Vol.97,No.1 DUKE MATHEMATICAL JOURNAL ©1999 THE FROBENIUS AND MONODROMY OPERATORS FOR CURVES AND ABELIAN VARIETIES ROBERTCOLEMANandADRIANIOVITA Contents Introduction............................................................. 1 PartI. Definitionsoftheoperators......................................... 3 1. DefinitionsofN andF forcurves....................................... 3 1.1. Themonodromyoperator.......................................... 3 1.2. TheFrobeniusoperator............................................ 4 2. N andF forAbelianvarieties.......................................... 6 2.1. ThemonodromyoperatorforAbelianvarieties....................... 7 2.2. TheFrobeniusoperatorforAbelianvarieties......................... 8 3. Equalityofthemonodromyoperators................................... 10 4. EqualityoftheFrobeniusoperators..................................... 14 PartII................................................................... 17 1. ReviewofFontaine’sringsandColmez’sintegration...................... 17 2. Theuniversalcoveringspace........................................... 20 3. Integrationofdifferentialformsofthesecondkindalongpaths............ 24 4. ExplicitdescriptionofFontaine’smonodromy............................ 27 5. ExplicitdescriptionofFontaine’sFrobenius.............................. 32 6. Thep-adicintegrationmapcommuteswiththemonodromies ............. 33 7. TheintegrationmapcommuteswiththeFrobenii......................... 35 Introduction. In this paper, we give explicit descriptions of Hyodo and Kato’s [HK]Frobeniusandmonodromyoperatorsonthefirstp-adicdeRhamcohomology groups of curves and Abelian varieties with semistable reduction over local fields of mixed characteristic. This paper was motivated by the first author’s paper [Co5], whereconjecturaldefinitionsoftheseoperatorsforcurveswithsemistablereduction were given. In “La structure de Hyodo-Kato pour les courbes” [LS], Le Stum also proposed formulas for these operators. We also prove that Le Stum’s operators are thesameasthoseofHyodoandKato. Thispaperisdividedintotwoparts.InPartI,writtenbythefirstauthor,wegivethe definitionsoftheFrobeniusandmonodromyoperatorsonthedeRhamcohomology ofAbelianvarietiesandofcurveswithsemistablereductionoveralocalfieldK. Received28January1997.Revisionreceived11November1997. 1991MathematicsSubjectClassification.Primary14K20;Secondary14K40. 171 172 COLEMAN AND IOVITA Suppose,forexample,thatAisanAbelianvarietywithsplitsemistablereduction over K. If we denote by K the maximal unramified subfield of K, we define a 0 canonicalandfunctorialK0-latticeinHd1R(A),denotedbyV0,andtwooperators(cid:10)A andNA onV0 suchthatwehavethefollowing: (i) (cid:10)A isσ-linear,whereσ istheabsoluteFrobeniusonK0; (ii) NA isK0-linearandNA(cid:10)A=p(cid:10)ANA. Theseoperatorsaredefinedintermsofthe“p-adicuniformizationcross”ofA: (cid:12) (cid:1)(cid:1) T (cid:2)(cid:2) G (cid:2)(cid:2) B, (cid:1)(cid:1) A where G is a semi-Abelian variety, T is a split torus, B is an Abelian variety with goodreduction,and(cid:12)isafreeAbeliangroupoffiniterank.Thusthediagrammakes senseintherigidanalyticcategory.ThemonodromyNA isdefinedasaresiduemap alongthetorusfollowedbyaboundarymap,andtheFrobenius(cid:10)A isdefinedusing the Frobenius operators on T and B and the p-adic integration of differential forms on A. On the other hand, if X is a semistable