One-point covers of elliptic curves and good reduction
James Phillips
math.AG math.NT


Wave breaking for the Stochastic Camassa-Holm equation
Paper submitted to the Physica D special issue Nonlinear Partial
Differential Equations in Mathematical Fluid Dynamics dedicated to Prof.
Edriss S. Titi on the occasion of his 60th birthday. Work partially
supported by the EPSRC Standard Grant EP/N023781/1.
    Dan Crisan
Department of Mathematics, Imperial College, London SW7 2AZ, UK. Email:
d.crisan@ic.ac.uk  and Darryl D Holm
Department of Mathematics, Imperial College, London SW7 2AZ, UK. Email:
d.holm@ic.ac.uk
     
========================================================================================================================================================================================================================================================================================================================





Raynaud gave a criterion for a branched G-cover of curves defined over a mixed-characteristic discretely valued field K with residue characteristic p to have good reduction in the case of either a three-point cover of ℙ^1 or a one-point cover of an elliptic curve. Specifically, such a cover has potentially good reduction whenever G has a Sylow p-subgroup of order p and the absolute ramification index of K is less than the number of conjugacy classes of order p in G. In the case of an elliptic curve, we generalize this to the case in which G has an arbitrarily large cyclic Sylow p-subgroup.





§ INTRODUCTION


Let f:Y → X be a G-Galois cover branched at r points. For the case of X = ℙ^1 and r=3, in <cit.>, Raynaud gives a criterion for such covers to have good reduction to characteristic p over mixed-characteristic discretely valued fields. Specifically, f has good reduction whenever p strictly divides the order of the Galois group, G, of this cover and the absolute ramification index of the field is less than the number of conjugacy classes of elements of order p in G. Motivating this was the surjection due to Grothendieck (<cit.>) of tame fundamental groups:


    pr: π_1(ℙ^1_K ∖{0,1,∞})^p-tame→π_1(ℙ^1_k ∖{0,1,∞})^tame.


Understanding the kernel of this map amounts to determining the p-tame three-point covers having good reduction.

In the same paper, Raynaud suggested that one may obtain a similar result for both three-point covers and one-point covers (that is to say, covers of elliptic curves branched over one point) whose Galois groups have arbitrarily large cyclic Sylow p-subgroups. Indeed, Raynaud's own result holds also in the one-point cover case, and the two cases often have analogous properties: for instance, both have the same étale fundamental group. Moreover, in <cit.>, Obus obtained this very result in the three-point case. We complete the analogy in the following:



Let G be a finite group with cyclic Sylow p-subgroup. Let k be an algebraically closed field of positive characteristic p, K_0 = Frac(W(k)), and K be a finite extension of K_0 such that the absolute ramification index of K is less than the number of conjugacy classes of order p in G. Let f: Y → E be a one-point G-cover defined over K, where E is an elliptic curve having good reduction. Then f has potentially good reduction.



Raynaud's proof relied on the study of the stable reduction of such covers. One obtains the stable model after at most a finite base change of the ground field. The Galois group of this extension is called the monodromy group of the cover and its p-part is called the wild monodromy. Raynuad showed that this wild monodromy of such an f is trivial, which sufficed to conclude potentially good reduction.

An analogous result holds for covers having cyclic Sylow p-subgroups, which is also due to Obus in <cit.>. He instead, however, concluded good reduction from the torsor structure of the cover at specific components. We follow a similar route. We must account for the fact that a cover of an elliptic curve need not ramify, so that we have a “borderline" case which does not arise in the case of ℙ^1. Such covers, however, are necessarily isogenies of elliptic curves, so this does not present much new difficulty.

In 2, we recall some facts regarding stable models of covers and their ramificaiton. In particular, we have Obus' (<cit.>) vanishing cycles formula, which places restrictions on the ramification based on the genus of the base curve and the number of branch points. In 3, we review the notion of multiplicative reduction. One can view reduction to characteristic p as “losing" information and multiplicative reduction as being a way of “recovering" some of this lost information. We also outline the construction of the auxiliary cover, a cover obtained from the original cover but whose Galois group will be of the form ℤ/p^s ⋊ℤ/m, where m is prime to p. We prove an analogue of a result of Wewers (<cit.>) on Jacobians of tame cyclic covers of elliptic curves to detect multiplicative reduction in this auxiliary cover. Finally, in 4, we adapt a result of Obus (<cit.>) to show that a one-point cover defined over a sufficiently small field (for instance, one described in Theorem 1) having bad reduction cannot have multiplicative reduction over certain components, allowing us to conclude good reduction.



