Coleman Integration Versus

Schneider Integration on Semistable Curves

To John Coates, on the occasion of his 60th birthday

Ehud de Shalit

Received: August 17, 2005 Revised: February 27, 2006

Abstract. The purpose of this short note is to clarify the relation between p-adic integration on curves with semistable reduction, and the filtered (Φ, N)-module attached to the curve, following the work of Coleman and Iovita.

2000 Mathematics Subject Classification:

Keywords and Phrases:

0.1. The filtered (Φ, N)-module attached to a semistable curve. Let K be a local field of characteristic 0 and residual characteristic p. Denote by O_{K} its ring of integers, and by κ its residue field. Denote by K_{0} the fraction field of the Witt vectors of κ, and by σ its Frobenius automorphism. Thus K/K_{0} is a finite, totally ramified extension. By a curve X over O_{K} we shall mean a proper flat scheme over O_{K} of relative dimension 1. We denote its generic fiber by X_{K} and its special fiber by X_{κ}. We assume that X has semistable reduction. This means that X is regular and X_{κ} is a reduced curve whose singularities are ordinary double points. [Some authors use a less restrictive definition, in which X need not be regular, but this will require some modifications in what we do below.] We assume also that X_{κ} is split: the irreducible components of the (geometric) special fiber, its singular points, and the two tangents at each singular point, are all defined over κ. This can be achieved if we replace K by a finite unramified extension. Let H = H_{dR}^{1}(X_{K}/K) be the first de-Rham cohomology of X_{K}. It can be identified with the space of differentials of the second kind on X_{K} modulo exact differentials. H is a finite dimensional vector space over K, and it carries the Hodge filtration (differentials of the first kind)

(0.1)  0 ⊂ F_{dR}^{1} = H^{0}(X_{K}, Ω^{1}) ⊂ F_{dR}^{0} = H.

Let X_{κ}^{×} be the log-scheme associated to the special fiber with its induced logstructure [Ill]. Let D = H_{crys}^{1}(X_{κ}^{×}/K_{0}) be its first log-crystalline cohomology [LS, H-K]. Recall that D is a finite dimensional vector space over K_{0}, which comes equipped with a σ-linear bijective endomorphism Φ (Frobenius) and a nilpotent endomorphism N (monodromy) satisfying the relation

(0.2)  NΦ = pΦN.

For every prime π of K Hyodo and Kato constructed a comparison isomorphism

(0.3)  ρ_{π} : D ⊗_{K_{0}} K ≃ H

and the following relation holds for any two choices of a uniformizer

(0.4)  ρ_{π′} = ρ_{π} ∘ exp (log(π′/π)N) .

Note that the exponential is in fact a finite sum because N is nilpotent. The structure (H, F_{dR}^{·}, D, Φ, N, ρ_{π}) is the filtered (Φ, N)-module attached to X.

0.2. The weight decomposition. Let ϕ = Φ^{f}, where f = [κ : 𝔽_{p}], be the relative Frobenius, which now acts linearly on D. Write q = p^{f} for the cardinality of κ. By [LS] (see [Mo] in higher dimensions) we have a weight decomposition

(0.5)  D = D^{0} ⊕ D^{1} ⊕ D^{2}

where ϕ acts on D^{i} with eigenvalues which are q-Weil numbers of weight i (algebraic integers whose absolute value in any complex embedding is q^{i/2}). From the relation Nϕ = qϕN we deduce that N must vanish on D^{0} and D^{1}, and must map D^{2} to D^{0}. In fact, it is known that it maps D^{2} isomorphically onto D^{0}. This is a special case of the p-adic monodromy-weight conjecture. By means of the isomorphism ρ_{π} we transport the weight decomposition to H,

(0.6)  H = H^{0} ⊕ H^{1} ⊕ H_{π}^{2}

where only the last summand, but not H^{0} or H^{1}, depends on π, because N vanishes on D^{0} and D^{1}. The weight filtration is defined by

(0.7)  F_{W}^{i} H = ∑_{j≤i} H^{j}.

