let f : P → P be a (mere) cover of degree d with monodromy group G = A d (respectively, S d ). Then the number of G -Galois structures on the Galoisclosure of f is exactly (respectively, ).Proof. The case that G = S d is clear, since conjugacy by S d is then the same asconjugacy by G . Suppose G = A d , and let X = { ( g , . . . , g r ) | Q i g i = 1 , h g i i = d } .Since the centralizer C S d ( A d ) of A d in S d is trivial, it follows that S d acts freelyon X , so the number of elements in X f ⊆ X corresponding to f as a mere cover is | S d | . Since the center of G = A d is trivial, G also acts freely on X , and X f breaksinto two equivalence classes of G -Galois covers, each of size | A d | . (cid:3) Stable reduction
In this section, we recall some generalities on stable reduction of Galois covers ofcurves, and prove a few simple lemmas as a prelude to our main calculations. Themain references for this section are [20] and [3]. Since these sources only considerthe case of G -Galois covers, we restrict to this situation here as well. Lemma 3.3implies that we may translate results on good or bad reduction of G -Galois coversto results on the stable reduction of the mere covers, so this is no restriction.Let R be a discrete valuation ring with fraction field K of characteristic zero andresidue field an algebraically closed field k of characteristic p >
0. Let f : V = P K → X = P K be a degree- p cover branched at r points Q = 0 , Q = 1 , . . . , Q r = ∞ over K with ramification type ( p ; C , . . . , C r ) . For the moment, we do not assume thatthe C i are the conjugacy classes of a single cycle. We denote the Galois closure of f by g : Y → P and let G be its Galois group. Note that G is a transitive subgroupof S p , and thus has order divisible by p . Write H := Gal( Y, V ), a subgroup of index p . We suppose that Q i Q j (mod p ), for i = j , in other words, that ( X ; { Q i } ) hasgood reduction as a marked curve. We assume moreover that g has bad reduction to OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 9 characteristic p , and denote by ¯ g : ¯ Y → ¯ X its stable reduction . The stable reduction¯ g is defined as follows. After replacing K by a finite extension, there exists a uniquestable model Y of Y as defined in [20]. We define X = Y /G . The stable reduction¯ g : ¯ Y := Y ⊗ R k → ¯ X := X ⊗ R k is a finite map between semistable curves incharacteristic p ; we call such maps stable G -maps . We refer to [20], Definition 2.14,for a precise definition.Roughly speaking, the theory of stable reduction proceeds in two steps: first, oneunderstands the possibilities for stable G -maps, and then one counts the numberof characteristic-0 covers reducing to each stable G -map.We begin by describing what the stable reduction must look like. Since ( X ; Q i )has good reduction to characteristic p , there exists a model X → Spec( R ) suchthat the Q i extend to disjoint sections. There is a unique irreducible component¯ X of ¯ X , called the original component , on which the natural contraction map¯ X → X ⊗ R k is an isomorphism. The restriction of ¯ g to ¯ X is inseparable.Let B ⊆ { , , . . . , r } consist of those indices i such that C i is not the conjugacyclass of a p -cycle. For i ∈ B , we have that Q i specializes to an irreducible component¯ X i = ¯ X of ¯ X . The restriction of ¯ g to ¯ X i is separable, and ¯ X i intersects the rest of¯ X in a single point ξ i . Let ¯ Y i be an irreducible component of ¯ Y above ¯ X i , and write¯ g i : ¯ Y i → ¯ X i for the restriction of ¯ g to ¯ Y i . We denote by G i the decomposition groupof ¯ Y i . We call the components ¯ X i (resp. the covers ¯ g i ) for i ∈ B the primitive tails (resp. the primitive tail covers ) associated with the stable reduction. The followingdefinition gives a characterization of those covers that can arise as primitive tailcovers (compare to [20], Section 2.2). Definition 4.1.
Let k be an algebraically closed field of characteristic p >
0. Let C be a conjugacy class of S p which is not the class of a p -cycle. A primitive tailcover of ramification type C is a G -Galois cover ϕ C : T C → P k defined over k whichis branched at exactly two points 0 , ∞ , satisfying the following conditions.(a) The Galois group G C of ϕ C is a subgroup of S p and contains a subgroup H of index p such that ¯ T C := T C /H has genus 0.(b) The induced map ¯ ϕ C : ¯ T C → P is tamely branched at x = 0, with conju-gacy class C , and wildly branched at x = ∞ .If ϕ is a tail cover, we let h = h ( ϕ ) be the conductor and pm = pm ( ϕ ) theramification index of a wild ramification point.We say that two primitive tail covers ϕ i : T i → P k are isomorphic if thereexists a G -equivariant isomorphism ι : T → T . Note that we do not require anisomorphism to send ¯ T to ¯ T .Note that an isomorphism ι of primitive tail covers may be completed into acommuting square T ι −−−−→ T ϕ y y ϕ P −−−−→ P . Note also that the number of primitive tail covers of fixed ramification type is finite.Since p strictly divides the order of the Galois group G C , we conclude that m is prime to p . The invariants h, m describe the wild ramification of the tail cover ϕ C . The integers h and m only depend on the conjugacy class C . In Section 5, wewill show this if C is the class of a single cycle or the product of 2 disjoint cycles, but this holds more generally. In the more general set-up of [20], Definition 2.9 itis required that σ := h/m < h, m ) = 1 (Lemma 5.1). Summarizing, we find that ( h, m ) satisfy:(4.1) m | ( p − , ≤ h < m, gcd( h, m ) = 1 . In the more general set-up of [20] there also exists so-called new tails, whichsatisfy σ >
1. The following lemma implies that these do not occur in our situation.
Lemma 4.2.
The curve ¯ X consists of at most r + 1 irreducible components: theoriginal component ¯ X and primitive tails ¯ X i for all i ∈ B .Proof. In the case that r = 3 this is proved in [20], Section 4.4, using that thecover is the Galois closure of a genus-0 cover of degree p . The general case is astraightforward generalization. (cid:3) It remains to discuss the restriction of ¯ g to the original component ¯ X . Asmentioned above, this restriction is inseparable, and it is described by a so-calleddeformation datum ([20], Section 1.3).In order to describe deformation data, we set some notation. Let ¯ Q i be the limiton ¯ X of the Q i for i B , and set ¯ Q i = ξ i for i ∈ B . Definition 4.3.