 §.§ Notation


Throughout this paper, a one-point cover will refer to a cover of an elliptic curve branched over one point; we typically assume this point to be the origin. A G-cover of K-curves refers to a finite map of curves f: Y → X, where both X and Y are smooth, projective, and geometrically integral K-curves and the extension K(Y)/K(X) is Galois with group G. When G has a cyclic Sylow p-subgroup, P, we denote by m_G the value |N_G(P)|/Z_G(P)|.

Given an arbitrary scheme morphism f:Y → X and a finite group G having a subgroup H with H ⊆Aut(Y/X), Ind_H^G f: Ind_H^G Y → X is the map obtained by taking the disjoint union of [G:H] copies of Y and applying f to each copy.



§ ACKNOWLEDGEMENTS


The author would like to thank Andrew Obus for his invaluable comments, discussions, and patience throughout this project.



§ STABLE REDUCTION




 §.§ The stable model


Let R be a complete discrete valuation ring with characteristic 0 fraction field K and and positive characteristic p residue field k. Let X be a curve defined over K having good reduction; we denote a fixed good model of X by X_R. Let f: Y → X be a G-Galois cover defined over K with prime-to-p branching indices, where G is a finite group with cyclic Sylow p-subgroup and Y is a curve of genus at least 2; we suppose that the branch points of f are defined over K and specialize to distinct points of X_k = X_R ×_R k.

We know from Deligne and Mumford (<cit.>, Corollary 2.7) that there is a finite extension of K for which we have a stable model Y^st of Y over the valuation ring of this extension; that is to say, the special fiber Y̅ has only ordinary double points as singularities and any irreducible component of Y̅ having genus 0 contains at least 3 marked points n our situation, (that is, ramification points or points of intersection with the rest of Y̅).

We combine this with the work of Raynaud in <cit.> and Liu in <cit.> to obtain the stable model of f, f^st: Y^st→ X^st, where Y^st is the stable model of Y, as above, X^st = Y^st / G, and the ramification points on the generic fiber of f specialize to distinct smooth points of Y̅. The cover f^st is defined over the valuation ring R^st of a minimal finite extension K^st of K.

We call the special fiber f̅: Y̅→X̅ the stable reduction of f. When Y̅ is smooth, we say that f has potentially good reduction; elsewise, we say that f has bad reduction.

We can view X^st as a blow-up of X_R ×_R R^st. We call the strict transform of the special fiber under this blow-up the original component, which we denote X̅_0.



 §.§ Ramification on the special fiber


The action of G on Y^st induces an action on Y̅. By <cit.>, Proposition 2.4.11, the inertia groups of this action at the generic points of Y̅ are p-groups. If V̅ is an irreducible component of Y̅, we will denote by I_V̅ and D_V̅ the inertia and decomposition groups, respectively, at the generic point of V̅.

In X̅, the inertia groups at the generic point of an irreducible component U̅ are conjugate p-groups; if they have order p^i, we call U̅ a p^i component. We say that U̅ is étale when i=0 and inseparable otherwise. This comes from the fact that, since Y̅ is reduced, the inertia arises from an inseparable extension of residue fields at this generic point. By <cit.>, Corollary 2.11, if U̅ and U̅^' are two intersecting components of X̅, then either I_U̅⊆ I_U̅^' or I_U̅^'⊆ I_U̅ (we stress that this relies crucially on the Sylow p-subgroup being cyclic).

We call an irreducible component U̅ a tail if it intersects the rest of X̅ at exactly one point; otherwise, it is an interior component. If, in addition, a tail contains the specialization of a branch point, we call this tail primitive; otherwise, we say that it is new. This follows the convention established, for example, in <cit.>. We index the new étale tails and primitive étale tails by the sets B_new and B_prim, respectively.

