Vector bundles on <span class="mathjax-formula">$p$</span>-adic curves and parallel transport

VECTOR BUNDLES ON p -ADIC CURVES AND PARALLEL TRANSPORT

B

C

DENINGER

A

WERNER

ABSTRACT. – We define functorial isomorphisms of parallel transport along étale paths for a class of vector bundles on ap-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve.

This may be viewed as a partial analogue of the classical Narasimhan–Seshadri theory of vector bundles on compact Riemann surfaces.

2005 Elsevier SAS

RÉSUMÉ. – Nous définissons des isomorphismes de « transport parallel » le long des chemins étales pour une classe de fibrés vectoriels sur une courbep-adique. Tous les fibrés de degré zéro avec reduction fortement semistable appartiennent à cette classe.

En particulier, ils donnent des représentations du groupe fondamental de la courbe. On peut voir ces résultats comme un analogue partiel de la théorie classique de Narasimhan et Seshadri concernant les fibrés holomorphes sur les surfaces de Riemann compactes.

2005 Elsevier SAS

0. Introduction

On a compact Riemann surface every finite dimensional complex representation of the fundamental group gives rise to a flat vector bundle and hence to a holomorphic vector bundle.

By a theorem of Weil, one obtains precisely the holomorphic bundles whose indecomposable components have degree zero [34]. It was proved by Narasimhan and Seshadri [28] that unitary representations give rise to polystable bundles of degree zero. Moreover, every stable bundle of degree zero comes from an irreducible unitary representation.

The present paper establishes a partial p-adic analogue of this theory, generalized to representations of the fundamental groupoid. The following is our main result. Recall that a vector bundle on a smooth projective curve over a field of characteristicpis called strongly semistable if the pullbacks ofEby all non-negative powers of the absolute Frobenius morphism are semistable. LetX be a smooth projective curve overQpand letobe the ring of integers in Cp. A modelXofXis a finitely presented flat and proper scheme overZpwith generic fibreX. The special fibreX_kis then a union of projective curves overk=Fp. We say that a vector bundle EonX_Cp=X⊗Cphas strongly semistable reduction of degree zero if the following is true:

Ecan be extended to a vector bundleE onX_o=X⊗o for some modelXof X such that the pullback of the special fibreEk ofEto the normalization of each irreducible component ofX_kis strongly semistable of degree zero. We say thatEhas potentially strongly semistable reduction of degree zero if there is a finite étale morphismα:Y →X of smooth projective curves such thatα^∗Ehas strongly semistable reduction of degree zero.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 0012-9593/04/2005 Elsevier SAS. All rights reserved

THEOREM. – LetEbe a vector bundle onX_Cpwith potentially strongly semistable reduction of degree zero. Then there are functorial isomorphisms of “parallel transport” along étale paths between the fibres ofE_Cp onX_Cp. In particular one obtains a representationρ_E,xofπ₁(X, x) onE_xfor every pointxinX(Cp). The parallel transport is compatible with tensor products, duals, internal homs, pullbacks and Galois conjugation.

The theorem applies in particular to line bundles of degree zero onX_Cp. In this case thep-part of the corresponding character ofπ₁(X, x)was already constructed by Tate using Cartier duality for thep-divisible group of the Abelian schemePic⁰_X/Z

p cf. [32, §4] and [9]. His method does not extend to bundles of higher rank.

Let us now discuss the contents of the paper in more detail. Afterwards we can sketch the proof of the theorem.

In the first section we investigate the categorySX,D consisting of finitely presented proper Zp-morphismsπ:Y →Xwhose generic fibre is a finite covering ofX which is étale outside of a divisorDonX. The important point is that for givenπinSX,D there is an objectπ:Y→X inSX,D lying overπwith better properties, e.g. cohomologically flat of dimension zero or even semistable. We also construct certain coveringsπusing the theory of the Picard functor which are used several times.