Let k be an algebraically closed field of characteristic p . A defor-mation datum is a pair ( Z, ω ), where Z is a smooth projective curve over k togetherwith a finite Galois cover g : Z → X = P k , and ω is a meromorphic differentialform on Z such that the following conditions hold.(a) Let H be the Galois group of Z → X . Then β ∗ ω = χ ( β ) · ω, for all β ∈ H. Here χ : H → F × p in an injective character.(b) The differential form ω is either logarithmic, i.e. of the form ω = d f /f , orexact, i.e. of the form d f , for some meromorphic function f on Z .Note that the cover Z → X is necessarily cyclic.To a G -Galois cover g : Y → P with bad reduction, we may associate a de-formation datum, as follows. Choose an irreducible component ¯ Y of ¯ Y above theoriginal component ¯ X . Since the restriction ¯ g : ¯ Y → ¯ X is inseparable and G ⊂ S p , it follows that the inertia group I of ¯ Y is cyclic of order p , i.e. a Sylow p -subgroup of G . Since the inertia group is normal in the decomposition group, thedecomposition group G of ¯ Y is a subgroup of N S p ( I ) ≃ Z /p Z ⋊ χ Z /p Z ∗ , where χ : Z /p Z ∗ → Z /p Z ∗ is an injective character. It follows that the map ¯ g factorsas ¯ g : ¯ Y → ¯ Z → ¯ X , where ¯ Y → ¯ Z is inseparable of degree p and ¯ Z → ¯ X isseparable. We conclude that the Galois group H := Gal( ¯ Z , ¯ X ) is a subgroup of Z /p Z ∗ ≃ Z / ( p − Z . In particular, it follows that(4.2) G ≃ I ⋊ χ H . The inseparable map ¯ Y → ¯ Z is characterized by a differential form ω on ¯ Z satisfying the properties of Definition 4.3, see [20], Section 1.3.2.The differential form ω is logarithmic if ¯ Y → ¯ Z is a µ p -torsor and exact if thismap is an α p -torsor. A differential form is logarithmic if and only if it is fixed bythe Cartier operator C and exact if and only if it is killed by C . (See for example[8], exercise 9.6, for the definition of the Cartier operator and an outline of these OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 11 properties.) Wewers ([19], Lemma 2.8) shows that in the case of covers branchedat r = 3 points the differential form is always logarithmic.The deformation datum ( Z, ω ) associated to g satisfies the following compatibili-ties with the tail covers. We refer to [20], Proposition 1.8 and (2) for proofs of thesestatements. For i ∈ B , we let h i (resp. pm i ) be the conductor (resp. ramificationindex) of a wild ramification point of the corresponding tail cover of type C i , asdefined above. We put σ i = h i /m i . We also use the convention σ i = 0 for i B .(a) If C i is the conjugacy class of a p -cycle then ¯ Q i is unbranched in ¯ Z → ¯ X and ω has a simple pole at all points of ¯ Z above ¯ Q i .(b) Otherwise, ¯ Z → ¯ X is branched of order m i at ¯ Q i , and ω has a zero oforder h i − Z above ¯ Q i .(c) The map ¯ Z → ¯ X is unbranched outside { ¯ Q i } . All poles and zeros of ω are above the ¯ Q i .(d) The invariants σ i satisfy P i ∈ B σ i = r − σ i ) is called the signature of the deformation datum ( Z, ω ). Proposition 4.4.
Suppose that r = 3 , . We fix rational numbers ( σ , . . . , σ r ) with σ i ∈ p − Z and ≤ σ i < , and P ri =1 σ i = r − . We fix points ¯ Q = 0 , ¯ Q =1 , . . . , ¯ Q r = ∞ on ¯ X ≃ P k . Then there exists a deformation datum of signature ( σ i ) , unique up to scaling. If further the ¯ Q i are general, the deformation datum islogarithmic and unique up to isomorphism.Proof. In the case that r = 3 this is proved in [20]. (The proof in this case is similarto the proof in the case that r = 4 which we give below.) Suppose that r = 4. Let B = { ≤ i ≤ r | σ i = 0 } . We write ¯ Q = λ ∈ P k \ { , , ∞} and σ i = a i / ( p − ω is the deformation datum associated with ¯ g , then a i = h i ( p − /m i . )It is shown in [3], Chapter 3, that a deformation datum of signature ( σ i ) consistsof a differential form ω on the cover ¯ Z of ¯ X defined as a connected component ofthe (normalization of the) projective curve with Kummer equation(4.3) z p − = x a ( x − a ( x − λ ) a . The degree of ¯ Z → ¯ X is m := p − p − , a , a , a , a ) . The differential form ω may be written as(4.4) ω = ǫ z d xx ( x − x − λ ) = ǫ x p − a ( x − p − − a ( x − λ ) p − − a z p x p ( x − p ( x − λ ) p d xx , where ǫ ∈ k × is a unit.To show the existence of the deformation datum, it suffices to show that onemay choose ǫ such that ω is logarithmic or exact, or, equivalently, such that ω isfixed or killed by the Cartier operator C . It follows from standard properties of theCartier operator, (4.4), and the assumption that a + a + a + a = 2( p −
1) that C ω = c /p ǫ (1 − p ) /p ω , where(4.5) c = min( a ,p − − a ) X j =max(0 ,p − − a − a ) (cid:18) p − − a a − j (cid:19)(cid:18) p − − a j (cid:19) λ j . Note that c is the coefficient of x p in x p − a ( x − p − − a ( x − λ ) p − − a . One easilychecks that c is nonzero as polynomial in λ . It follows that ω defines an exact differential form if and only if λ is a zero of the polynomial c . This does not happenif { , , λ, ∞} is general.We assume that c ( λ ) = 0. Since k is algebraically closed, we may choose ǫ ∈ k × such that ǫ p − = c . Then C ω = ω , and ω defines a logarithmic deformation datum.It is easy to see that ω is unique, up to multiplication by an element of F × p . (cid:3) The tail covers
In Section 4, we have seen that associated with a Galois cover with bad reductionis a set of (primitive) tail covers. In this section, we analyze the possible tail coversfor conjugacy classes e = p and e - e of S p . Recall from Section 2 that these areconjugacy classes which occur in the 3-point covers obtained as degeneration of thepure-cycle 4-point covers.The following lemma shows the existence of primitive tail covers for the conjugacyclasses occurring in the degeneration of single-cycle 4-point covers (Theorem 2.3). Lemma 5.1. (a)
Let ≤ e < p − be an integer. There exists a primitivetail cover ϕ e : T e → P k of ramification type e . Its Galois group is A p if e is odd and S p if e is even. The wild branch point of ϕ e has inertiagroup of order p ( p − / gcd( p − , e −
1) =: pm e . The conductor is h e :=( p − e ) / gcd( p − , e − . (b) In the case that e = p − , there exists a primitive tail cover ϕ e of ramifica-tion type e , with Galois group F p ⋊ F ∗ p . The cover is totally branched abovethe wild branch point and has conductor h p − = 1 . (c) Let ≤ e ≤ e ≤ p − be integers with e + e ≤ p . There is a primitive tailcover ϕ e ,e : T e ,e → P k of ramification type e - e . The wild branch pointof ϕ e ,e has inertia group of order p ( p − / gcd( p − , e + e −
2) =: pm e ,e .The conductor is h e ,e := ( p + 1 − e − e ) / gcd( p − , e + e − . In all three cases, the tail cover is unique with the given ramification when con-sidered as a mere cover.Proof.