Now let x be a point of intersection of two components U̅ and U̅^' with I_U̅^'⊆ I_U̅ and let y be a point lying above x, the intersection point of two components V̅ and V̅^' lying above U̅^' and U̅, respectively. Following <cit.>, we call the conductor of higher ramification (as in <cit.>, for instance) of the extension of complete discrete valuation rings 𝒪̂_U̅,x↪𝒪̂_V̅,y the effective ramification invariant at x, denoted σ_x. If x is on a tail X̅_b we will often simply write σ_b for σ_x.

We recall the following, which places some bounds on these invariants; it is originally found in <cit.> in the form of Lemmas 2.20 and 4.2:




The effective ramification invariants σ_b are positive and lie in 1/m_Gℤ. If σ_b is a new tail, we have σ_b ≥ 1+1/m_G



In the case in which G = ℤ/p^s ⋊ℤ/m and f: Y → E is a G-cover branched at the points P_1, …, P_r, r > 1, where E is an elliptic curve having good reduction over R, we can be more precise as to the fractional parts of the σ_b, to be denoted ⟨σ_b ⟩. We can decompose f into an étale ℤ/p^n-cover Y → Z and a ℤ/m-cover g: Z → E given, birationally, by z^m = f_r · f_u, where f_r and f_u are rational functions on E corresponding to, respectively, a divisor in the form ∑_i=1^r a_i P_i and an m-divisible divisor on E. The following is a generalization of <cit.>, Proposition 1.8, originally found in <cit.>, Proposition 3.6. It was originally proven in the case in which the base curve is ℙ^1, but can be proven identically in the case in which the base curve is an elliptic curve E.




Let f: Y → E be a G = ℤ/p^s ⋊ℤ/m-cover branched in r ≥ 1 points and let f^ss be a fixed semistable model for f; let X̅_b be an étale tail of E̅^ss containing the specialization of a unique branch point x_i and intersecting an inseparable component of E̅^ss. Then ⟨σ_b ⟩ = a_i/m, where the former denotes the fractional part of σ_b.



In <cit.>, Corollary 3.4.4, we find the vanishing cycles formula, which relates the invariants σ_b and the genus of X in the case in which G has a Sylow p-subgroup of the form ℤ/p. This formula is generalized to the case in which G has a cyclic Sylow p-subgroup in <cit.>, Theorem 3.14; we recall this version below:

 


Let f: Y → X be a G-Galois cover branched at r points having bad reduction, where G has cyclic Sylow p-subgroup; let g be the genus of X; let B_et be an indexing set for the étale tails, each having effective ramification invariant σ_b. Then we have


    2g - 2 + r = ∑_b ∈ B_et (σ_b - 1)




In particular, when f is an elliptic curve, we have 


    r = ∑_b ∈ B_new (σ_b - 1) + ∑_b ∈ B_primσ_b





§ REDUCTION TYPES


We maintain the assumptions of 2; namely, that f: Y → E is a G-cover branched at r ≥ 1 points with prime-to-p branching indices, G having a cyclic Sylow p-subgroup, and Y having genus at least 2. Now we will consider the torsor structure of the reduction of one-point covers f: Y → X = E. These will eventually play a key part in the proof of the main theorem.



 §.§ Multiplicative reduction


Supposing that f has bad reduction, we fix a semistable model f^ss: Y^ss→ E^ss. Let U̅ be an inseparable component of E̅^ss and let V̅ be a component of Y̅^ss lying above W̅ having decomposition and inertia group D_V̅ and I_V̅, respectively. The group D_V̅ has a normal subgroup of order p, since I_V̅ is normal in D_V̅ and I_V̅ is cyclic. Then <cit.>, Corollary 2.4, implies that D_V̅ has a maximal normal prime-to-p subgroup with D_V̅/N ≅ℤ/p^j ⋊ℤ/m_D_V̅ for some j>0; we denote by P the Sylow p-subgroup of D_V̅/N.

Let η be the generic point of V̅/N. The group P acts on 𝒪̂_Y_R/N, η; we denote this ring by B and its subring fixed by P by A. The action of P on the residue field of B is trivial, so that B is a totally ramified extension of A. We say that f has multiplicative reduction when B/A has the structure of a μ_p^j-torsor.