In the second section we define and investigate categoriesB_XCp,D andB_X

Cp,D involving a divisorD onX and also an analogous categoryB_Xo,D for a fixed modelXofX. These are defined as follows. The categoryB_Xo,D consists of all vector bundlesEonX_osuch that for all n1there is a coveringπinSX,D withπ^∗E trivial modulopⁿ. In theorem 16 it is proved that forEto lie inB_Xo,Dit suffices thatπ^∗_kEkis trivial whereπkis the special fibre of someπ.

Next,B_XCp,Dconsists of all bundles which are isomorphic to the generic fibre of a bundleE inB_Xo,D for some modelXof X. These categories are additive and stable under extensions.

Finally, we defineB_X

Cp,Das the category of vector bundles onX_Cpwhose pullback alongαlies inB_YCp,α∗Dfor some finite morphism α:Y →X between smooth projective curves which is étale overX\D. We obtain an additive category which is closed under extensions and contains all line bundles of degree zero. All vector bundles inBare semistable of degree zero.

The third section is devoted to the definition and study of certain isomorphisms of parallel transport along étale paths in U =X \D for the bundles in the category B_X

Cp,D. In more technical terms, we construct an exact⊗-functorρfromB_X

Cp,D to the category of continuous representations of the étale fundamental groupoidΠ₁(U)onCp-vector spaces. The basic idea is this: Consider a bundleE inB_Xo,D and for a givenn1letπ:Y →Xbe an object ofSX,D

such thatπ_n^∗Enis a trivial bundle onYn. Here the indexndenotes reduction modulopⁿ. Consider pointsxandxinX(Cp) =X(o)and choose a pointyinY =Y_Cpabovex. For an étale pathγ fromxtox, i.e. an isomorphism of fibre functors, letγybe the corresponding point abovex. For a “good” coverπwe have isomorphisms

Exn y∗

∼n

←−−Γ(Yn, π^∗_nEn)

(γy)∗

∼n

−−−→ Ex_n.

We define the parallel transportρ_E(γ) :Ex−→ E∼ x as the projective limit of the mapsρ_E,n(γ) = (γy)^∗_n◦(y^∗_n)⁻¹. This parallel transport is then extended toB_XCp,DandB_X

Cp,D. We also prove that the functor mapping a bundleEinB_X

Cp,D to its fibre in a pointx∈U(Cp)is faithful.

Using a Seifert–van Kampen theorem for étale groupoids we show that for a bundleEwhich is inBfor two disjoint divisors, one actually obtains a parallel transport along all étale paths inX.

e ◦

The proof of the theorem above starts with a characterization of those vector bundles on a purely one-dimensional proper scheme over a finite fieldFqwhose pullback to the normalization of each irreducible component is strongly semistable of degree zero: These are exactly the bundles whose pullback by a finite surjective morphism to a purely one-dimensional proper Fq-scheme becomes trivial. For vector bundles on smooth projective curves over finite fields this characterization is due to Lange and Stuhler [22]. Hence we have to lift finite covers in characteristicpto characteristic zero. The main point here is to construct a morphism of models whose reduction factors over a given power of Frobenius. In fact our method allows us to construct two coveringsπinSX,D andπ˜ inS_X,^D for two disjoint divisorsDandD such that π_k^∗Ek andπ˜^∗_kEk are both trivial. By the above theory, one gets the parallel transport on all of X_Cp. In the case of good reduction M. Raynaud has shown us a direct proof of this fact, cf.

Theorem 20.

For Mumford curves, Faltings [17] associates a vector bundle on X to every K-rational representation of the Schottky group and proves that every semistable vector bundle of degree zero arises in this way. It was shown by Herz [21] that his construction is compatible with ours.

Recently Faltings has announced a p-adic version of non-Abelian Hodge theory [18]. He proves an equivalence of categories between vector bundles onX_Cpendowed with ap-adic Higgs field and a certain category of “generalized representations” which contains the representations ofπ1(X, x)as a full subcategory. His methods are different from ours. In particular Faltings uses his theory of almost étale extensions. The main open problem in Faltings’ approach is to characterize the Higgs bundles corresponding to actual representations ofπ₁(X, x). He shows that with zero Higgs field, line bundles of degree zero and their successive extensions come from π1(X, x)-representations and suggests that perhaps all semistable vector bundles of degree zero are obtained in this way. The main theorem of our paper shows that this is true if in addition the bundle has potentially strongly semistable reduction.