Our goal is to explain the weight decomposition of H in terms of the generic fiber only, using two transcendental processes in rigid analysis - Schneider and Coleman integration. The main theorem is a reformulation of the work of Coleman and Iovita [Co-I1], and only the presentation, and a few trivial observations, are new. We have a vague hope that similar techniques might help to understand the weight decomposition, and in particular the monodromyweight conjecture, for the cohomology of higher dimensional varieties as well. For p-adically uniformized varieties this was done in [dS], see also [Ito]. We also note that [Co-I2] and [GK] treat Frobenius and monodromy in more general situations, the first reference in cohomology of curves with coefficients, and the second in higher dimensions.

0.3. Schneider integration. To describe the main theorem, consider the degeneration complex Δ of X (also called the dual graph of the special fiber). Its vertices Δ_{0} are the irreducible components of X_{κ}. Its edges Δ_{1} are the singular points of X_{κ} (recall that the special fiber is assumed to be split). Each singular point, being an ordinary double point, determines two distinct analytic branches. The irreducible components on which these analytic branches lie, which may be the same, are the two end points of the edge. An orientation of an edge is an ordering of the two analytic branches at the singularity. We denote by Δ⃗_{1} the set of oriented edges, and by Δ⃗_{1}(v) the oriented edges originating at a vertx v. Note that if the two end points of an edge e are distinct, at most one of the oriented edges ε, ε̄ lying over e may belong to Δ⃗_{1}(v), but if e is a loop based at v, then both of them belong there. We introduce the space of harmonic 1-cochains on Δ which we denote by C_{har}^{1}(Δ). These are the maps f : Δ⃗_{1} → K satisfying

(0.8)  (i) f(ε̄) = −f(ε), (ii) ∀v ∑_{ε∈Δ⃗_{1}(v)} f(ε) = 0.

There is a canonical isomorphism

(0.9)  ν : C_{har}^{1}(Δ) ≃ H^{1}(Δ, K),

which sends a harmonic cochain to the singular cohomology class that it represents. Let X^{an} denote the rigid analytic curve (over K̄) attached to XK̄. There is a well known “retraction” map r : X^{an} → |Δ|. The inverse image under r of a vertex v consists of an affinoid with good reduction X_{v}. The reduction of X_{v} is the smooth part of Y_{v}, the irreducible component of X_{κ} labelled by v. The inverse image under r of an open edge e is an annulus X_{e}, isomorphic to

(0.10)  {z| |π| < |z| < 1} ,

and an orientation of e determines an orientation of the annulus. All the points in X_{e} reduce in the special fiber to the singular point labelled by e. If ω is a regular differential, and ε is an orientation on e, res_{ε}(ω) will denote the residue of ω with respect to a local parameter z on X_{e} compatible with ε. Clearly

(0.11)  resε̄ω = −res_{ε}ω,

and the rigid analytic Cauchy theorem guarantees that

(0.12)  ∑_{ε∈Δ⃗_{1}(v)} res_{ε}(ω) = 0.

Defining

(0.13)  c_{ω}(ε) = res_{ε}(ω)

we obtain a harmonic 1-cochain c_{ω}. This definition extends without any difficulty to differentials of the second kind. Indeed, such a differential may be locally (Zariski) modified by an exact differential to make it regular, so on each X_{e} we may assume it is regular and define its residue as before. Cauchy’s theorem still holds, so c_{ω} is harmonic. Since all the residues along annuli of exact differentials vanish, we get a well defined map

(0.14)  H → C_{har}^{1}(Δ), [ω] ↦ c_{ω}.

Passing to H^{1}(Δ, K) we obtain the Schneider class

(0.15)  S_{ω} = ν(c_{ω}) ∈ H^{1}(Δ, K)

of ω.