curve over K, then we can define a Frobenius(cid:10)X andamonodromyNX,asin[Co5].WeprovethatifJ istheJacobian of X, then (cid:10)X = (cid:10)J and NX = NJ, where we identify H1 (X) and H1 (J). In dR dR order to prove the identities of these operators, one needs to work with de Rham cohomologyanddualityfor1-motives.TheseareinvestigatedinSection3. In Part II, written by the second author (it contains essentially the main results of [I]), the filtered Frobenius monodromy module attached to H1 (A), where A is dR a split, semistable Abelian variety as in Part I, is compared to the filtered Frobenius monodromymoduleDst(V(ˆA))providedbyFontaine’stheory,whereV(A)=Tp(A) ⊗ZpQp and∧meansQp-lineardual.ThemainresultofSectionIIisthefollowing: Thep-adicintegrationpairing (cid:5),(cid:6):T (A)×H1 (A)→B+, p dR dR defined by P. Colmez in [Cz], induces an isomorphism of filtered Frobenius mon- odromy modules between the K -structure of H1 (A), as defined in Section I, and 0 dR D (V(ˆA)).Ourmaintoolisthe“universalcoveringspace”ofA(K)definedby st (cid:1) (cid:2) A(cid:1)(K):=lim A(K),[p] × A(K), ← A(K) where[p]isthemultiplicationbyp-isogenyonA.ItturnsoutthatA(cid:1)(K)Q:=A(cid:1)(K) ⊗ZQ is naturally a semistable representation of the Galois group of K over K, FROBENIUS AND MONODROMY OPERATORS 173 and one can define a map U : T(K) → Dst(A(cid:1)(K)Q) which plays the role of a “coresiduemap”alongthetorusT.Intheendweareabletoprove,usingtheresults in[Co7],thatFontaine’smonodromyoperatoronD (V(A))isessentiallyinducedby st Grothendieck’smonodromypairing(afterappropriateidentifications).Asacorollary, weprovethefollowingtheorem. Theorem 1. Let A be an Abelian variety over the local field K. Then Tp(A) is crystallineifandonlyifAhasgoodreduction. Here K is allowed to be any complete discrete valued field of characteristic zero and any perfect residue field of characteristic p. The “if” part of this statement is knownthroughworkofJ.-M.Fontaine[Fo3],andthe“onlyif”partwasconjectured by Fontaine in [Fo1]. The conjecture was proved in [Fo1] if the ramification degree of K is less than p−1. It was also proved in [Mk] if A is potentially a product of JacobiansandtheresiduefieldofK isfinite. Acknowledgements. ThesecondauthorherebyexpresseshisgratitudetoG.Stevens forhisgenerosityinsharinghisideasandinsightswithhim.Thesecondauthorvisited theUniversityofCaliforniaatBerkeleyduringworkonthispaper,andhewouldlike tothankthisinstitutionforitshospitality.Bothauthorsthanktherefereeforthevery carefulreadingofthemanuscript. Note. Any reference made in Part I or II to some section or result refers to a sectionorresultinthatpart.Exceptionsareexplicitlystated. PartI. Definitionsoftheoperators. HerewerecallthedefinitionsofFrobenius and monodromy operators on the de Rham cohomology groups of curves given in [Co5], give definitions of such operators for Abelian varieties, and prove that for Jacobianstheyareequivalent. LetK beafiniteextensionofQp,K0 bethemaximalunramifiedsubextensionof K,R betheringofintegersinK,k betheresiduefieldofR,andv bethevaluation of K, which is 1 on a uniformizing parameter of R. Suppose f = [k : Fp]. Let k¯ be an algebraic closure of k and σ be the Frobenius automorphism of k¯/k. We also useσ todenotetheliftingofthisautomorphismtoanautomorphismofW(k¯)/W(k). Also, fix a branch log of the p-adic logarithm defined over K. For a rigid space S overK andanaturalnumberi,Hi (S)denotestheithdeRhamcohomologygroup dR ofS overK. 