Let 2 ≤ e ≤ p − ϕ e as theGalois closure of the degree- p cover ¯ ϕ e : ¯ T e := P → P given by(5.1) y p + y e = x, ( x, y ) x. One easily checks that this is the unique degree- p cover between projective lineswith one wild branch point and the required tame ramification.The decomposition group G e of T e is contained in S p . We note that the nor-malizer in S p of a Sylow p -subgroup has trivial center. Therefore the inertia group I of a wild ramification point of ϕ e is contained in F p ⋊ χ F ∗ p , where χ : F ∗ p → F ∗ p is an injective character. Therefore it follows from [6], Proposition 2.2.(i) thatgcd( h e , m e ) = 1. The statement on the wild ramification follows now directly fromthe Riemann–Hurwitz formula (as in [20], Lemma 4.10). Suppose that e is odd.Then m e = ( p − / gcd( p − , e −
1) divides ( p − /
2. Therefore in this case boththe tame and the wild ramification groups are contained in A p . This implies thatthe Galois group G e of ϕ e is a subgroup of A p .To prove (a), we suppose that e = p −
1. We show that the Galois group G e of ϕ e is A p or S p . Suppose that this is not the case. Proposition 3.1 implies that e = p − r −
1. Moreover, G e is a subgroup of PΓL (2 r ) ≃ PSL (2 r ) ⋊Z /r Z . Thenormalizer in PΓL (2 r ) of a Sylow p -subgroup I is Z /p Z⋊Z / r Z . The computationof the wild ramification shows that the inertia group I ( η ) of the wild ramification OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 13 point η is isomorphic to Z /p Z ⋊ Z / p − Z . Therefore PΓL (2 r ) contains a subgroupisomorphic to I ( η ) if and only if p = 17 = 2 + 1, in which case I ( η ) = N PΓL (2 r ) ( I ).We conclude that if G e S p , A p then e = 15 and p = 17. However, in this lastcase one may check using Magma that a suitable specialization of (5.1) has Galoisgroup A . As before, we conclude that G e ≃ A .Now suppose that e = p −
1. It is easy to see that the Galois closure of ¯ ϕ p − is inthis case the cover ϕ p − : P → P obtained by dividing out F p ⋊ F ∗ p ⊂ PGL ( p ) =Aut( P ). This proves (b).Let e , e be as in the statement of (c). As in the proof of (a), we define ϕ e ,e as the Galois closure of a non-Galois cover ¯ ϕ e ,e : ¯ T e ,e → P of degree p . Thecover ¯ ϕ e ,e , if it exists, is given by an equation(5.2) F ( y ) := y e ( y − e ˜ F ( y ) = x, ( x, y ) x, where ˜ F ( y ) = P p − e − e i =0 c i y i has degree p − e − e . We may assume that c p − e − e =1. The condition that ¯ ϕ e ,e has exactly three ramification points y = 0 , , ∞ yieldsthe condition F ′ ( y ) = γt e − ( t − e − . Therefore the coefficients of ˜ F satisfy therecursion(5.3) c i = c i − e + e + i − e + i , i = 1 , . . . , p − e − e . This implies that the c i are uniquely determined by c p − e − e = 1. Conversely, itfollows that the polynomial F defined by these c i has the required tame ramification.The statement on the wild ramification follows from the Riemann–Hurwitz formula,as in the proof of (a). (cid:3) It remains to analyze the number of tail covers, and their automorphism groups.Due to the nature of our argument, we will only need to carry out this analysisfor the tails of ramification type e . From Lemma 5.1, it follows already that themap ϕ C : T C → P is unique. However, part of the datum of a tail cover is anisomorphism α : Gal( T C , P ) ∼ → G C . For every τ ∈ Aut( G C ), the tuple ( ϕ, τ ◦ α ) also defines a tail cover, which is potentially non-equivalent. Modification by τ leaves the cover unchanged as a G C -Galois cover if and only if τ ∈ Inn( G C ).However, the weaker notion of equivalence for tail covers implies that τ leaves thecover unchanged as a tail cover if and only if τ can be described as conjugation byan element of N Aut( T ) ( G C ). Thus, the number of distinct tail covers correspondingto a given mere cover is the order of the cokernel of the map N Aut( T C ) ( G C ) → Aut( G C )given by conjugation. Denote by Aut G C ( ϕ C ) the kernel of this map, or equivalentlythe set of G C -equivariant automorphisms of T C . It follows finally that the numberof tail covers corresponding to ϕ C is(5.4) | Aut( G C ) || Aut G C ( ϕ C ) || N Aut( T C ) ( G C ) | . Finally, denote by Aut G C ( ϕ C ) ⊂ Aut G C ( ϕ C ) the subset of automorphisms whichfix the chosen ramification point η . We now simultaneously compute these auto-morphism groups and show that in the single-cycle case, we have a unique tailcover. Lemma 5.2.