0.4. Coleman integration. To define the Coleman class of a differential of the second kind ω we use Coleman’s p-adic integration [Co], [Co-dS]. Let Δ̃ be the tree which is the universal covering of Δ, and

(0.16)  X̃^{an} = X^{an} ×_{Δ} Δ̃

the rigid analytic curve which is the fiber product of X^{an} with Δ̃ over Δ (the map from X^{an} to Δ being the retraction map r). We shall denote by Γ the group of deck transformations of the covering Δ̃ → Δ (or, equivalently of X̃^{an} → X^{an}). We shall continue to denote by r also the map from X̃^{an} to |Δ̃|. Let X̃(ṽ), for ṽ ∈ Δ̃_{0}, be the inverse image under r of the star of a vertex ṽ. (The star is the union of the vertex with the open edges originating at ṽ. Note that a loop in Δ based at v lifts in Δ̃ to two distinct edges starting at ṽ. If we assume that Δ has no loops, then X̃(ṽ) is isomorphic to X(v), the inverse image of the star of v in X^{an}.) In Coleman’s language X̃(ṽ) is a wide open space, and (if v is the image of ṽ in Δ) X̃ṽ ≈ X_{v} is an underlying affinoid with good reduction in X̃(ṽ). One can define a Coleman primitive F_{π,ṽ} of ω in X̃(ṽ). It is a locally analytic function which satisfies dF_{π,ṽ} = ω, and is uniquely determined up to an additive constant by its behavior under a rigid analytic (overconvergent) lifting of Frobenius to X̃ṽ. As the notation suggests, F_{π,ṽ} depends (on the annuli surrounding X̃ṽ in X̃(ṽ)) on the choice of π. For a given π, though, and neighboring vertices ṽ, ũ of Δ̃, F_{π,ṽ} − F_{π,ũ} is constant on the annulus where it is defined. Since Δ̃ is a tree, it is possible to choose the constants in such a way that the F_{π,ṽ} glue to give a primitive F_{π} of ω on all of X̃^{an}. Since ω is Γ-invariant,

(0.17)  C_{π,ω}(γ) = γ(F_{π}) − F_{π} = F_{π} ∘ γ^{−1} − F_{π}

is constant for every deck transformation γ ∈ Γ. The homomorphism

(0.18)  C_{π,ω} ∈ H^{1}(Γ, K) = H^{1}(Δ, K)

is the obstruction to descending F_{π} to X^{an}. It vanishes if and only if F_{π} lives on X^{an}, not merely on X̃^{an}, namely if and only if we can “Coleman integrate” ω on X. We shall prove that

(0.19)  C_{π′,ω} − C_{π,ω} = − log(π′/π)S_{ω}.

Our reformulation of the paper [Co-I] can now be stated as follows.

Theorem 0.1. (1) One has a canonical identification H/F_{W}^{1} H ≃ C_{har}^{1}(Δ) via the residue homomorphism. In other words, the cochains c_{ω} give us all the harmonic cochains, and c_{ω} = 0 if and only if S_{ω} = 0, if and only if ω ∈ F_{W}^{1} H.

(2) One has a canonical identification F_{W}^{0} = H^{0} = H^{1}(Δ, K) and C_{π,ω} is the projection of [ω] onto H^{0} relative to the decomposition H = H^{0} ⊕ H^{1} ⊕ H_{π}^{2}.

(3) The map ν corresponds to the monodoromy isomorphism

(0.20)  ν : H/F_{W}^{1} H ≃ H^{0}

derived from N.

Corollary 0.2. The subspace H^{1} is characterized as the space of differentials of the second kind for which a global Coleman primitive exists on X, regardless of π.

Proof. Indeed, in view of the relation between C_{π,ω}, C_{π′,ω} and S_{ω}, the following are equivalent: (i) C_{π,ω} = 0 for all π (ii) C_{π,ω} = 0 for two π whose ratio is not a root of unity, (iii) C_{π,ω} = 0 for some π and S_{ω} = 0. In view of the theorem, the last property is equivalent to [ω] ∈ H^{1}. □

Another corollary is the following. Denote by g(X_{K}) the genus of the curve and by g(Δ) the genus of Δ.

Corollary 0.3. For generic (all but finitely many) π the image of C_{π,ω} is all of H^{1}(Δ, K), and the dimension of the space of Coleman-integrable differentials of the second kind modulo exact differentials is 2g(X_{K}) − g(Δ).