The present preprint improves and replaces the second part of [8]. The first part of [8] will be published as [9].

Finally we would like to draw the reader’s attention to possibly related works of Berkovich [2, §9] on p-adic integration, of Ogus and Vologodsky on non-Abelian Hodge theory in characteristicpand of Vologodsky [33] on Hodge structures on fundamental groups.

1. Categories of “coverings”

In this section we introduce simplified and generalized versions of the categories of coverings that were used in [8] to define thep-adic representations attached to certain vector bundles.

In the following, a variety over a field k is a geometrically irreducible and geometrically reduced separated scheme of finite type overk. A curve is a one-dimensional variety. Let R be a valuation ring with quotient fieldQof characteristic zero. For a smooth projective curveX overQconsider a modelXofX overR, i.e. a finitely presented, flat and proper scheme over specRtogether with an isomorphismX=X⊗RQ. For a divisorDonX we writeX\Dfor X\suppD.

Consider the following category SX,D. Objects are finitely presented proper R-morphisms π:Y →Xwhose generic fibreπQ:YQ→X is finite and such that

πQ:π_Q⁻¹(X\D)→X\D is étale.

We setSX=SX,∅. In this case the generic fibreπ_Qis a finite étale covering. A morphism from π₁:Y1→Xtoπ₂:Y2→XinSX,Dis given by a morphismϕ:Y1→ Y2such thatπ₁=π₂◦ϕ.

Note thatϕis finitely presented and proper and thatϕQis finite, and étale overX\D.

If such a morphism exists, we say that π₁ dominates π₂. If in addition ϕ_Q induces an isomorphism of the local rings in two generic points we say that π₁ strictly dominates π₂. In the case where Y1Q andY2Q are both smooth projective curves this means that ϕ_Q is an isomorphism.

It is clear that finite products and finite fibre products exist in SX,D. Moreover, for every morphismf:X→Xof models overRand every divisorDonX, the fibre product induces a functorf⁻¹:SX,D→ SX,f^∗D.

We frequently use the fact that any non-constant morphism of a reduced and irreducible schemeZto a discrete valuation ring is flat, cf. [25, Corollary 4.3.10]. Besides, note that ifZ is flat and of finite presentation overRwith irreducible and reduced generic fibre, thenZis also irreducible and reduced by [25, Proposition 4.3.8].

We define the full subcategory

S_X,D^good⊂ SX,D

to consist of those objects inSX,Dwhose structural morphismλ:Y →specRis flat and satisfies λ_∗OY=OspecRuniversally and whose generic fibreλ_Q:YQ→specQis smooth. In particular YQ is geometrically connected and hence a smooth projective curve, which implies thatY is irreducible and reduced.

Let S_X,D^ss denote the full subcategory of SX,D consisting of all π:Y → X such that λ:Y →specRis a semistable curve whose generic fibreYQis a smooth projective curve overQ.

Recall thatλ:Y →specRis a semistable curve iffλis flat and for alls∈specRthe geometric fibreYsis reduced with only ordinary double points as singularities, see [7] or [25, Section 10.3].

Note that sinceYQis irreducible and reduced, the schemeYis irreducible and reduced as well. If Ris a discrete valuation ring, thenYis normal sinceYQis normal, see [25, Proposition 10.3.15].

THEOREM 1. – Assume that the base ringRis a discrete valuation ring.

(1) The categoryS_X,D^ss is a full subcategory ofS_X,D^good.

(2) The objects Y →X of S_X,D^ss have the property that Pic⁰_Y/R exists as a semi-Abelian scheme which is isomorphic to the identity component of the Néron model of the Abelian varietyPic⁰_YQ/Q.

(3) For any discrete valuation ringR dominatingR setX=X⊗RR and letD be the inverse image ofDinX. The natural base extension functorSX,D→ SX,DmapsS_X,D^good intoS_X^good,D andS_X,D^ss intoS_X^ss,D. (More generally this is true for valuation ringsRand R.)