Following <cit.>, we have an equivalent formulation which will be of use to us. We say that B/A, as above, is of μ_p^j-type if Frac(B) is a Kummer extension of Frac(A) given by the adjunction of a p^j^th root of a unit in B that does not reduce to a p^th power in the residue field of A. Similarly, we say that B/A is potentially of μ_p^j-type if the base-change by some field, K^', is of μ_p^j-type. The former is equivalent to B/A having the structure of a μ_p^j-torsor.

Now we assume that G is of the form ℤ/p^s ⋊ℤ/m and is branched at the points P_1, …, P_r, r>1. As we saw in 2, we can decompose f into an étale ℤ/p^s-cover Y → Z and a branched ℤ/m-cover, Z → E.

Let χ: ℤ/m →μ_m(K) be a character of order m. We have the reduction χ̅ of χ, which gives a map ℤ→𝔽_p^×. We then may use χ̅ to obtain a semidirect product, ℤ/p ⋊ℤ/m. The p-cyclic cover above Z is étale, so corresponds to a nontrivial class in H^1_et(Z, ℤ/p)_χ. Letting J_Z denote the Jacobian of Z, we have a canonical isomorphism H^1_et(Z, ℤ/p)_χ≅ J_Z[p]_χ(-1). Moreover, we have that J_Z[p]_χ(-1) = Hom_𝔽_p(μ_p((̅K)), J_Z[p]_χ). Choosing a p^th root of unity in a fixed algebraic closure K̅ of K, we identify J_Z[p]_χ(-1) with J_Z[p]_χ. We have the following result concerning the structure of the Jacobian, Jac(Z̅)[p]_χ̅, of the special fiber Z̅, which we denote J̅[p]_χ̅:

 


Let g̅: Z̅→E̅ be an m-cyclic cover given, birationally, by the equation z^m = f_r · f_u, where f_r and f_u are rational functions on E corresponding to, respectively, a divisor in the form ∑_i=1^r a_i P_i and an m-divisible divisor on E, ∑_j=1^ρ m b_j Q_j with branch locus {P_1, …, P_r }, r ≥ 2, satisfying ∑ a_i = m. Then J̅[p]_χ̅≅( ℤ/p )^r-1×μ_p.



 

The action of ℤ/m on g̅_*𝒪_Z̅ arising from that on Z̅ gives us the decomposition

    g̅_*𝒪_Z̅ = ⊕ℒ_ψ̅,
 
where ψ̅ runs through all characters ℤ→ k^×, into isotypical line bundles.

The rational function z will have zeros of order b_j at each Q_j and a pole of order 1/m( ∑ a_i + m ∑ b_j ) = 1+∑ b_j at ∞. This function is a rational section of χ̅, so we get deg(ℒ_χ̅) = ∑ b_j - (1+∑ b_j) = -1. Similarly, z^m-1 has zeros of order a_i - 1 at each x̅_i, zeros of order b_j(m-1) at each P_j and a pole of order (1+∑ b_j)(m-1) at ∞, so that deg(ℒ_χ̅^-1) = (m-r) + (m-1)∑ b_j - (1+∑ b_j)(m-1) = 1-r. So H^1(Z̅, 𝒪_Z̅)_χ̅ and H^1(Z̅, 𝒪_Z̅)_χ̅^-1 have respective dimensions 1 and r-1.

Combining <cit.>, Corollary 5.11, and <cit.>, Theorem 2.1, we may compute the rank of J̅[p]_χ̅ as a group scheme as 

    _k(H^1_dR(Z̅)_χ̅) = _k(H^1(Z̅, 𝒪_Z̅)_χ̅) + _k(H^0(Z̅, Ω_Z̅/k)_χ̅) = r

since H^0(Z̅, Ω_Z̅/k)_χ̅ is dual to H^1(Z̅, 𝒪_Z̅)_χ̅^-1.

There is a canonical k-linear isomorphism Lie(J[p]) ≅ H^1(Z̅, 𝒪_Z̅); it is compatible with the ℤ/m action, so we also have an isomorphism of χ̅-eigenspaces. So Lie(J[p]_χ̅) has dimension 1 and J[p]_χ̅ is isomorphic to (ℤ/p )^r-1× G, where G is either μ_p or α_p. Since J̅[p]_χ̅^-1 is dual to J̅[p]_χ̅, we also have that J̅[p]_χ̅^-1 is isomorphic to (μ_p )^r-1× G^', where G^' will be dual to G. Since Lie(J[p])_χ̅^-1 has dimension r-1, we must have G^' = ℤ/p, and so J̅[p]_χ̅ = ( ℤ/p )^r-1×μ_p, as we wanted.



