Stable Maps of Curves

Robert F. Coleman

Received: October 31, 2002 Revised: February 21, 2003

Abstract. Let h: X → Y be a finite morphism of smooth connected complete curves over C_{p}. We show h extends to a finite morphism between semi-stable models of X and Y.

2000 Mathematics Subject Classification: 11G20 Keywords and Phrases: Stable models, wide opens

Let p be a prime. It is known that if C is a smooth proper curve over a complete subfield K of C_{p}, there exists a finite extension L of K in C_{p} and a model of the base extension of C to L over the ring of integers, R_{L}, of L whose reduction modulo the maximal ideal has at worst ordinary double points as singularities. In fact, if g(C), the genus of C, is at least 2 or g(C) = 1 and C has a model with good reduction, there is a minimal such model, which is called the stable model. Indeed, if L′ is any complete extension of L in C_{p}, the base extension of a stable model over R_{L} is the stable model over R_{L′}.

Liu and Lorenzini showed [L-L; Proposition 4.4(a)] that a finite morphism of curves extends to a morphism of stable models, but the extension is not in general finite. E.g., Edixhoven has show that the natural map from X_{0}(p^{2}) to X_{0}(p) does not in general extend to a finite morphism of stable models [E] (see also [C-M]). However, we show,

Theorem. Suppose h: X → Y is a finite morphism of smooth connected complete curves over C_{p}. Then there are semi-stable models X and Y of X and Y over the ring of integers of C_{p} such that h extends to a finite morphism from X to Y.

(We work over C_{p} to avoid having to worry about base extensions and because reduced affinoids over C_{p} have reduced reductions.)

In fact, when X has a stable model (i.e., g(X) ≥ 2 or g(X) = 1 and X has good reduction) and either X/Y is Galois, or the model has irreducible reduction and Y has a stable model, one can take X to be the stable model for X. In the latter case Y will be the stable model for Y, which, a fortiori, will have irreducible reduction.

Abbes has informed us that this result also follows from results of Raynaud (in particular Proposition 5 of [R] (and its corollary)).

We say such a morphism is semi-stable, and stable if it is the minimal object in the category of semi-stable morphisms from X to Y (which may not exist). Terminology and Notation

If Z is a rigid space A(Z) will denote the ring of analytic functions on Z and A^{0}(Z) the subring of functions whose spectral norm is bounded by 1.

If X is an affinoid over C_{p}, \overline{X} = SpecA^{0}(X)/𝔪A^{0}(X), where 𝔪 is the maximal ideal of R_{p} and if x ∈ \overline{X}(F̄_{p}), R_{x} will denote the corresponding residue class in X. (Residue classes are also called formal fibers.)

By a regular singular point on a curve we mean a singular point which is an ordinary double point. If C is a curve, let S(C) denote the set of irregular singular points on C.

1. Wide Opens In this section, we review and extend the results on wide open spaces discussed in [RLC].

A (smooth one-dimensional) wide open is a rigid space conformal to C −D where C is a smooth complete curve and D is a finite disjoint union of affinoid disks in C, which contains at least one in each connected component. A wide open disk is the complement of one affinoid disk in P^{1} (it is conformal to B(0, 1)) and a wide open annulus is conformal to the complement of two disjoint such disks (it is conformal to A(r, 1) where r ∈ |C_{p}|, 0 < r < 1).

An underlying affinoid Z of a wide open W is an affinoid subdomain Z of W such that W \Z is a finite disjoint union of annuli none of which is contained in an affinoid subdomain of W. An end of W is an element of the inverse limit of of the set of connected components of W \Z where Z ranges over subaffinoids of W.

We slightly modify the definition of basic wide open given in [RLC] and say a wide open W is basic if it has an underlying affinoid Z such that \overline{Z} is irreducible and has at worst regular singular points.

Suppose X is a smooth one dimensional affinoid over C_{p} and x ∈ \overline{X}. Because A^{0}(R_{x}) is the completion of A^{0}(X) at x, we have,

Lemma 1.1. Then x is a smooth point of \overline{X} if and only if R_{x} is a wide open disk and a regular singular point if and only if R_{x} is a wide open annulus.

We have the following generalization of Proposition 3.3(ii) of [RLC],

Lemma 1.2. The residue class, R_{x}, is a connected wide open and its ends can naturally be put in 1-1 correspondence to the branches of \overline{X} through x.

Proof. That R_{x} is connected is a consequence of Satz 6 of [B].