Proof. Since S is surjective, so is C_{π} for all but finitely many π. □

The Hodge filtration did not play any role so far. The position of the differentials of the first kind in H with respect to the weight decomposition is mysterious. It is known that they are transversal to H^{0}, and that together with F_{W}^{1} H they span H, but their intersection with H^{1} can be large or small. All we can say is the following.

Corollary 0.4. For generic π, the dimension of the Coleman-integrable differentials of the first kind is g(X_{K}) − g(Δ). The dimension of the space of differentials of the first kind for which a Coleman primitive exists for all π is

(0.21)  g(X_{K}) − 2g(Δ) ≤ dim(H^{1} ∩ F_{dR}^{1}) ≤ g(X_{K}) − g(Δ).

Proof. Since F_{dR}^{1} maps onto C_{har}^{1}(Δ) under the residue map, S|_{F_{dR}^{1}} is still surjective, so the first assertion is proved as in the previous corollary. The upper bound in the second assertion follows from it, while the lower bound is obvious by counting dimensions. □

Remark 0.1. In [Cz], Colmez defines primitives for every differential of the second kind on X_{K}, regardless of the type of reduction. His primitives are independent of a choice of π, and in general do not coincide with Coleman’s primitives, except for the case of good reduction. He embeds the curve in its Jacobian, and uses the group structure of the latter to extend his integral from a neighborhood of the origin to the whole Jacobian. As an example, the reader may keep in mind the case of a Tate elliptic curve, of multiplicative period q_{E}. Colmez’ primitive of the differential of the first kind in this case would be the same as Coleman’s primitive, based on a branch of the p-adic logarithm which vanishes on q_{E}. It is clear that for curves of higher genus no branch of the logarithm conforms to all the periods. It is precisely the consideration of Coleman’s theory, as opposed to Colmez’, that gives us the possibility to identify the weight decomposition in the generic fiber (granted a choice of π is fixed).

1. The proof

1.1. Establishing the relation between C_{π,ω} and S_{ω}. Denote by log_{π} the unique logarithm on K̄^{×} for which log_{π}(π) = 0. We recall that Coleman’s primitive F_{π,ṽ} on the wide open X̃(ṽ) satisfies the following. If ε̃ = (ṽ, ũ) is an oriented edge of Δ̃, and X̃ε̃ the corresponding oriented annulus in X̃^{an}, and if z is a local parameter on X̃ε̃, then we may expand

(1.1)  ω|X̃ε̃ = ∑ a_{n}z^{n}dz

and, up to an additive constant,

(1.2)  F_{π,ṽ}|X̃ε̃ = ∑_{n≠−1} a_{n}(n + 1)^{−1}z^{n+1} + a_{−1} log_{π}(z).

Since

(1.3)  log_{π′}(z) − log_{π}(z) = − log(π′/π)ord_{K}(z)

we get that (again, up to a constant)

(1.4)  F_{π′} − F_{π}|X̃ε̃ = − log(π′/π)res_{ε}(ω)ord_{K}(z).

On an affinoid X̃ṽ, Coleman’s primitive is independent of π, up to a constant. Let γ ∈ Γ and normalize F_{π′} and F_{π} so that they agree on X̃ṽ. On X̃_{γ^{−1}(ṽ)} we shall then have

(1.5)  F_{π′} − F_{π} = − log(π′/π) ∑_{ε∈(ṽ,γ^{−1}(ṽ))} res_{ε}ω,

where the sum is over the oriented edges leading from ṽ to γ^{−1}(ṽ). This sum is just S_{ω}(γ), because S_{ω} is obtained from c_{ω} via the connecting homomorphism

(1.6)  C_{har}^{1}(Δ) = C_{har}^{1}(Δ̃)^{Γ} → H^{1}(Γ, K)

which is associated with the short exact sequence

(1.7)  0 → K → C̃_{har}^{0}(Δ̃) → C_{har}^{1}(Δ̃) → 0,

where C̃_{har}^{0}(Δ̃) are the 0-cochains on the tree satisfying the mean value property. It follows that

(1.8)  C_{π′,ω}(γ) − C_{π,ω}(γ) = − log(π′/π)S_{ω}(γ),

as we had to prove.