We say that f: Y → E is of multiplicative type when ∑_i=1^r a_i = m, as is the case in the previous proposition. The following is analogous to <cit.>, Proposition 3.7 and gives the reasoning behind the terminology multiplicative type:

 


Let f: Y → X = E be a G = ℤ/p^s ⋊ℤ/m-Galois cover of multiplicative type. Then any semistable model f^ss: Y^ss→ X^ss of f has multiplicative reduction above every inseparable component of X̅^ss.

 



If s=1, then by Proposition <ref> and <cit.>, p. 190, the p-cyclic cover Y → Z reduces to a μ_p-cover above the generic point of each component, and so will have multiplicative reduction above every component.

If s>1, then f has a quotient ℤ/p ⋊ℤ/m-cover, h, which will also be of multiplicative type, so that any semistable model of h will have multiplicative reduction over every component of X̅^ss. In particular, every component of X̅^ss will be a p^s-component for f. Let K^ss be a field of definition for f^ss, a semistable model for f; since K has algebraically closed residue field, <cit.>, Lemma 2.2 implies that f^ss has multiplicative reduction over all of X̅, as we wanted.





 §.§ The auxiliary cover
 Raynaud's proof of potentially good reduction in <cit.> the case of a G-cover having a ℤ/p Sylow p-subgroup relies on the construction of the auxiliary cover. Obus (<cit.>) generalized this construction to the case of a cyclic Sylow p-subgroup, whose properties we recall here (cf. <cit.>, <cit.>):




Over some finite extension K^' of K, we construct the auxiliary cover of f, a G^aux-Galois cover f^aux: Y^aux→ X^aux, with G^aux≤ G, having the following properties:




  (i) (X^aux)^ss = X^st and X̅^aux = X̅.


  (ii) There is an étale neighborhood Z of the union of the inseparable components of X̅ with the property that, as covers, f^st×_X^st Z ≅(Ind_G^aux^G (f^aux)^st)×_X^st Z.


  (iii) The branch locus of f^aux consists of a branch point x_b of index m_b for each étale tail X̅_b of X̅ for which m_b > 1. If X̅_b is primitive, so that it contains a branch point of f, x_b is this corresponding branch point. If X̅_b is new, x_b specializes to a smooth point.


  (iv) Given an étale tail X̅_b of X̅ and an irreducible component V̅_b of Y̅^aux above X̅_b with effective ramification invariant σ_b^aux, we have σ_b^aux = σ_b.


  (v) If N is the maximal prime-to-p normal subgroup of G^aux, then G^aux/N ≅ℤ/p^s ⋊ℤ/m_G^aux with s ≥ 1.





We denote by G^st the quotient in (v); we then call the G^str-quotient cover of f^aux the strong auxiliary cover, denoted f^str: Y^str→ X^str. The upper-numbered filtration for the  higher ramification groups is unaffected by taking quotients, so both the auxiliary cover and the strong auxiliary cover will have the same effective ramification invariants as the original cover. By passing to either cover, we allow ourselves the convenience of studying the ramification with a simplified Galois group at the expense of a potentially larger branch locus. 

Strong auxiliary covers arising from one-point covers have the following property:




If the strong auxiliary cover of a one-point cover with prime-to-p branching ramifies, it is of multiplicative type.





Let X̅_b be an étale tail of X̅^str with effective ramification invariant σ_b. We have seen that passing to the strong auxiliary cover preserves effective ramification invariants, so by Proposition <ref> and Theorem <ref>, ∑⟨σ_b ⟩ = 1. If there were only one étale tail, this would mean that σ_b = 1 and f^str would be unramified, a contradiction. So we assume there are at least two étale tails. Then by Lemma <ref>, each σ_b is a non-integer, so that each étale tail has m_b > 1. So, applying Proposition <ref>, we see that ∑⟨σ_b ⟩ = ∑a_i/m = 1, so that ∑ a_1 = m and f^str is of multiplicative type.