Theorem A-1 of [pAI] and its proof naturally generalizes to semi-dagger algebras. These are quotients of the rings of series ∑_{I,J} a_{I,J}x^{I}y^{J} in K[x_{1}, …, x_{n}, y_{1}, …, y_{m}], where K is a complete non-Archimedean valued field, such that there exists r_{∈R} > 1 so that

lim_{s(I,J)→∞} |a_{I,J}|r^{s(J)} = 0,

where I and J range over Z_{≥0}^{n} and Z_{≥0}^{m} and s(M), where M is a multi-index, is the sum of its entries. What this implies in our context is that if R is an affinoid over C_{p} whose reduction is equal to the normalization of \overline{X} and S is the set of points of \overline{R} above T, the singular points of \overline{X}, then the rings

lim_{→}_{M} A(R\M) and lim_{→}_{N} A(X\N)

are isomorphic, where M ranges over the subaffinoids of ⋃_{s∈S} R_{s} and N ranges over the subaffinoids of ⋃_{x∈T} R_{x}. Since ⋃_{s∈S} R_{s} is a union of wide open disks which correspond to the set, B, of branches of \overline{X} through points in T, this, in turn, implies that there exists a subaffinoid N of ⋃_{x∈T} R_{x} such that ⋃_{x∈\overline{T}} R_{x} − N is a finite union of wide open annuli which correspond to the elements of B. One can now glue affinoid disks to ⋃_{x∈T} R_{x} to make a smooth complete curve, using [B; Satz 6.1] and the direct image theorem of [K], as in the proof of Proposition 3.3 of [RLC]. The result follows.

It follows from Lemmas 3.1 and 3.2 of [RLC] that,

Lemma 1.3. if A and B are disjoint wide open annuli or disjoint affinoids in a smooth curve C over C_{p}, then A is disconnected from B in C.

If W is a wide open space

H_{DR}^{1}(W) = Ω_{W}^{1} /dA(W),

where Ω_{W}^{1} is the A(W)-module of rigid analytic differentials on W. It follows from Theorem 4.2 of [RLC] that H_{DR}^{1}(W) is finite dimensional over C_{p}.

Lemma 1.4. Suppose f: W → V is a finite morphism of wide opens. Then, if W is a disk or annulus, the same is true for V.

Proof. Suppose W is a disk and f has degree d. Then V has only one end. Suppose ω is a differential on V. Then f^{∗}ω = dg for some function g ∈ A(W) since dim H_{DR}^{1}(W) = 0, in this case. Hence ω = d Tr (g/d). Thus H_{DR}^{1}(V) = 0. Let C be a proper curve obtained by glueing a wide open disk D to V along the end, as in the proof of Proposition 3.3 of [RLC ]. From the Meyer-Vietoris long exact sequence, we see that C has genus zero and as B := D\V is an affinoid disk V = C\B is a wide open disk.

The argument in the case where W is an annulus is similar, except one has to use residues.

Proposition 1.5. Suppose f: X → Y is a finite map of smooth one dimensional affinoids over C_{p}. Then, if the reduction of X has only regular singular points, the same is true of Y.

Proof. We know the map f induces a finite morphism f̄: X̄ → Ȳ. Let y be a point of Ȳ. Let x ∈ X̄ such that f̄(x) = y. Then f restricts to a finite morphism R_{x} → R_{y}. But R_{x} is a disk or annulus. It follows from Lemma 1.2 that R_{y} is a wide open and hence by Lemma 1.4 is a disk or annulus, as well. Hence y is either smooth or regular singular by Lemma 1.1.

This implies the well known result that if h: X → Y is a finite morphism of curves and X has good reduction so does Y. In fact, it implies the result of Lorenzini-Liu, [L-L; Corollary 4.10], that, in this case if g(Y) ≥ 1, h extends to a finite morphism between the unique models of good reduction. It also implies that if X has a stable model with irreducible reduction, so does Y.

Lemma 1.6. Suppose ϕ: X → Y is a non-constant rigid morphism of smooth one dimension affinoids over C_{p}. Suppose G is a finite group acting on X such that ϕ^{σ} = ϕ for σ ∈ G and \overline{X} = ⋃_{σ⊂G} V^{σ} where V is an irreducible component of \overline{X}. Then ϕ surjects onto an affinoid subdomain of Y.