(4) For any finite number of objectsπi:Yi→XinSX,D there exists a finite extensionQ/Q such that the objectsπ_i⊗RR ofSX,D are all dominated by a single object ofS_X^good,D

and even ofS_X^ss,D. HereRis a discrete valuation ring inQdominatingR.

(5) For any object π:Y →XofSX,D there exists an extension of discrete valuation rings R/Ras in (4) such thatπ⊗RRis strictly dominated by an object ofS_X^good,Dand even of S_X^ss,D.

Proof. – (1) Letπ:Y →Xbe an object ofS_X,D^ss . By assumption the geometric fibres ofYover specRare reduced. Together with the flatness ofλ:Y →specRit follows from [13, 7.8.6] that λis cohomologically flat in dimension zero. This means that the formation ofλ_∗(O)commutes with arbitrary base changes. Sinceλis proper the sheafλ_∗(O)onspecRis coherent and hence given by the finitely generated R-moduleΓ(specR, λ_∗(O)) = Γ(Y,O). Since Y is integral, this module is torsion free, hence free, so thatλ_∗(O) =O_spec^r _R for somer1. SinceYQ is a smooth curve, it follows thatr= 1. Taken together we find that the equationλ_∗(O) =OspecR

holds universally.

e ◦

(2) Since Y has semistable reduction overspecR it follows from [3, 9.4, Theorem 1] that Pic⁰_Y/R is a smooth separatedR-scheme which is semi-Abelian. By [3, 9.7, Corollary 2] the connected component of the Néron model ofPic⁰_YQ/Qis canonically isomorphic toPic⁰_Y/R.

(3) Note here that semistability is by definition preserved under base change.

(4) Since finite products exist inSX,D assertion (4) follows from assertion (5).

(5) I. Let us first prove the claim for the category S_X,D^good. This proof will be taken up in a G-equivariant context in Theorem 4 below. LetQbe a finite extension field ofQsuch thatYhas aQ-rational point overX\Dand such that the irreducible components ofYQare geometrically irreducible. LetRbe a discrete valuation ring inQdominatingR. SetYR=Y ⊗RR. Choose an irreducible component ofYQ containing aQ-rational point over X\D and letY^∗ be its closure inYR with the reduced scheme structure. ThenY^∗ is integral and we can pass to its normalizationY which is finite overY^∗ by [14, (7.8.6)].Yis a proper, flatR-scheme. Since Y ⊗ R Q is the normalization of Y_Q^∗ it has a Q-rational point. By Lipman’s resolution of singularities, there is an irreducible regularR-schemeY^∨together with a properR-morphism Y^∨→Y which is an isomorphism on the generic fibre.Y^∨is obtained by repeatedly blowing up the singular locus followed by normalization. This process becomes stationary after finitely many steps (see [24] and also [25, 8.3.44]). Hence we obtain a regular, irreducible schemeY^∨, which is proper and flat overR, together with a proper morphismY^∨→Xstrictly dominating π⊗RR. TheQ-rational point in the generic fibre ofY^∨induces a section ofY^∨→specRby properness. Now we apply a theorem of Raynaud to deduce thatY^∨ is cohomologically flat in dimension0, see [29, Théorème (8.2.1), (ii)⇒(iv)] or [25, 9.1.24 and 9.1.32]. ThusY^∨→X lies inS_X^good,D.

II. Alternatively, at least if the residue field ofRis perfect the claim forS_X,D^goodcould be proved by using instead of Raynaud’s theorem a theorem of Epp. ReplacingY byYandRbyR(of I) we may assume thatY is normal and thatYQ is a smooth projective curve overQ. Using [10, Theorem 2.0], it can be shown that there are a finite extensionQofQand a discrete valuation ringRinQdominatingRsuch that the normalizationYofY ⊗RRhas geometrically reduced fibres. As in the proof of part (1) it follows that the objectπ˜:Y → Y ⊗ RR→X⊗RR=X strictly dominatingπ=π⊗RR:Y ⊗RR→Xis inS_X^good,D.