Let ≤ e ≤ p − be an integer. (a) The group
Aut G e ( ϕ e ) (resp. Aut G e ( ϕ e ) ) is cyclic of order ( p − e ) / (resp. h e ) if e is odd and p − e (resp. h e ) is e is even. (b) There is a unique primitive tail cover of type e .Proof. First note that the definition of Aut G e ( ϕ e ) implies that any element inducesan automorphism of any intermediate cover of ϕ e , and in particular induces auto-morphisms of ¯ T e and P . Choose a primitive ( p − e )th root of unity ζ ∈ ¯ F p . Then µ ( x, y ) = ( ζ e x, ζy ) is an automorphism of ¯ T e . One easily checks that µ generatesthe group of automorphisms of ¯ T e which induces automorphisms of P under ϕ e ,and that furthermore T e is Galois over P / h µ i , so in particular every element of µ lifts to an automorphism of T e . Taking the quotient by the action of µ , we obtaina diagram(5.5) ¯ T e −−−−→ ¯ T ′ e = ¯ T e / h µ i ¯ ϕ e y y ¯ ψ e P −−−−→ P / h µ i . Since we know the ramification of the other three maps, one easily computesthat the tame ramification of ¯ ψ e is e -( p − e ). Let ψ e : T ′ e → P be the Galois closureof ¯ ψ e .We now specialize to the case that e is odd. Since G e = A p does not containan element of cycle type e -( p − e ), it follows that the Galois group G ′ of ψ e is S p . Therefore it follows by degree considerations that the cover T e → T ′ e is cyclicof degree ( p − e ) /
2. Denote by Q the Galois group of the cover T e → P / h µ i .This is a group of order p !( p − e ) /
2, which contains normal subgroups isomorphicto A p and Z / p − e Z , respectively. It follows that Q = Z / p − e Z ⋊ S p . Note thatAut G e ( ϕ e ) is necessarily a subgroup of Q . In fact, it is precisely the subgroup of Q which commutes with every element of A p ⊆ Q . One easily checks that thesemidirect product cannot be split, and that Aut G e ( ϕ e ) is precisely the normalsubgroup Z / p − e Z , that is the Galois group of T e over T ′ e .To compute Aut G e ( ϕ e ) we need to compute the order of the inertia group of awild ramification point of T e in the map T e → T ′ e . Since a wild ramification pointof T ′ e has inertia group of order p ( p −
1) = pm e gcd( p − , e − G e ( ϕ e ) hasorder h e = ( p − e ) / gcd( p − , e − e is odd.For (b), we simply observe that since Q ⊂ N Aut( T e ) ( G e ), we have | Aut( G e ) || Aut G e ( ϕ e ) || N Aut( T e ) ( G e ) | ≤ p ! p − e | Q | = 1 , so the tail cover is unique, as desired.We now treat the case that e is even. For (a), if e < p −
1, the Galois group of¯ ψ e is equal to the Galois group of ¯ ϕ e , which is isomorphic to S p . We conclude thatthe degree of T e → T ′ e is p − e in this case, and the group Q defined as above is adirect product Z / ( p − e ) Z × S p . Similarly to the case that e is odd, we concludethat Aut G e ( ϕ e ) (resp. Aut G e ( ϕ e )) is cyclic of order p − e (resp. h e ) in this case, asdesired. On the other hand, if e = p −
1, we have that p − e = 1, hence µ is trivial,and we again conclude that (a) holds. Finally, (b) is trivial: if e < p −
1, the Galoisgroup of ϕ e is S p and Aut( S p ) = S p . Therefore there is a unique tail cover in this OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 15 case. The same conclusion holds in the case that e = p −
1, since G p − ≃ F p ⋊ χ F ∗ p and Aut( G p − ) = G p − . The statement of the lemma follows. (cid:3) Remark . In the case of e - e tail covers, there may in fact be more than onestructure on a given mere cover. However, we will not need to know this numberfor our argument. 6. Reduction of -point covers In this section, we (almost) compute the number of 3-point covers with badreduction for ramification types ( p ; e - e , e , e ). More precisely, we compute thisnumber in the case that not both e and e + e are even. In the remaining case, weonly compute this number up to a factor 2, which is good enough for our purposes.Although we restrict to types of the above form, our strategy applies somewhatmore generally. The results of this section rely on the results of Wewers [20], whogives a precise formula for the number of lifts of a given special G -map (Section 4)in the 3-point case.We fix a type τ = ( p ; e - e , e , e ) satisfying the genus-0 condition P i e i = 2 p +2.We allow e or e to be p , although this is not the case that ultimately interests us;see below for an explanation. We do however assume throughout that we are not inthe exceptional case τ = (5; 2-2 , , g i of type C i , for i such that C i = p . Moreover, by Proposition4.4 we have a (unique) deformation datum, so we know there exists at least onespecial G -map ¯ g of type τ . Lemma 5.2 implies moreover that the number of special G -maps is equal to the number of e - e tail covers. Wewers ([20], Theorem 3)shows that there exists a G -Galois cover g : Y → P in characteristic zero with badreduction to characteristic p , and more specifically with stable reduction equal tothe given special G -map ¯ g . Moreover, Wewers gives a formula for the number ˜ L (¯ g )for lifts of the given special G -map ¯ g .In order to state his formula, we need to introduce one more invariant. LetAut G (¯ g ) be the group of G -equivariant automorphisms of ¯ Y which induce the iden-tity on the original component ¯ X . Choose γ ∈ Aut G (¯ g ), and consider the restric-tion of γ to the original component ¯ X . Let ¯ Y be an irreducible component of ¯ Y above ¯ X whose inertia group is the fixed Sylow p -subgroup I of G . As in (4.2),we write G = I ⋊ χ H ⊂ F p ⋊ χ F ∗ p for the decomposition group of ¯ Y . Wewers([20], proof of Lemma 2.17) shows that γ := γ | ¯ Y centralizes H and normalizes I ,i.e. γ ∈ C N G ( I ) ( H ). Since ¯ Y | ¯ X = Ind GG ¯ Y and γ is G -equivariant, it follows thatthe restriction of γ to ¯ X is uniquely determined by γ . We denote by n ′ ( τ ) theorder of the subgroup consisting of those γ ∈ C N G ( I ) ( H ) such that there exists a γ ∈ Aut G (¯ g ) with γ | ¯ Y = γ . Our notation is justified by Corollary 6.3 below.Wewers ([20], Corollary 4.8) shows that(6.1) | ˜ L (¯ g ) | = p − n ′ ( τ ) Y i ∈ B h C i | Aut G Ci (¯ g C i ) | . The numbers are as defined in Section 4. (Note that the group Aut G Ci (¯ g C i ) isdefined differently from the group Aut G (¯ g ).)To compute the number of curves with bad reduction, we need to compute thenumber n ′ ( τ ) defined above. As explained by Wewers ([20, Lemma 2.17]), one mayexpress the number n ′ ( τ ) in terms of certain groups of automorphisms of the tail covers. However, there is a mistake in the concrete description he gives of Aut G (¯ g )in terms of the tails, therefore we do not use Wewers’ description. For a correctedversion, we refer to the manuscript [17].The difficulty we face in using Wewers’ formula directly is that we do not knowthe Galois group G e - e of the e - e tail. This prevents us from directly computingboth the number of e - e tails, and the term n ′ ( τ ). We avoid this problem by usingthe following trick. We first consider covers of type τ ∗ = ( p ; e - e , ε, p ), with ε = p +2 − e − e , which all have bad reduction. This observation lets us compute n ′ ( τ ∗ )from Wewers’ formula. We then show that for covers of type τ = ( p ; e - e , e , e ),the number n ′ ( τ ) essentially only depends on e and e , allowing us to compute n ′ ( τ ) from n ′ ( τ ∗ ). A problem with this method is that in the case that the Galoisgroups of covers with type τ and τ ∗ are not equal, the numbers n ′ ( τ ) and n ′ ( τ ∗ )may differ by a factor 2. Therefore in this case, we are able to determine the numberof covers of type τ with bad reduction only up to a factor 2.In Lemma 2.2, we have counted non-Galois covers, but in this section, we dealwith Galois covers. Let G ( τ ) be the Galois group of a cover of type τ . This group iswell-defined and either A p or S p , by Corollary 3.2. We write γ ( τ ) for the quotientof the number of Galois covers of type τ by the Hurwitz number h ( τ ). By Lemma3.3, it follows that γ ( τ ) is 2 if G is A p and 1 if it is S p . The number γ ( τ ) willdrop out from the formulas as soon as we pass back to the non-Galois situation inSection 7.We first compute the number n ′ ( τ ∗ ). We note that by Corollary 3.2, the Galoisgroup G ( τ ∗ ) of a cover of type τ ∗ is A p if e + e is even and S p otherwise. Inparticular, we see that G ( τ ) = G ( τ ∗ ) unless e + e and e are both even. Inthis case we have that G ( τ ) = S p and G ( τ ∗ ) = A p . Recall from Lemma 5.2 thatthere is a unique tail cover for the single-cycle tails. We denote by N e - e thenumber of e - e tails, and by Aut e - e the group Aut G e e (¯ g e - e ) for any tail cover¯ g e - e as in Lemma 5.1. Note that since ¯ g e - e is unique as a mere cover, and thedefinition of Aut G e e (¯ g e - e ) is independent of the G -structure, this notation iswell-defined. We similarly have from (6.1) that | ˜ L (¯ g ) | depends only on τ , so wewrite ˜ L ( τ ) := | ˜ L (¯ g ) | for any special G -map ¯ g of type τ . Lemma 6.1.
Let τ ∗ be as above. Then n ′ ( τ ∗ ) = (1 + δ e ,e ) N e - e ( p − p − , e + e − γ ( τ ∗ ) | Aut e - e | . Here δ e ,e is the Kronecker δ .Proof. Lemma 2.2 implies that the Hurwitz number h ( τ ∗ ) equals ( p + 1 − e − e ) / e = e and ( p + 1 − e − e ) otherwise. Since all covers of type τ ∗ have badreduction, h ( τ ∗ ) γ ( τ ∗ ) is equal to N e - e · ˜ L ( τ ∗ ). The statement of the lemma followsby applying Lemmas 5.1.(c), 5.2, and (6.1). (cid:3) We now analyze n ′ in earnest. For convenience, for i ∈ B we also introduce thenotation g Aut G i (¯ g i ) for the group of G -equivariant automorphisms of the induced tailcover Ind GG i (¯ g i ). Recall also that ξ i is the node connecting ¯ X to ¯ X i . We note that n ′ may be analyzed tail by tail, in the sense that given γ ∈ C N G ( I ) ( H ), we havethat γ lifts to Aut (¯ g ) if and only if for each i ∈ B , there is some γ i ∈ g Aut G i (¯ g i )whose action on ¯ g − i ( ξ i ) is compatible with γ . The basic proposition underlyingthe behavior of n ′ is then the following: OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 17
Proposition 6.2.
Suppose G = S p or A p , and we have a special G -map ¯ g : ¯ Y → ¯ X .Then: (a) For i ∈ B , the G -equivariant automorphisms of ¯ g − ( ξ i ) form a cyclic group. (b) Given an element γ ∈ C N G ( I ) ( H ) and i ∈ B , there exists γ i ∈ g Aut G i (¯ g i ) agreeing with the action of γ on ¯ g − ( ξ i ) if and only if there exists γ ′ i ∈ g Aut G i (¯ g i ) having the same orbit length on ¯ g − ( ξ i ) as γ has.Proof. For (a), if ˜ ξ i is a point above ξ i lying on the chosen component ¯ Y , oneeasily checks that a G -equivariant automorphism γ of ¯ g − ( ξ i ) is determined bywhere it sends ˜ ξ i , which can in turn be represented by an element g ∈ G chosen sothat g ( ˜ ξ i ) = γ ( ˜ ξ i ). Note that γ = g ; in fact, if γ, γ ′ are determined by g, g ′ , thecomposition law is that γ ◦ γ ′ corresponds to g ′ g . Such a g yields a choice of γ ifand only if we have the equality of stabilizers G ˜ ξ i = G g (˜ ξ i ) . Now, any h ∈ G ˜ ξ i isnecessarily in G , and using that I ⊆ G ˜ ξ i , we find that we must have gIg − ⊆ G .But I contains the only p -cycles in G , so we conclude g ∈ N G I . However, since I fixed ˜ ξ i , the choices of G may be taken modulo I , so we conclude that they lie in N G I/I . Finally, since G = S p or A p , we have that N G I/I is cyclic, isomorphic to Z / ( p − Z if G = S p and to Z / ( p − ) Z if G = A p .(b) then follows immediately, since the actions of both γ and γ ′ i on ¯ g − ( ξ i ) liein the same cyclic group; we can take γ i to be an appropriate power of γ ′ i . (cid:3) Corollary 6.3.
For τ as above, n ′ ( τ ) is well defined.Proof. We know that G = S p or A p , and we also know by Proposition 4.4 andLemma 5.1 that the deformation datum is uniquely determined, and so are the tailcovers, at least as mere covers. But the description of n ′ ( τ ) given by Proposition6.2 is visibly independent of the G -structure on the tail covers, so we obtain thedesired statement. (cid:3) We can now obtain the desired comparison of n ′ ( τ ) with n ′ ( τ ∗ ). Proposition 6.4.