Let f: Y → E be a branched one-point cover with bad reduction and prime-to-p branching indices. Then the stable model of f has multiplicative reduction over the original component.





By Propositions <ref> and <ref>, any semistable model of the strong auxiliary cover, f^str, has multiplicative reduction over the original component. Since f^str is a prime-to-p quotient of the auxiliary cover f^aux, any semistable model of f^aux will also have multiplicative reduction over the original component. The cover, f, will be isomorphic to the disjoint union of copies of f^aux over an étale neighborhood of the original component, so that the stable model of f will also have multiplicative reduction over the original component, as we wanted.





§ POTENTIALLY GOOD REDUCTION


Now let f: Y → X be a cover as in 2 with the additional requirement that X ≅ E, an elliptic curve having good reduction over K.




Let f: Y → E be an étale cover defined over K of the elliptic curve E, where E has good reduction over K. Then f has good reduction over K.





By the Riemann-Hurwitz formula, Y must also be an elliptic curve, so f is an isogeny. Isogenous elliptic curves either both have good or bad reduction over a given field (see, for instance, <cit.>, Corollary 7.2); since E has good reduction, so must Y, so that f has good reduction, as we wanted.






Let f: X → E be one-point cover with abelian Galois group G. Then f is, indeed, an étale cover f: X → E.





The étale fundamental group π_1(E ∖{O}) of E ∖{O} has the presentation {a,b,c   |   aba^-1b^-1c=1}. The map π_1(E ∖{O}) → G factors via the abelianization π_1(E ∖{O}) →π_1(E ∖{O})^ab = {ab   |   aba^-1b^-1 = 1} = π_1(E), so that f corresponds to an element of π_1(E). So the map f is, indeed, étale, as we wanted.



In particular, one-point covers having abelian Galois groups must have good reduction.

The following is analogous to <cit.>, Lemma 4.5:




Let f: Y → E be a ramified cover, branched at r ≥ 1 points, with tame branching indices; let V̅ be an irreducible component above the original component above the original component with decomposition group D_V̅. Then m_D_V̅ > 1.





It suffices to work with the strong auxiliary cover f^str, which, in light of Lemma <ref>, we assume has more than one branch point. It will be enough to show that the decomposition group of a component above the original component is not a p-group.

The cover f^str has a ℤ/m_G_str-quotient cover given (as, for instance, in 2) birationally by z^m = f_r · f_u, where the rational function f_r corresponds to the divisor ∑_1^ρ a_i P_i on E, with 0 < a_i < m. It then suffices to show that the reduction of f_r is not an m^th power in k(X).

The branch points of f specialize to distinct points on the special fiber, so we have at least r different residue classes among them. By Proposition <ref> and Theorem <ref>, ∑_i=1^ρ a_i = m · r, so that some subset of the a_i satisfy 0 < ∑ a_i < m. So the factor f_r does not reduce to an m^th power, as we wanted.



The following is the key step in the proof of good reduction and is analogous to <cit.>, Proposition 5.1. We recall the statement; the proof is identical in our situation, with Lemma <ref> assuming the role of <cit.>, Lemma 4.5.




Let G be a finite group with nontrivial cyclic Sylow p-subgroup; let k be an algebraically closed field of characteristic p; let K_0 = Frac(W(k)) and let K be a finite extension with e(K) < p-1/m_G, where e(K) is the absolute ramification index of K; let f: Y → E be a G-cover defined over K, where E is an elliptic curve having good reduction over the valuation ring R of K, branched at the distinct K-points {x_1, …, x_r }, with r ≥ 1; suppose that the branch points of f specialize to distinct points on the special fiber of the good model of E over R. If f has bad reduction, then the stable model of f does not have multiplicative reduction over the original component.



We then obtain the following by applying this to the r=1 case.



Let G be a finite group with cyclic Sylow p-subgroup. Let K be as in Proposition <ref> and let f: Y → E be a one-point G-cover defined over K. Then f has potentially good reduction.





If f is unbranched, then f has good reduction by Lemma <ref>. Otherwise, <cit.>, Lemme 4.2.13, implies that the branching indices of f are tame. If f were to have bad reduction, Corollary <ref> would imply that f has multiplicative reduction over the original component, contrary to Proposition <ref>. So f has potentially good reduction.



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