III. We now prove that after base extension every objectπofSX,D is strictly dominated by an object ofS_X,D^ss . In view of part (1) this gives a third proof for the assertion on S_X,D^good. We constructY^∨→Xas in I. SinceY^∨is irreducible, regular and proper and flat overR, a result of Lichtenbaum [23] implies that Y^∨ is projective over R. According to [26, Theorem 0.2], there is a finite extensionQ^† ofQ and a discrete valuation ringR^† inQ^† dominatingRand a semistable modelY^†ofY^∨⊗RQ^† together with a morphismY^†−→ Y^{ϕ ∨}⊗RR^†overspecR^†. The composition

Y^†−→ Y^{ϕ ∨}⊗RR^†→X^†=X⊗RR^†

defines an object ofS_X^ss†,D^† which strictly dominatesπ^†=π⊗RR^†. 2

The next result is used later to prove that certain categories of vector bundles are stable under extensions and contain all line bundles of degree zero.

As before let R be a discrete valuation ring with quotient field Q of characteristic zero.

Consider a smooth projective curve of nonzero genusXoverQwith aQ-rational pointxand a semistable modelXofXoverspecR. Fix someN1and define an étale coveringα:Y →X

by the cartesian diagram

Y ⁱ

α

Alb_X/Q

N

X ^ix Alb_X/Q

Hereixis the canonical immersion into the Albanese variety corresponding to the rational point x. Note thatY is geometrically connected and hence a smooth projective curve.

PROPOSITION 2. – In the above situation, there exist

• a finite extensionQ/Qand a discrete valuation ringR inQdominatingR;

• a semistable modelYofY=Y ⊗QQoverspecR;

• a morphism

π:Y→X=X⊗RR such that the following assertions hold:

(a) The generic fibreπ_Q ofπ isα=α⊗QQ. (b) There is a commutative diagram

Pic⁰_X/R

π^∗

N

Pic⁰_Y/R

Pic⁰_X/R g

for some morphismgwithg(0) = 0, where0denotes the zero section overspecR. Remark. – After proving the proposition, we saw that in [18] Faltings uses a similar construction to make Higgs bundles onp-adic curves “small”.

Proof. – Let Y1 be the normalization of Xin the function field Q(Y) of Y. Then Y1 is a model of Y which is equipped with a morphism π₁:Y1→X. According to [14, 7.8.3 (vi)]

the morphismπ1 is finite. We will viewπ1 as an object ofSX. For an extension R/R as in Theorem 1 part (5) there exists an objectπ:Y→X of SX^ss strictly dominating π₁⊗RR. Changing the identification ofY⊗RQwithY=Y ⊗QQif necessary, we may assume that the generic fibre ofπisα=α⊗QQ.

The origin inAlbX/Q=Pic⁰_X/Qand the pointxofX define aQ-rational pointyofY with i(y) = 0. Let

i_y:Y →Alb_{Y /Q}

be the corresponding immersion. By the universal property of the Albanese variety, there is a unique morphism f: AlbY /Q→AlbX/Q which is necessarily a homomorphism such that f◦iy=i.

e ◦

Applying the functorPic⁰__⊗Q/Q to the commutative diagram Y

α iy

Alb_{Y /Q}

f

AlbX/Q

N

X ^ix Alb_X/Q

we obtain the following commutative diagram, wheref=f⊗QQ: Pic⁰_Y/Q

Pic⁰_X/Q fˆ

Pic⁰_X/Q α^∗

N=Nˆ

LetN be the Néron model ofPic⁰_Y/Q overspecR and letN⁰be its identity component. By Theorem 1 part (2) we know thatPic⁰_X/RandPic⁰_Y/R exist as smooth and separated schemes and thatPic⁰_Y/Ris isomorphic toN⁰. By the universal property of the Néron model, the natural map

Mor_R

Pic⁰_X/R,N _∼

−→Mor_Q

Pic⁰_X/Q,Pic⁰_Y/Q

(1)

is bijective. Hencefˆhas a unique extension to a morphismg: Pic⁰_X/R→ N. By construction, the compositiong◦Nhas generic fibrefˆ◦N=α^∗. Sinceαis the generic fibre ofπ:Y→X, the induced homomorphism

π^∗: Pic⁰_X/R→Pic⁰_Y/R

has generic fibreα^∗as well. Using the Néron property (1) it follows thatg◦N is equal to the composition

Pic⁰_X/R−−→π ^∗ Pic⁰_Y/R=N⁰→ N.