Let τ = ( p ; e - e , e , e ) be a type satisfying the genus- condi-tion, and let τ ∗ be the corresponding modified type. Then if G ( τ ) = G ( τ ∗ ) we have n ′ ( τ ) = n ′ ( τ ∗ ) . Otherwise, n ′ ( τ ) ∈ { n ′ ( τ ∗ ) , n ′ ( τ ∗ ) } .Proof. Let γ be a generator of C N G ( I ) ( H ). We ask which powers of γ extend toan element of Aut G (¯ g ), and we analyze this question tail by tail. Fix a tail ¯ X i , andsuppose that it is a single-cycle tail of length e := e i . The crucial assertion is that γ itself (and hence all its powers) always extends to ¯ X i .First suppose that e < p − G = G i = S p , and g Aut G (¯ g i ) =Aut G i (¯ g i ). Now, γ acts on the fiber of ξ i with orbit length ( p − /m e = gcd( p − , p − e ). On the other hand, by Lemma 5.1 we have that h e = ( p − e ) / gcd( p − , e − γ i ∈ Aut G i (¯ g i ) is a generator, then the orderof γ i is p − e , and also that Aut G i (¯ g i ) has order h e . We conclude that an orbit of γ i has length gcd( p − , e −
1) = gcd( p − , p − e ), and thus by Proposition 6.2 that γ extends to ¯ X i , as claimed.The next case is that e is odd, and G = A p . This proceeds exactly as before,except that both orbits in question have length gcd( p − , p − e ) /
2. Now, suppose e is odd, but G = S p . Then the orbit length of γ is gcd( p − , p − e ). We have g Aut G (¯ g i ) equal to the G -equivariant automorphisms of Ind S p A p (¯ g i ). These contain induced copies of the G -equivariant automorphisms of ¯ g i , so in particular we knowwe have elements of orbit length gcd( p − , e − /
2. However, in fact one also hasa G -equivariant automorphism exchanging the two copies of ¯ g i , and whose squareis the generator of the A p -equivariant automorphisms of ¯ g i . One may think ofthis as coming from the automorphism constructing in Lemma 5.2 inducing theisomorphism between the two different A p -structures on the tail cover. We thushave an element of g Aut G (¯ g i ) of orbit length gcd( p − , e − γ extends to thetail in this case as well.Finally, if e = p − m i = p − γ acts as the identity on the fiberof ξ i . The claim is trivially satisfied in this case.It follows that extending γ to the e -tails imposes no condition when e < p , andof course we do not have tails in the case that e = p . Therefore the only non-trivialcondition imposed in extending γ is the extension to the e - e -tail.In the case that G ( τ ) = G ( τ ∗ ) we conclude the desired statement from Propo-sition 6.2, since the orbit lengths in question are clearly the same in both cases.Suppose that G ( τ ) = G ( τ ∗ ). This happens if and only if both e + e and e areeven. In this case we have that G ( τ ) = S p and G ( τ ∗ ) = A p . Here, we necessarilyhave that e + e , e , e are all even, so the only conditions imposed on either n ′ ( τ )or n ′ ( τ ∗ ) come from the e - e tail. Since the orbit of γ is twice as long in the caseof τ , the answers can differ by at most a factor of 2 in this case, as desired. (cid:3) Let 2 ≤ e ≤ e ≤ e ≤ e < p be integers with P i e i = 2 p + 2 and e + e ≤ p .The following corollary translates Proposition 6.4 into an estimate for the numberof Galois covers of type τ = ( p ; e - e , e , e ) with bad reduction. Theorem 1.2 is aspecial case. Corollary 6.5.
Let τ = ( p ; e - e , e , e ) with τ = (5; 2 - , , . The number ofmere covers of type τ with bad reduction to characteristic p is equal to ( δ ( τ )( p + 1 − e − e ) if e = e ,δ ( τ )( p + 1 − e − e ) / if e = e , where δ ( τ ) ∈ { , } , and δ = 1 unless e + e and e are both even.Proof. We recall that the number of Galois covers of type τ with bad reduction isequal to N e - e · ˜ L ( τ ∗ ). It follows from Lemma 5.1.(c) and (6.1) that this number is γ ( τ ∗ ) n ′ ( τ ∗ ) n ′ ( τ ) ( p + 1 − e − e ) if e = e ,γ ( τ ∗ ) n ′ ( τ ∗ ) n ′ ( τ ) ( p + 1 − e − e ) / e = e . The definition of the Galois factor γ ( τ ) implies that the number of mere covers oftype τ with bad reduction is γ ( τ ∗ ) γ ( τ ) n ′ ( τ ∗ ) n ′ ( τ ) ( p + 1 − e − e ) if e = e ,γ ( τ ∗ ) γ ( τ ) n ′ ( τ ∗ ) n ′ ( τ ) ( p + 1 − e − e ) / e = e . Proposition 6.4 implies that n ′ ( τ ) /n ′ ( τ ∗ ) ∈ { , } , and is equal to 1 unless e + e , e , e are all even. Moreover, if n ′ ( τ ) = n ′ ( τ ∗ ) then γ ( τ ∗ ) /γ ( τ ) = 2. Thestatement of the corollary follows from this. (cid:3) OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 19
Remark . Similar to the proof of Corollary 6.5, one may show that every genus-0 three-point cover of type ( p ; e , e , e ) has bad reduction. We do not includethis proof here, as a proof of this result using linear series already occurs in [14],Theorem 4.2. 7. Reduction of admissible covers
In this section, we return to the case of non-Galois covers, and use the results ofSection 6 to compute the number of “admissible covers with good reduction”. Westart by defining what we mean by this. As always, we fix a type ( p ; e , e , e , e )with 1 < e ≤ e ≤ e ≤ e < p satisfying the genus-0 condition P i e i = 2 p + 2.As in Section 2, we consider admissible degenerations of type ( p ; e , e , ∗ , e , e ),which means that Q = λ ≡ Q = ∞ (mod p ). Recall from Section 2 that inpositive characteristic not every smooth cover degenerates to an admissible cover,as a degeneration might become inseparable. The number of admissible covers(even counted with multiplicity) is still bounded by the number of smooth covers,but equality need not hold. Definition 7.1.
We define h adm p ( p ; e , e , ∗ , e , e ) as the number of admissiblecovers of type ( p ; e , e , ∗ , e , e ), counted with multiplicity, over an algebraicallyclosed field of characteristic p .The following proposition is the main result of this section. Proposition 7.2.