1. DefinitionsofN andF forcurves 1.1. Themonodromyoperator. SupposeXisaconnected,smooth,completecurve overKwitharegularsemistablemodel(cid:1)overRsuchthattheirreduciblecomponents ofitsreduction(cid:1)¯ aresmooth.Wealsosupposeforsimplicityofexpositionthatthere areatleasttwoofthemandthatthey,aswellasthesingularpointsof(cid:1)¯,aredefined over k. For a subscheme Y of (cid:1)¯, let XY denote the tube of Y considered as a rigid 174 COLEMAN AND IOVITA subspaceofX. WeadoptthenotationofLeStum[LS].LetGr((cid:1))bethegraphwithorientededges definedasfollows.TheverticesV((cid:1))ofGr((cid:1))aretheirreduciblecomponentsof(cid:1)¯. Let(cid:1)¯ndenotethenormalizationof(cid:1)¯.Letm:(cid:1)¯n→(cid:1)¯ bethenaturalmap.Theedges E((cid:1)) of Gr((cid:1)) are the symbols [x,y], where x and y are points on (cid:1)¯n(k¯) whose images(cid:1)¯(k¯)arethesame.WesetA([x,y])equaltotheimageofthecomponentof (cid:1)¯n on which x lies, and we set B([x,y]) equal to the image in (cid:1)¯ of the component on which y lies. Then, if e∈E((cid:1)), e is an edge from A(e) to B(e). We also define aninvolutionτ ofE((cid:1))byτ([x,y])=[y,x]. Ife=[x,y]∈E((cid:1)),wesetXe=Xm(e).Wenotethat(cid:2)={XA:A∈V((cid:1))}isan admissiblecoverofXbybasicwideopens.Wenotethatsince(cid:1)isregular,anypoint inX(K)iscontainedinauniqueelementof(cid:2). Let (cid:3) (cid:3) X0= XA and X1= Xe. A∈V((cid:1)) e∈E((cid:1)) Let ι be the involution on X1, which takes a point in Xe ⊂X1 to the corresponding pointinXτ(e).ForamoduleM onwhichι∗ acts,M± ={m∈M :ι∗m=±m}. Wehavealongexactsequence (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) a − ∂ b − →H0 X0 −→H0 X1 −→H1 (X)→H1 X0 −→H1 X1 →. (1.1) dR dR dR dR dR Foreache∈E((cid:1)),wehaveanaturalresiduemap Rese:H1 (Xe)→H0 (Xe). dR dR (cid:4) (See [Co3].) We set Res = e∈E((cid:1))Rese : Hd1R(X1) → Hd0R(X1). This map takes H1 (X1)+ toH0 (X1)−. dR dR WedefineanoperatorN(cid:1) onH1 (X)tobethecomposition dR (cid:1) (cid:2) (cid:1) (cid:2) H1 (X)−→ρ H1 X1 + −−R−es→H0 X1 −−→∂ H1 (X), dR dR dR dR whereρisthemapobtainedfromrestriction.LetH((cid:1))denoteH0 (X1)−/a(H0 (X0)). dR dR WeultimatelyseethatN(cid:1) andtheimageofH((cid:1))inH1 (X)areindependentofthe dR model(cid:1). 1.2. TheFrobeniusoperator. Againweusetheexactsequence(1.1). LetX† denotethedaggercompletionofX0 alongthenonsingularlocusNSof(cid:1)¯. Wenotethat   (cid:5) (cid:5) X = X − X  NS A B A∈V((cid:1)) A(cid:22)=B isanunderlyingaffinoid(see[Co3])ofX0.LetY beasmooth,completecurvewitha model(cid:3)withgoodreductionobtainedfromX0 byglueinginopendiskstotheends FROBENIUS AND MONODROMY OPERATORS 175 ofX0 (theconnectedcomponentsofX0−X ∼=X1).Thenwehaveacommutative NS diagram where the rows are exact (for the bottom row, see [M, Thm. 4.1]) and the verticalarrowsareisomorphisms(see[B1]and[BC]): (cid:1) (cid:2) (cid:1) (cid:2) 0 (cid:2)(cid:2) H1 (Y) (cid:2)(cid:2) H1 X0 (cid:2)(cid:2) H1 X1 (cid:2)(cid:2) H2 (Y) dR dR dR dR (cid:1)(cid:1)(cid:1) (cid:2) (cid:1)(cid:1)(cid:1) (cid:2) (cid:1)(cid:1) (cid:1)(cid:1) 0 (cid:2)(cid:2) H1† Y† (cid:2)(cid:2) H1† X† (cid:2)(cid:2) KE((cid:1)) (cid:2)(cid:2) KV((cid:1)). DaggercohomologygiveslinearFrobeniusendomorphismsofthetermsinthebottom row(see[MW,Thm.8.5]and[M,Thm.4.3]),whichismultiplicationbyq onKE((cid:1)) andKV((cid:1)) andistheextensionbyscalarsofthefthpowerofcrystallineFrobenius