In particular we get thatg(0) =g(N(0)) = 0where 0 denotes the zero sections of Pic⁰_X/R

respectivelyPic⁰_Y/R. Since the special fibre of Pic⁰_X/R is connected, it follows that g is a morphism

g: Pic⁰_X/R→ N⁰= Pic⁰_Y/R

withg◦N=π^∗as desired. 2

We fix an algebraic closureQpofQpand consider finite extensionsQp⊂K⊂Qp. The rings of integers will be denoted byo_Kando_K=Zp.

The following corollary of Theorem 1 will be used constantly.

COROLLARY 3. – LetXbe a smooth projective curve overQpandDa divisor onX. LetX be a model ofXoverspecZp.

(1) Given any finite number of objects π_i:Yi →XinSX,D (respectively given one object π₁:Y1→XinSX,D) there is a finite extensionKofQpand a curveX_K/K with model X_oK/o_Kand a divisorD_KofX_Ksuch that the following hold: We haveX=X_K⊗KK and D=D_K ⊗K and X=X_oK ⊗oK Zp and there is an objectπ_oK:YoK→X_oK of S_X^good_o

K,DK and even of S_X^ss_o

K,DK such thatπ=πoK⊗oKZp dominates all πi inSX,D

(respectively dominatesπ1strictly).

(2) The categoryS_X,D^ss is a full subcategory ofS_X,D^good.

(3) Any finite number of objectsπi:Yi→Xin SX,D are dominated by a common object π:Y →XofS_X,D^goodand even ofS_X,D^ss . Every single objectπ1:Y1→XinSX,D is strictly dominated by an object ofS_X,D^goodand even ofS_X,D^ss .

Proof. – Part (1) follows from Theorem 1, (4), (5) using noetherian descent as in [14, §8, in particular (8.8.3) and (8.10.5)], together with [14, (17.7.8)] to descend to the categorySX1,D1

for someX₁/o_K1with divisorD₁whereK₁⊃Qpis a finite extension.

(2) Similarly as above, every objectπ:Y →XofS_X,D^ss descends to an objectπoK:YoK→ X_oKofSXoK,DK whereK⊃Qpis finite such thatYoK/o_Kis flat. Since the geometric fibres of YoK/o_KandY/Zpcan be identified, it follows thatπoK:YoK→X_oKis inSX^ssoK,DKand hence inS_X^good_o

K,DKby Theorem 1(1). Thereforeπ=π_oK⊗oKZplies inS_X,D^goodby Theorem 1(3).

Part (3) follows by combining (1) and Theorem 1(3). 2

Later we will construct a canonical parallel transport for certain vector bundles. The proof that it is well defined requires the following theorem. LetT_X,D be the following category. Objects are finitely presented proper G-equivariant morphisms π:Y →Xover specZp whereG is a finite (abstract) group which actsZp-linearly from the left onYand trivially onX. Moreover the generic fibreπ_Q

pis finite and its restrictionY_Qp\π^∗D→X\Dis an étaleG-torsor.

A morphism from theG1-equivariant morphismπ1:Y1→Xto theG2-equivariant morphism π2:Y2→X inT_X,D is given by a morphismϕ:Y1→ Y2 withπ1=π2◦ϕ together with a homomorphismγ:G1→G2of groups such thatϕisG1-equivariant ifG1acts onY2viaγ.

This definition generalizes the category T_X=T_X,φ used in [8, §5]. There is an obvious forgetful functorT_X,D→ SX,D. The full subcategoryT^good_X,D ofT_X,D consists of those objects which are mapped to objects ofS_X,D^good.