Proof. Let x ∈ X. Because ϕ is non-constant we can find an element f of A(Y) such that f(ϕ(x)) = 0 and |ϕ^{∗}f|_{X} = 1. We can and will replace Y with the affinoid subdomain \{y ∈ Y: |f(y)| ≤ 1\}. Then \overline{ϕ}|_{V} is non-constant. Since \overline{ϕ}\overline{X} = \overline{ϕ}V and V is irreducible, \overline{ϕ} factors through the inclusion of an irreducible component S of \overline{Y}. Let S^{0} be the complement in S of the other irreducible components of \overline{Y}. Then, Z = red^{−1}S^{0} is an affinoid subdomain of Y whose reduction is S^{0}. Let X′ be the affinoid subdomain of X, ϕ^{−1}Z. This is just X minus a finite number of residue classes stable under the action of G so its reduction is the union of the G-conjugates of V′ = V \\overline{ϕ}^{−1}(S\S^{0}) which is irreducible. Suppose s ∈ \overline{ϕ}\overline{X} ∩ S^{0}. We claim that R_{s} ⊂ ϕ(X).

Suppose y_{0} ∈ R_{s}\ϕ(X). Since the class group of Z is torsion, there exists an h ∈ A(Z) such that y_{0} is the only zero of h. Because \overline{Z} is irreducible, we can also suppose |h|_{Z} = 1. Since h(y_{0}) = 0, it follows that |h(y)| < 1 for y ∈ R_{s}. Let g = ϕ^{∗}h ∈ A^{0}(X′). If y_{0} ∉ ϕ(X), 1/g ∈ A(X′) but |1/g|_{X′} = |c| > 1, for some c ∈ C_{p}, since s is in the image of \overline{ϕ}\overline{X}′. However,

|g(1/cg)|_{X′} = |c^{−1}| < 1 = |g|_{X′}|(1/cg)|_{X′}.

This implies \overline{g}_{V} = 0 or \overline{(1/cg)}_{V} = 0, but as X′ = ⋃_{σ∈G} V′^{σ}, this implies the contradiction that \overline{g} = 0 or \overline{(1/cg)} = 0.

We will finish the proof by showing X = X′.

Let Y′ be the affinoid obtained by glueing in disks to red^{−1}S at the ends of the wide open red^{−1}S\S^{0} corresponding to irreducible components of \overline{Y} distinct from S. The reduction of this affinoid is naturally isomorphic to S. Then as ϕ factors through the inclusion of red^{−1}S in Y we naturally obtain a morphism ϕ′: X → Y′. Since by construction, for each point s ∈ S\S^{0}, there is a point in the residue class of Y′ not in the image of X the above argument implies no point in this residue class is in the image of X and so ϕ′(X) in the reduction inverse in Y′ of S^{0}. As this latter is naturally isomorphic to Z, X = X′.

Suppose W is a wide open annulus. If σ: W → W is a rigid analytic morphism, define ρ(σ) by

ρ(σ) Resω = Res σ^{∗}ω.

The restriction of ρ to the group of rigid analytic automorphisms of W is a homomorphism from Aut(W) onto {±1}. We say σ is orientation preserving if ρ(σ) = 1.

Lemma 1.7. Suppose G is a finite group of rigid automorphisms of the wide open annulus W = A(r, 1) of order m. Then there is a rigid morphism ϕ: W → V of degree m such that A(W)^{G} = ϕ^{∗}A(V), where V = A(r^{m}, 1) if G is orientation preserving and V = A(B(0, 1)) if not.

Proof. First, if σ ∈ G

σ^{∗}T = c_{σ}T^{ρ(σ)}h_{σ}(T),

where h_{σ}(T) ∈ A(W), |h_{σ}(t) − 1| < 1, for t ∈ W, and c_{σ} ∈ C_{p},

|c_{σ}| = \{ \begin{matrix}1 & if ρ(σ) = 1 \\ r & if ρ(σ) = −1 \end{matrix} .

Let G^{o} = Ker ρ and n = |G^{o}|. Let S = ∏_{τ∈G^{o}} τ^{∗}T. Then

S(T) = T^{n}g(T),

where |g(t)| = 1. Let α: W → A(r^{n}, 1), be the map

t ↦ S(t).

It is easy to see this map has degree n and R := α^{∗}A(A(r^{n}, 1)) ⊆ A(W)^{G}. In particular, R is an integral domain and its fraction field is K^{G} where K is the fraction field of A(W). Since, R and A(W) are Dedekind domains, it follows that R = A(W)^{G_{0}}. If G is orientation preserving, G = G^{o} and taking ϕ = α completes the proof, in this case.

Suppose now G is not orientation preserving. Then G/G^{o} has order 2. Using, the result of the last paragraph we can replace W with A(r^{n}, 1) and assume G^{o} is trivial. Let G = {1, σ} and

U(T) = T + σ^{∗}T = T + c_{σ}T^{−1}h_{σ}(T).