The assumptions on the type τ = ( p ; e , e , e , e ) are as above.Then h adm p ( p ; e , e , ∗ , e , e ) > h ( p ; e , e , e , e ) − p, and h adm p ( p ; e , e , ∗ , e , e ) = h ( p ; e , e , e , e ) − p unless e + e and e are both even.Proof. We begin by noting that in the case τ = (5; 2 , , ,
4) corresponding to theexceptional case of Corollary 3.2, the assertion of the proposition is automatic since h (5; 2 , , ,
4) = 8 <
10. We may therefore assume that τ = (5; 2 , , , p , i.e. that remain separable.We first consider the pure-cycle case, i.e. the case of Theorem 2.3.(a). Let m be an integer satisfying the conditions of loc. cit. We write f : V → X forthe corresponding admissible cover. Recall from Section 2 that ¯ X consists of twoprojective lines X , X intersecting in one point. Choose an irreducible component Y i of Y above X i , and write f i : Y i → X i for the restriction. These are coversof type ( d ; e , e , m ) and ( d ; m, e , e ) with d i ≤ p , respectively. The admissiblecover f has good reduction to characteristic p if and only if both three-point covers f i have good reduction.It is shown in [14], Theorem 4.2, that a genus-0 three-point cover of type( d ; a, b, c ) with a, b, c < p has good reduction to characteristic p if and only ifits degree d is strictly less than p . Since the degree d of the cover f is always atleast as large as the other degree d , it is enough to calculate when d < p . The Riemann–Hurwitz formula implies that d = ( m + e + e − /
2. Therefore thecondition d < p is equivalent to the inequality e + e + m ≤ p − . Since we assumed the existence of an admissible cover with ρ an m -cycle, itfollows from Theorem 2.3.(a) that m ≤ d + 1 − e − e = 2 p + 1 − e − e . We findthat d < p unless m = 2 p + 1 − e − e . We also note that the lower bound for m is always less than or equal to the upper bound, which is 2 p + 1 − e − e . We thusconclude that there are 2 p + 1 − e − e admissible covers with bad reduction.We now consider the case of an admissible cover with ρ an e - e -cycle (Theorem2.3.(b)). Let f : V → X be such an admissible cover in characteristic 0, asabove. In particular, the restriction f (resp. f ) has type ( d ; e , e , e - e ) (resp.( d ; e - e , e , e )). We write g for the Galois closure of f , and g i for the corre-sponding restrictions. Let G i be the Galois group of g i . The assumptions on the e i imply that p does not divide the order of Galois group of g , therefore g hasgood reduction to characteristic p . Moreover, the cover g is uniquely determinedby the triple ( ρ − , g , g ). If e = e , the gluing is likewise uniquely determined,while if e = e there are exactly 2 possibilities for the tuple ( g , g , g , g ) for agiven triple ( ρ − , g , g ). Therefore to count the number of admissible covers withbad reduction in this case, it suffices to consider the reduction behavior of the cover g : Y → X .Corollary 6.5 implies that whether or not e equals e , the number of admissiblecovers with bad reduction in the 2-cycle case is equal to ( p + 1 − e − e ) unless e + e and e are both even, and bounded from above by 2( p + 1 − e − e ) always.We conclude using Theorem 2.3 that the total number of admissible covers withbad reduction counted with multiplicity is less than or equal to(2 p + 1 − e − e ) + 2( p + 1 − e − e ) = p + ( p + 1 − e − e ) < p, and equal to (2 p + 1 − e − e ) + ( p + 1 − e − e ) = p unless e + e and e are both even. The proposition follows. (cid:3) Remark . Theorem 4.2 of [14] does not need the assumption d = p . Thereforethe proof of Proposition 7.2 in the single-cycle case shows the following strongerresult. Let ( d ; e , e , e , e ) be a genus-0 type with 1 < e ≤ e ≤ e ≤ e < p .Then the number of admissible covers with a single ramified point over the nodeand bad reduction to characteristic p is( d − p + 1)( d + p + 1 − e − e )when either d + 1 ≥ e + e or d + 1 − e < p . Otherwise, all admissible covers havebad reduction. 8. Proof of the main result
In this section, we count the number of mere covers with ramification type( p ; e , e , e , e ) and bad reduction in the case that the branch points are generic.Equivalently, we compute the p -Hurwitz number h p ( p ; e , e , e , e ).Suppose that r = 4 and fix a genus-0 type τ = ( p ; e , e , e , e ) with 2 ≤ e ≤ e ≤ e ≤ e < p . We let g : Y → X = P K be a Galois cover of type τ defined overa local field K as in Section 4, such that ( X ; Q i ) is the generic r -marked curve of OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 21 genus 0. It is no restriction to suppose that Q = 0 , Q = 1 , Q = λ, Q = ∞ , where λ is transcendental over Q p . We suppose that g has bad reduction to characteristic p , and denote by ¯ g : ¯ Y → ¯ X the stable reduction. We have seen in Section 4 thatwe may associate with ¯ g a set of primitive tail covers (¯ g i ) and a deformation datum( ¯ Z , ω ). The primitive tail covers ¯ g i for i ∈ B = { , , , } are uniquely determinedby the e i (Lemma 5.1).The following proposition shows that the number of covers with bad reductionis divisible by p in the case that the branch point are generic. Proposition 8.1.
Suppose that ( X = P K ; Q i ) is the generic r = 4 -marked curve ofgenus zero. Then the number of mere covers of X of ramification type ( p ; e , e , e , e ) with bad reduction is nonzero and divisible by p .Proof. Since the number of Galois covers and the number of mere covers differby a prime-to- p factor, it suffices to prove the proposition for Galois covers. Theexistence portion of the proposition is proved in [3], Proposition 2.4.1, and thedivisibility by p in Lemma 3.4.1 of loc. cit. (in a more general setting). We brieflysketch the proof, which is easier in our case due to the simple structure of thestable reduction (Lemma 4.2). The idea of the proof is inspired by a result of [19],Section 3.We begin by observing that away from the wild branch point ξ i , the primitivetail cover ¯ g i is tamely ramified. Therefore we can lift this cover of affine curves tocharacteristic zero.Let X = P R be equipped with 4 sections Q = 0 , Q = 1 , Q = λ, Q = ∞ ,where λ ∈ R is transcendental over Z p . Then (4.3) defines an m -cyclic cover Z → X . We write Z → X for its generic fiber. Proposition 4.4 implies theexistence of a deformation datum ( ¯ Z , ω ). Associated with the deformation datumis a character χ : Z /m Z → F × p defined by χ ( β ) = β ∗ z/z (mod z ). The differentialform ω corresponds to a p -torsion point P ∈ J ( ¯ Z )[ p ] χ on the Jacobian of ¯ Z . Seefor example [18] (Here we use that the conjugacy classes C i are conjugacy classesof prime-to- p elements. This implies that the differential form ω is holomorphic.)Since P i =1 h i = 2 m and the branch points are generic, we have that J ( ¯ Z )[ p ] χ ≃ Z /p Z × µ p ([4], Proposition 2.9) After enlarging the discretely valued field K , ifnecessary, we may choose a p -torsion point P ∈ J ( Z ⊗ R K )[ p ] χ lifting P . Itcorresponds to an ´etale p -cyclic cover W → Z . The cover ψ : W → X is Galois,with Galois group N := Z /p Z ⋊ χ Z /m Z . It is easy to see that ψ has bad reduction,and that its deformation datum is ( ¯ Z , ω ).By using formal patching ([16] or [20]), one now checks that there exists a map g R : Y → X of stable curves over Spec( R ) whose generic fiber is a G -Galois coverof smooth curves, and whose special fiber defines the given tails covers and thedeformation datum. Over a neighborhood of the original component g R is theinduced cover Ind GN Z → X . Over the tails, the cover g R is induced by the liftof the tail covers. The fact that we can patch the tail covers with the cover over X follows from the observation that h i < m i (Lemma 5.1), since locally there aunique cover with this ramification ([20], Lemma 2.12). This proves the existencestatement.The divisibility by p now follows from the observation that the set of lifts P ofthe p -torsion point P ∈ J ( ¯ Z )[ p ] χ corresponding to the deformation datum is a µ p -torsor. (cid:3) We are now ready to prove our Theorem 1.1, as well as a slightly sharper versionof Theorem 1.3.