THEOREM 4. – For any object π:Y → X in SX,D there are a finite group G and a G-equivariant morphism π:Y→X defining an object of T^good_X,D which admits a morphism ϕ:Y→ Y withπ◦ϕ=π. In other words, every object ofSX,D is dominated by the image of an object inT^good_X,D.

Proof. – Let us first show that every objectπ:Y →XofT_X,D is dominated by an object of T^good_X,D. By noetherian descent we can assume that there is a finite extensionKofQpinQpwith ring of integersRsuch thatπdescends to the objectπR:YR→X_RinT_XR,DK. Denote byGthe group acting onYRoverX_Rsuch thatYK\π^∗_KDK→XK\DKis an étaleG-torsor. Now we follow the construction in the proof of Theorem 1(5) I and consider a geometrically irreducible component ofYKcontaining aK-rational point overXK\DK, whereKis a finite extension ofKinQp. Denote byH⊆Gthe stabilizer of this component. ThenH acts in a natural way onY^∗, and also onYandY^∨. ThereforeY^∨→X⊗RR, whereR is the ring of integers in K, is an object ofT^good_X

R,DK dominatingπR. By base-change toZp, our claim follows. Hence

e ◦

it suffices to show that there is an objectπofT_X,D which dominatesπ. By Corollary 3(1), we may assume that we haveπ=πoK ⊗Zp withπoK:YoK→X_oK inS_X^good_o

K,DK. LetY_K be the smooth projective curve whose function field is the Galois closure ofK(YK)overK(XK). The Galois groupGacts onY_K overXK. The morphismY_K →XK is finite and overXK\DK

it defines a Galois covering with groupG. Consider the normalizationYoK ofYoK inK(Y_K ).

By [14, (7.8.3) (vi)] the morphismYoK→ YoK is finite. HenceYoK→ YoK→X_oK defines an object ofSXoK,DKwith generic fibreYK=Y_K →XK.

By the proof of [26, Lemma 2.4], there exists a modelY_oK ofY_K overo_K endowed with an action ofGextending the action onY_K together with a morphismϕoK:YoK→ YoKwhich is an isomorphism on the generic fibre.

Letπ_oK:YoK→X_oKbe the compositionπ_oK=πoK◦ϕoK:YoK→ YoK→X_oK. SinceYoK

is reduced,G-equivariance of the generic fibre ofπ_oK impliesG-equivariance ofπ_oK, cf. [12, 7.2.21].

Now put

Y:=YoK⊗oKZp and π=π_oK⊗oKZp:Y→X.

Then π:Y → X is an object of T_X,D such that ϕ= ϕ_oK ⊗oK Zp:Y → Y satisfies π◦ϕ=π. 2

2. Two categories of vector bundles onp-adic curves

LetVecS be the category of vector bundles on a schemeS. For a bundleE we often write Efor its locally free sheaf of sectionsO(E). Letobe the ring of integers inCp=Qˆp and set o_n=o/pⁿo=Zp/pⁿZp. For every o-schemeY we setYn=Y ⊗oo_n. LetX be as before a smooth projective curve overQpand setX_Cp=X⊗_Q

pCp.

First of all, we show that vector bundles onX_Cpcan be extended to vector bundles on suitable models. The elegant argument in the proof was communicated to us by M. Raynaud.

THEOREM 5. – For every vector bundleE onX_Cp and every modelXofX there exists a modelXofX dominatingXsuch thatEextends to a vector bundle onX_o. IfXis smooth, then Ecan be extended to a vector bundle onX_oitself.

Proof. – We can extendE to a quasi-coherent sheafF of finite presentation onX_o, see [20, Appendix, Corollary 2 to Proposition 2]. LetJ ⊂ OXo be therth Fitting ideal ofF, wherer is the rank ofE. SinceF is of finite presentation,J is quasi-coherent of finite type. Besides, J · OX_Cp is equal to the Fitting ideal ofE, hence toOX_Cp. Therefore there exists somen1 such thatpⁿOXo⊂ J. By approximating the local generators ofJ with elements inOXmodulo pⁿ, we see that J descends to an ideal J0⊂ OX. Let ϕ:X→X be the blowing-up of J0. Since J0 is of finite type, ϕ is of finite presentation, so that ϕis a map in SX inducing an isomorphism on the generic fibre. The base change mapϕo:X_o→X_o is the blowing-up ofJ. Henceϕ⁻_o¹(J)OX_o is invertible. Sinceϕ⁻_o¹(J)OX_o is therth Fitting idealFr(ϕ^∗_oF)ofϕ^∗_oF, we can apply [30, (5.4.3)] to deduce thatϕ^∗_oF/Annϕ^∗_oF(Fr(ϕ^∗_oF))is locally free of rankron X_o. Hence it gives rise to a vector bundleEonX_owith generic fibreE.