Now, if we define ϕ: W → B(0, 1) to be the morphism

t ↦ U(t),

we can apply the same argument, as above, to complete the proof.

Remark. One can show:

Proposition. Suppose p is odd G is a finite group of automorphisms of A(r, R). Then there is a natural homomorphsm of G into Aut G_{m}^{\overline{F}_{p}} whose kernel is the unique p-Sylow subgroup of G^{o}. Moreover, the exact sequence,

1 → G^{o} → G → G/G^{o} → 1,

splits.

For example: Suppose p = 3, 1 > |r| > |27| and V = A(r, 1) . Let s be the parameter on A^{1}. Then the integral closure of A(V) in the splitting field of X^{3} + sX = s over K(V) is the ring of analytic functions on an annulus W which is an étale Galois cover of V. If G is the Galois group, G = G^{o} ≅ S_{3}.

2. Semi-stable Coverings A semi-stable covering of a curve C is a finite admissible covering D of C by connected wide opens such that

(i) if U ≠ V ∈ D, U ∩ V is a finite collection of disjoint wide open annuli, (ii) if T, U, V ∈ D are pairwise distinct, T ∩ U ∩ V = ∅. (iii) for U ∈ D, if

U^{u} = U\( ⋃_{V ∈D} ; V ≠U V ),

U^{u} is a non-empty affinoid whose reduction is irreducible and has at worst regular singular points.

In particular, if U ∈ D, U is a basic wide open and U^{u} is an underlying affinoid of U. We let E(U) denote the set of connected components of U\U^{u}. These are all wide open annuli.

Proposition 2.1. Semi-stable models of C whose reductions have at least two components correspond to semi-stable covers of C.

Proof. Suppose C is a semi-stable model for C whose reduction \overline{C} has at least two components. Let I_{C} denote the set of irreducible components of \overline{C}. If Z ∈ I_{C} let Z^{0} = Z\ ⋃_{A∈I_{C}} ; A≠Z A and W_{Z} := red^{−1}Z. As every singular point of

\overline{C} is regular it follows from Lemma 1.2 that W_{Z} is a basic wide open with underlying affinoid red^{−1}Z^{0} and {W_{Z}: Z ∈ I_{C}} is a semi-stable cover.

Conversely, suppose D is a semi-stable cover of C. For U, V ∈ D, let Z_{U} = Spf A^{0}(Z) and Z_{U,V} = Spf (U ∩ V). Then the formal schemes Z_{U} glue by means of the glueing data

Z_{U,V} → Z_{U} ∐ Z_{V}

into a model S_{D} of C.

If we have semi-stable coverings D_{X} and D_{Y} such that for every W, V ∈ D_{X}, h(W) ∈ D_{Y} and there exist W′, V′ ∈ D_{X} such that h(W)∩h(V) = h(W′ ∩V′), then f extends to a finite morphism from S_{D_{X}} to S_{D_{Y}}. We say h induces a finite morphism of semi-stable covers from D_{X} to D_{Y}.

3. Proof of Theorem First, let h′: X′ → Y be the Galois closure of h with Galois group G. Let D be a semistable cover X’ of such that Y ∉ C where

C = {h′(U): U ∈ D}.

Then we claim C is a semi-stable cover of Y. Clearly C is a finite admissible open cover. By Lemma 1.7, if W ∈ D and A ∈ E(W), h′(A) is a wide open disk or annulus. Since h′(W) ≠ Y, h′(A) cannot be a disk for all A ∈ E(W). It follows by a glueing argument, as in the proof of Lemma 1.2, that h′(W) is a connected wide open. Now suppose U, V ∈ D and h′(W) ≠ h′(V). We must show h′(W) ∩ h′(V) is a finite union of disjoint annuli. First, we remark that h′(W^{u}) and h′(V^{u}) are disjoint affinoids in Y, using Lemma 1.6. Suppose A is a component of W ∩ V so A ∈ E(W) ∩ E(V). Suppose (x_{n}) is a sequence of points in A. If x_{n} → W^{u}, h′(x_{n}) → h′(W^{u}) and if x_{n} → V^{u}, h′(x_{n}) → h′(V^{u}). It follows that h′(A) is an annulus. Also, we know that if U is a connected component of h′(W)∩h′(V), U = ⋃_{σ∈S} h′(A_{σ}) where the A_{σ} are in E(W)∩E(V^{σ}), for some subset S of G. Now it follows from Lemma 1.3 that if S has more than one element and σ ∈ S there must be a τ ∈ S such that τ ≠ σ and A_{σ} ∩ A_{τ} ≠ ∅. Then A_{σ} ∪ A_{τ} is an annulus, arguing as in the proof of Corollary 3.6a of [RLC] (A_{σ} ∪ A_{τ} ≠ Y since W^{u} ∩ (A_{σ} ∪ A_{τ}) = ∅). The fact that A_{σ} and A_{τ} are connected to both W^{u} and V^{u} implies A_{τ} = A_{σ} ∪ A_{τ} = A_{σ}. We conclude that all the A_{σ} equal U, for σ ∈ S and so U is a wide open annulus.