on (cid:1) (cid:2) (cid:1) (cid:2) H1† Y† ∼=H1 (cid:3)¯,K ⊗ K. cris 0 K0 ByvirtueoftheRiemannhypothesis,thesequence (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) 0→H1† Y† →H1† X† →Ker KE((cid:1))→KV((cid:1)) →0 hasacanonicalFrobeniusequivariantsplitting.WecannowputaK /Frobeniusstruc- 0 ture(bythisexpression,wemeanaK -sublatticeandaσ-linearFrobeniusmorphism) 0 onH1 (X0)byputtingcompatibleonesonallthetermsinthebottomrowapartfrom dR H1†(X†).Moreover,Hd1R(Y)isnaturallyisomorphictoHc1ris((cid:3)¯,K0)⊗K0K and (cid:1) (cid:2) KE((cid:1))∼= W(k¯)E((cid:1)) Gal(W(k¯)/W(k))⊗ K. W(k) WeactuallywantandgetaK /Frobeniusstructureonthekernelofthecomposition 0 (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) − H1 X0 →H1 X1 →H1 X1 , dR dR dR whichwecallH1 (X0)+.Inparticular,wehaveanexactsequence dR (cid:1) (cid:2) (cid:1) (cid:2) + + 0→H1 (Y)→H1 X0 →H1 X1 →H2 (Y). dR dR dR dR Remark. In[Co5],wedefinedanotherK /FrobeniusstructureonH1 (X0)using 0 dR logstructures.Itisprobablyequivalent,butwehavenotproventhat. Hence,togetaK -latticeV inH1 (X)andaFrobeniusoperatorF onV ,allwe 0 0 dR 0 havetodoissplit (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) − 0→H((cid:1))→H1 (X)→Ker H1 X0 →H1 X1 →0. dR dR dR Weaccomplishthisusingp-adicintegration(see[Co2]and[Cds]).Let(cid:4)denotethe full subcategory of the category of rigid spaces whose objects consist of basic wide opens(see[Co3]). Summarizingresultsof[Co2]and[Cds],wehavethefollowingtheorem. 176 COLEMAN AND IOVITA Theorem1.1. Thereexistsauniqu(cid:10)efunctorfrom(cid:4)tothecategoryofhomomor- phismsbetweenvectorspaces,W → ,where W (cid:11) :/1 (W)→(cid:5)locan(W)/K, W W W suchthat (cid:11) (cid:11) dz dz=zmodK and =log(z)modK, z A1 Gm wherezisthestandardparameterontheaffinelineoverK. Note. Itfollowsthatifω∈/1 (W) W (cid:11) d ω=ω W andifω=df,wheref isrigid-analytic,then (cid:11) df =fmodK. W IfωA∈/1X(XA)forA∈V((cid:1))andfe∈(cid:5)X(Xe)fore∈E((cid:1)),then (cid:1) (cid:2) {ω } ,{f } A A∈V((cid:1)) e e∈E((cid:1)) denotesthe1-hypercochainonX ofthecomplex/. withrespecttothecovering(cid:2), X XA (cid:24)→ωA, and (XA,XB)(cid:24)→gA,B. Here gA,B ∈(cid:5)X(XA∩XB) is the function such thatgA,B|Xe =feife∈E((cid:1))issuchthatA(e)=AandB(e)=B.Thehypercochain isahypercocycleifandonlyif (cid:1) (cid:2) ωA(e)−ωB(e) |Xe =dfe, ∀e∈E((cid:1)). Let w∈Hd1R(X//1X) and ({ωA},{fe}) be a 1-hypercocycle of (cid:5)X with respect to (cid:2),whichrepresentsit.Wemaya(cid:10)nddosupposethatfτ(e)=−fe.ForeachA∈V((cid:1)), lets(ωA)bearepresentativeof WωA.Then (cid:1) (cid:1) (cid:2) (cid:1) (cid:2)(cid:2) X (cid:24)→f − s ω −s ω e e A(e) B(e) representsanelementofH0 (X1)−,welldefinedmodulotheimageofH0 (X0).Let dR dR I ((cid:1))denotethemapthatsendswtotheimageofthisclassinH1 (X).Thisisthe log dR desiredsplitting. 2. N andF forAbelianvarieties 2.1. The monodromy operator for Abelian varieties. Now let A be an Abelian FROBENIUS AND MONODROMY OPERATORS 177 schemeoverK withsemistablereduction.Thenwehavethe“uniformizationcross” T (cid:1)(cid:1) (cid:12) (cid:2)(cid:2) G π (cid:2)(cid:2) A, (cid:1)(cid:1) B whereT isatorus,(cid:12)isadiscretegroup,BisanAbelianschemewithgoodreduction, andGisanextensionofBbyT,allconsideredasrigidgroups.Theverticalsequence extendstoasequenceofgroupschemeswithgoodreductionoverR.Wehaveanexact sequence 0→(cid:6)om((cid:12),K)→H1 (A)→H1 (G)→0. (2.1) dR dR The map from (cid:6)om((cid:12),K) to the kernel of H1 (A)→H1 (G) is described as fol- dR dR lows.Suppose(cid:2)isanadmissiblecoveringofAand (cid:1) (cid:2) {ω :U ∈(cid:2)},{f :U,V ∈(cid:2),U (cid:22)=V} U UV . isa1-hypercocyclefor/ ,whichdeterminesanelementofKer(H1 (A)→H1 (G)). A dR dR ThismeanstherearefunctionshU onπ−1U forU ∈(cid:2)suchthat dhU =π∗ωU and hU−hV =π∗fUV. Nowletγ ∈(cid:12)andgU =γ∗hU−hU.