If X is smooth over Zp, then Pic⁰_X/Z

p(o) = Pic⁰_X/Q

p(Cp), so that every line bundle of degree0 extends to a line bundle on X_o. Besides, Xcarries a line bundle N whose generic fibre has rank one. Hence every line bundle onX_Cp can be extended toX_o. The general case follows by induction on the rank ofE. Namely, there is an exact sequence of vector bundles 0→E1→E→E2→0onX_CpwhererkEi<rkEfori= 1,2. By hypothesis,E1andE2can

be extended toE1andE2onX₀. By flat base change we have an isomorphism Ext¹_Xo(E2,E1)⊗oCp−→∼ Ext¹_XCp(E2, E1).

This implies that E is isomorphic to the generic fibre of a vector bundleE onX_o. Note here that extensions of locally free sheaves are locally free because the cohomology of affine schemes vanishes. 2

DEFINITION 6. –

(a) For a modelXofX overZp and a divisorDinX the categoryB_Xo,D is defined to be the full subcategory ofVec_Xoconsisting of vector bundlesE onX_o=X⊗_Zpowith the following property: For everyn1there is an objectπ:Y →XofSX,Dsuch thatπ^∗_nEn

is a trivial bundle onYn. Hereπ_n,YnandEnare the reductionsmodpⁿofπ,YandE. (b) The full subcategoryB_XCp,D ofVec_XCp consists of all vector bundles onX_Cpwhich are

isomorphic to a bundle of the formj^∗EwithE inB_Xo,Dfor some modelXofX. Herej is the open immersion ofX_CpintoX_o.

(c) The full subcategoryB_X

Cp,D ofVec_XCp consists of all vector bundlesE onX_Cpsuch thatα^∗_CpEis inB_YCp,α∗Dfor some finite coveringα:Y →XofXby a smooth projective curveY overQpsuch thatαis étale overX\D.

Remarks. –

(a) ForD=∅we simply writeB_XoforB_Xo,D, etc.

(b) In [8, §6] a categoryB_Xowas defined as above, but using coverings inT_Xinstead ofSX. It follows from Theorem 4 that both definitions give the same category. Consequently, also the categoryB_XCp is the same as the one defined in [8, Definition 19].

LEMMA 7. – The categoryB_XCp,D consists of all vector bundles isomorphic toj^∗EwithE inB_Xo,DandXa semistable model ofXoverZp.

Proof. – Given any modelXofX, there is a semistable modelY ofX strictly dominatingX.

This follows from Corollary 3(3) applied toπ1= idX. Since the pullback of bundles onXtoY mapsB_Xo,DtoB_Yo,Dby Proposition 9 below, the assertion follows. 2

LEMMA 8. – Letf:X→Xbe a morphism of smooth, projective curves overQp. For every modelXofX there exists a modelXofX and aZp-linear morphismf˜:X→X such that the diagram

X

f˜

X

f

X X is commutative.

Proof. – Sincef is proper, it is either surjective or mapsX to a closed point ofX. In the second case, because of properness any model Xof X will do. Hence we can assume thatf is surjective, hence finite. There is a finite extension K of Qp inQp such that f descends to a morphismf_K:X_K→X_K of smooth, proper curves overK and such thatX descends to a modelX_oK ofX_K .

DefineX_oK as the normalization of the reduced and irreducible schemeX_oK in the function fieldK(XK)of XK, and letf˜oK:X_oK→X_o

K be the corresponding finite morphism. Since

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