Suppose U, V, W ∈ D are such that h′(U), h′(V), h′(W) are distinct. If y ∈ h′(U) ∩ h′(V) ∩ h′(W), there exist σ, τ ∈ G and x ∈ U ∩ V^{σ} ∩ W^{τ} such that y = h′(x). But this implies U, V^{σ}, W^{τ} are not distinct which in turn implies h′(U), h′(V), h′(W) are not distinct.

We must show for U ∈ D, h′(U)^{u}, which equals

h′(U)\( ⋃_{V ∈C} ; V ≠h′(U) V ),

is an affinoid whose reduction is irreducible and only has regular singular points. Now,

h′(U)^{u} = h′( ⋃_{σ∈G} (U^{σ}\ ⋃_{A∈E(U^{σ})∩E(V)} ; V ∈D,V ≠U^{τ} ,τ∈G A)

and also

= h′(U^{u}) ∪ ⋃_{A∈E(U)∩E(U^{σ})} ; σ_{≠1}∈G, h′(A).

It follows from the first equality that h′(U)^{u} is an affinoid using Lemma 1.6 and Proposition 3.3 of [RLC]. Its reduction is irreducible as the reduction U^{u} is, and from Proposition 1.5 it has at worst only regular singular points. Finally, since all the h′(A) are disks or annuli by Lemma 1.7 whose ends are connected to h′(U^{u}), h(U)^{u} is an affinoid and these h′(A) must, by Lemma 1.1, be smooth or regular singular classes of h′(U)^{u}, and thus, in particular, h′(U)^{u} must have irreducible reduction. Thus C is a semi-stable cover and clearly h′ induces a finite map of semi-stable covers from D to C.

We also know X′/X is Galois and if r: X′ → X is the corresponding morphism,

E := {r(U): U ∈ D}

does not contain X so is a semi-stable cover of X and r induces a finite map of semi-stable covers from D to E. It follows that h induces a finite map of semistable covers from E to C and hence extends to the corresponding semi-stable models.

Now we must explain how we can find a cover D of X′ with the required properties. If X′ has a stable model X, then X is preserved by G. Let D be a wide open disk in X′ such that D^{σ} ∩ D = ∅ for all σ_{≠1} ∈ G and B an affinoid ball in D. Let X′ be the minimal semi-stable refinement of X such that no two elements of {D^{σ}: σ ∈ G} are contained in the same residue class. Let E = ⋃_{σ∈G} B^{σ}. Then we can take for D,

\{(red^{−1}Z)\E: Z is an irreducible component of \overline{X}′\} ∪ \{D^{σ}: σ ∈ G\}.

If g(X′) ≤ 1 and the set of ramified points S ⊂ X′ contains at least 3 − 2g(X′) elements we do the same thing starting with the minimal semi-stable model with the property that S injects into the smooth points of the reduction of this model. The remaining cases are easier.

References

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[pAI] Coleman, R., Torsion points on curves and p-adic Abelian integrals, Ann. Math. 121 (1985), 111-168.

[RLC] , Reciprocity Laws on Curves, Compositio, 72, (1989) 205-235. [C-M] and W. McCallum, Stable Reduction of Fermat Curves and

Local Components of Jacobi Sum Hecke Characters, vvJ. reine angew. Math. 385 (1988) 41-101.

[E] Edixhoven, S., Stable models of modular curves and applications, PhD thesis.

[K] Kiehl, R., Der Endlichkeitssatz fúr eigenliche Abbildungen in der nichtarchimedischen Funktionentheorie, Math. 2, (1967) 191-214.

[L-L] Liu, Q. and D. Lorenzini, Models of curves and finite covers, Compositio,

118, (1999) 62-102.

[R] Raynaud, M., p-groupes et reduction semi-stable des courbes, Grothendieck Festschrift III, Birkhäuser (1990) 179-197.

Robert F. Coleman

Department of Mathematics University of California California, USA 94530 coleman@math.berkeley.edu 226

Documenta Mathematica · Extra Volume Kato (2003)