Theorem 8.2.
Let p be an odd prime and k an algebraically closed field of charac-teristic p . Suppose we are given integers ≤ e ≤ e ≤ e ≤ e < p . There exists adense open subset U ⊂ P k such that for λ ∈ U the number of degree- p covers withramification type ( e , e , e , e ) over the branch points (0 , , λ, ∞ ) is given by theformula h p ( e , . . . , e ) = min i ( e i ( p + 1 − e i )) − p. Furthermore, unless both e + e and e are even, every such cover has good degen-eration under a degeneration of the base sending λ to ∞ .Proof. Proposition 8.1 implies that the number of covers with ramification type( p ; e , e , e , e ) and bad reduction is at least p . This implies that the genericHurwitz number h p ( e , . . . , e ) is at most min i ( e i ( p + 1 − e i )) − p . Proposition7.2 implies that the number of admissible covers in characteristic p strictly largerthan min i ( e i ( p + 1 − e i )) − p . Since the number of separable covers can onlydecrease under specialization, we conclude that the generic Hurwitz number equalsmin i ( e i ( p + 1 − e i )) − p . This proves the first statement, and the second followsimmediately from Proposition 7.2 in the situation that e + e and e are not botheven. (cid:3) Remark . By using the results of [3] one can prove a stronger result than Theorem8.2. We say that a λ ∈ P k \ { , , ∞} is supersingular if it is a zero of the polynomial(4.5) and ordinary otherwise. Then the number of covers in characteristic p of type( p ; e , e , e , e ) branched at (0 , , λ, ∞ ) is h p ( p ; e , e , e , e ) if λ is ordinary and h p ( p ; e , e , e , e ) − λ is supersingular. To prove this result, one needs tostudy the stable reduction of the cover π : ¯ H → P λ of the Hurwitz curve to theconfiguration space. We do not prove this result here, as it would require too manytechnical details. References [1] Guy Boccara,
Cycles comme produit de deux permutations de classes donn´ees , Discrete Math-ematics (1982), no. 2-3, 129–142.[2] Irene Bouw, Construction of covers in positive characteristic via degeneration , Proceedingsof the AMS (to appear), arXiv:0709.2036.[3] ,
Pseudo-elliptic bundles, deformation data, and the reduction of Galois covers , Ha-bilitation thesis.[4] ,
Reduction of the Hurwitz space of metacyclic covers , Duke Mathematical Journal (2004), no. 1, 75–111, arXiv:math.AG/0204043.[5] Irene Bouw and Stefan Wewers,
Stable reduction of modular curves , Modularcurves and abelian varieties, Progress in Mathematics, vol. 224, Birkh¨auser, 2004,arXiv:math.AG/0210363, pp. 1–22.[6] ,
The local lifting problem for dihedral groups , Duke Mathematical Journal (2006),no. 3, 421–452, arXiv:math.AG/0409395.[7] Peter J. Cameron,
Finite permutation groups and finite simple groups , Bulletin of the LondonMathematical Society (1981), no. 1, 1–22.[8] Philippe Gille and Tam´as Szamuely, Central simple algebras and Galois cohomology , Cam-bridge Studies in Advanced Mathematics, no. 101, Cambridge University Press, 2006.[9] Alexandre Grothendieck,
Revetements etales et groupe fondamental , SGA, no. 1, Spring-Verlag, 1971.[10] Joe Harris and David Mumford,
On the Kodaira dimension of the moduli space of curves ,Inventiones Mathematicae (1982), 23–88. OME 4-POINT HURWITZ NUMBERS IN POSITIVE CHARACTERISTIC 23 [11] Richard Levingston and Donald E. Taylor,
The theorem of Marggraff on primitive permuta-tion groups which contain a cycle , Bulletin of the Australian Mathematical Society (1976),no. 1, 125–128.[12] Fu Liu and Brian Osserman, The irreducibility of certain pure-cycle Hurwitz spaces , AmericanJournal of Mathematics (2008), no. 6, 1687–1708, arXiv:math.AG/0609118.[13] Peter M¨uller,
Reducibility behavior of polynomials with varying coefficients , Israel Journal ofMathematics (1996), 59–91.[14] Brian Osserman, Rational functions with given ramification in characteristic p , CompositioMathematica (2006), no. 2, 433–450, arXiv:math.AG/0407445.[15] , Linear series and existence of branched covers , Compositio Mathematica (2008),no. 1, 89–106, arXiv:math.AG/0507096.[16] Michel Raynaud,
Sp´ecialisations des revˆetements en caract´eristique p >
0, Ann. Sci. ´EcoleNorm. Sup. (1999), no. 1, 87–126.[17] Bj¨orn Selander, Counting three-point G -covers with a given special G -deformation datum ,preprint.[18] Jean-Pierre Serre, Sur la topologie des variet´es alg´ebriques en caract´eristique p , Symposiuminternacional de topolog´ıa algebraica, 1958, pp. 24–53.[19] Stefan Wewers, Reduction and lifting of special metacyclic covers , Annales Scientifiques del’´Ecole Normale Sup´erieure (2003), no. 1, 113–138, arXiv:math.AG/0105052.[20] , Three points covers with bad reduction , Journal of the AMS16