If we assume theorem 1, then (1.8) follows also from (0.4) and the fact that N^{2} = 0 (and vice versa). In the rest of this chapter we shall show how to derive the main theorem from the paper [Co-I].

1.2. The weight filtration on H. Determining the weight filtration on H in terms of the general fiber, and finding an expression for N, do not require a choice of π, or the use of Coleman integration. These will be needed only for the weight decomposition, to be considered in the next section. By GAGA, H can be identified with rigid de-Rham cohomology H_{dR}^{1}(X_{K}^{an}). For simplicity let us assume from now on that Δ contains no loops, so we identify the wide open set X̃(ṽ) with its image X(v) in X^{an}. The covering U = {X(v)} of X^{an} is admissible and acyclic, defined over K. It follows that we may identify H with the space of rigid 1-hyper-cocycles

(1.9)  \{ \begin{matrix}(ω_{v}, f_{ε}); ω_{v} ∈ Ω(X(v)), f_{ε} ∈ O(X_{ε}), df_{ε} = ω_{v} − ω_{u}|_{X_{ε}} \\ if ε connects u to v, and fε̄ = −f_{ε} \end{matrix} \} ,

modulo the space of rigid 1-hyper-coboundaries: elements of the type (df_{v}, f_{v} − f_{u}) for f_{v} ∈ O(X(v)). Specifically, if ω is a differential of the second kind, we pick rational functions g_{v} so that ω_{v} = ω − dg_{v} is holomorphic on X(v), and put f_{ε} = g_{u} − g_{v}. The class [ω] is then represented by (ω_{v}, f_{ε}). Since X_{v} is an affinoid with good reduction, the Frobenius morphism ϕ (of degree q) lifts to characteristic 0, to a rigid analytic mapping ϕ_{v} of X_{v} to itself. This rigid analytic Frobenius is overconvergent: there exists a strict neighborhood X_{v} ⊂ X′_{v} ⊂ X(v) such that ϕ_{v} extends to a morphism of X′_{v} to X(v). This X′_{v} can (and will) be chosen to consist of X_{v} together with an open annulus for each edge originating at v, and then the inclusion X′_{v} ⊂ X(v) induces isomorphism on de Rham cohomology. We can therefore regard ϕ_{v}^{∗} as an endomorphism of H_{dR}^{1}(X(v)). In fact, if we let Y_{v}^{0} be the smooth part of Y_{v}, the reduction of X_{v}, H_{dR}^{1}(X(v)) is nothing but the Monsky-Washnitzer cohomology of Y_{v}^{0} (tensored with K) and ϕ_{v}^{∗} is its Frobenius. It is independent of the lifting. The roots of the characteristic polynomial of ϕ_{v}^{∗} on H_{dR}^{1}(X(v)) have weights 1 or 2. Moreover, there is an exact sequence

(1.10)  0 → F_{W}^{1} H_{dR}^{1}(X(v)) → H_{dR}^{1}(X(v)) →^{res} (⊕_{ε∈Δ⃗_{v}}K)_{0} → 0

where res is the residue map, and F_{W}^{1} is the weight 1 subspace. The subscript 0 on the quotient means that we take only those elements in the direct sum whose coordinates add up to 0. On the weight 2 quotient ϕ_{v}^{∗} acts by multiplication by q. Let F_{W}^{0} H be the subspace of H represented by classes [(0, k_{ε})], where the k_{ε} are constants. It is thus isomorphic to H^{1}(Δ, K). Coleman and Iovita prove that under the Hyodo-Kato isomorphism this subspace is the image of the weight zero part of D (combine Lemma I.4.2 and Theorem II.5.4 of their paper). The quotient

(1.11)  H/F_{W}^{0} H

is the image of H in H_{dR}^{1}(X̃^{an}) under pullback. It is the space of differentials of the second kind on X^{an}, modulo those which become exact on X̃^{an}. The residue map gives the filtration

(1.12)  0 → gr_{W}^{1} H = F_{W}^{1} H/F_{W}^{0} H → H/F_{W}^{0} H →^{res} C_{har}^{1}(Δ) → 0,

which is the direct sum, over the vertices of Δ, of the short exact sequences recorded above. The surjectivity of the global residue map results from a dimension counting. Once again, Coleman and Iovita prove that under the Hyodo-Kato isomorphism, the Frobenius structure of H/F_{W}^{0} H is the one described above, rigid analytically, in terms of the ϕ_{v}^{∗}. (Compare how they define, in Section I.1, the Frobenius structure on

(1.13)  Ker(H_{dR}^{1}(X^{0}) → H_{dR}^{1}(X^{1})^{−}),

which is our H/F_{W}^{0} H, and apply their Theorem II.5.4.) It follows that F_{W}^{1} is indeed the weight 1 filtration, and C_{har}^{1} = gr_{W}^{2} H. Finally, that the monodromy operator is derived from the isomorphism ν between C_{har}^{1}(Δ) and H^{1}(Δ, K) also follows from [Co-I] (combine the description of N in Section I.1.1 and the commutative diagram on p.185). This checks all the statements of our main theorem, except for the identification of the weight decomposition in terms of Coleman integration.

1.3. The weight decomposition on H. Fix a choice of π. In Section I.1 of [Co-I] the authors describe a splitting of the projection H → H/F_{W}^{0} H, whose image is H^{1} ⊕ H_{π}^{2}. Recall that an element of H/F_{W}^{0} H is a collection {[ω_{v}]} of classes [ω_{v}] ∈ H_{dR}^{1}(X(v)), such that for any oriented edge ε, connecting u to v, res_{ε}ω_{u} = res_{ε}ω_{v}. Let F_{π,v} be the Coleman integral of ω_{v} on X(v), described above, which is determined up to a constant. Since the residues of ω_{u} and ω_{v} on X_{ε} agree, the function

(1.14)  f_{π,ε} = F_{π,v} − F_{π,u} ∈ O(X_{ε})

is rigid analytic in the annulus. The 1-hyper-cocyle (ω_{v}, f_{π,ε}) is well-defined up to a coboundary, and its class in H gives the desired splitting. It is now easy to check that C_{π} vanishes on classes ω which are in the image of this splitting. Indeed, suppose the differential of the second kind ω is such that

(1.15)  ω = ω_{v} + dg_{v}

for a meromorphic function g_{v} on X(v), and g_{u} − g_{v} = f_{π,ε} = F_{π,v} − F_{π,u} on X_{ε}. Then F_{π,u} + g_{u} agree on the annuli, hence glue to give a well defined Coleman meromorphic function F_{π,ω} on X^{an}, which is a global primitive of ω. It follows that C_{π,ω} = 0. On the other hand, if we start with a 1-hypercocycle (0, k_{ε}) where the k_{ε} are constants, and if ω is a differential of the second kind for which there are meromorphic functions g_{v} on X(v) such that ω = dg_{v} there, and g_{v}− g_{u} = k_{ε} for an edge connecting u to v, then [(k_{ε})] ∈ H^{1}(Δ, K) is the obstruction to integrating ω globally on X^{an}, hence is equal to C_{π,ω}.

These computations show that C_{π} annihilates H^{1} ⊕ H_{π}^{2}, and is the identity map on H^{0}. This completes the proof of Theorem 1.

1.4. Relation to the Neron model of the Jacobian. Even though the primitive F̃_{π,ω} of a differential of the second kind ω need not descend to X^{an}, we may use it to define the integral

(1.16)  ∫_{D} ω

over certain divisors of degree 0, namely those who specialize in X_{κ} to divisors which avoid the singular points and are of degree 0 on each of the irreducible components Y_{v} separately. This is because such a divisor D intersects each affinoid X_{v} in a divisor D_{v} of degree 0, while F_{π,v} is well defined, up to a constant, and independently of π, on X_{v}. Observe that the divisors in question are precisely those whose classes in P ic^{0} represent the connected component J^{0} of the Neron model of the Jacobian of X_{K}.

References

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Ehud de Shalit Institute of Mathematics Hebrew University deshalit@math.huji.ac.il