(Thismakessensebecauseγ preservesπ−1U.) Butnow dgU =0 and gU−gV =0. Itfollowsthat{gU}correspondstoanelementkγ ∈K.Thecorrespondenceγ (cid:24)→kγ istheonewewant. IfhisahomomorphismfromGmintoT andα∈H1 (T),weset(α,h)=Res(h∗α). dR This determines an isomorphism from Hd1R(T) onto (cid:6)omZ((cid:6)om(Gm,T),K). Now thereisaperfectpairing (cid:6)om(Gm,T)×(cid:6)om(T,Gm)→Z, andsowehaveanisomorphismof(cid:6)omZ((cid:6)om(Gm,T),K)with(cid:6)om(T,Gm)⊗ZK. CalltheisomorphismfromHd1R(T)onto(cid:6)om(T,Gm)⊗ZKdeterminedbytheabove, ResT. Let G0, T0, and Gm0 denote the formal completions of G, T, and Gm along their specialfibers.Thenwehaveanisomorphism f :G(K)/G0(K)→T(K)/T0(K). 178 COLEMAN AND IOVITA Moreover,ifh∈(cid:6)om(T,Gm),hinducesamorphismfromT0 toGm0.Thusifv isa valuationonK,γ ∈(cid:12)⊂G,andh∈(cid:6)om(T,Gm),wehaveanelement(γ,h)∈Q, (cid:12) (cid:13) (cid:1) (cid:1) (cid:2)(cid:2) v h f γ modG0 . Thisdeterminesanondegeneratepairing (cid:12)×(cid:6)om(T,Gm)→Q and thus an isomorphism from (cid:6)om(T,Gm)⊗K onto (cid:6)om((cid:12),K) (see [ReP]). Fi- nally,letNA denotethecomposition H1 (A)→H1 (T)−−Re−s→T (cid:6)om(T,Gm)⊗K →(cid:6)om((cid:12),K)→H1 (A). dR dR dR Wenotethatwehavedescribedmaps (cid:6)om((cid:12),K)→H1 (A)→(cid:6)om(T,Gm)⊗K. (2.2) dR 2.2. TheFrobeniusoperatorforAbelianvarieties. Weonlyhavetosplittheexact sequence 0→(cid:6)om((cid:12),K)→H1 (A)→H1 (G)→0 (2.3) dR dR andthenputK /Frobeniusstructureson(cid:6)om((cid:12),K)andonH1 (G). 0 dR First we describe the splitting. Suppose A# is the universal vectorial extension of AandG∗ isthepullbackofA# toG.Wehave V V (cid:1)(cid:1) (cid:1)(cid:1) (cid:12) (cid:2)(cid:2) G∗ (cid:2)(cid:2) A# (cid:1)(cid:1) (cid:1)(cid:1) (cid:12) (cid:2)(cid:2) G (cid:2)(cid:2) A, whereV isthevectorialgroupschemeV(H1((cid:5)A))overK.IfH isagroupscheme over a field L, we let InvL(H) denote the K-space of invariant differentials on H overL.Wehave H1 (A)∼=InvK(A#)∼=InvK(G∗). dR (SeeTheorem1.2.2of[Co6].) Usingtheargumentof[Bou,Sect.7.6],oneobtainsthefollowingtheorem. Theorem2.1. If ω is an invariant differential on G∗ over K, there is a unique primitiveλω ofω onG∗(Cp),whichisahomomorphismsuchthattherestrictionof λω toT iscontainedin (cid:14) (cid:15) log◦h:h∈(cid:6)om(T,Gm) ⊗K. FROBENIUS AND MONODROMY OPERATORS 179 We use this to split (2.3) as follows. Suppose α ∈ H1 (A) corresponds to the dR invariantdifferentialω onG∗.Thenα goestothehomomorphism h :γ ∈(cid:12)(cid:24)→λ (γ). α ω Ifαistheimageofanelementhof(cid:6)om((cid:12),K),thenω=dg,whereg∈(cid:6)omK(G∗, Ga) such that g(γ)=h(γ). It follows that hα =h. Thus the map Ilog(A):α(cid:24)→hα is a splitting. Now we let (cid:6)om((cid:12),Z)⊗K ∼=(cid:6)om((cid:12),K), and for a∈W(k), γ ∈(cid:12), weset Fγ⊗a=γ⊗aσ. ItremainstodetermineaK -structureforH1 (G).TheschemesT,G,andB have 0 dR modelswithgoodreductionoverR.IfX isoneoftheseschemes,letX† denotethe dagger completion of X along the special fiber of its model. In particular, B† =B. Thenwehave 0 (cid:2)(cid:2) H1 (B) (cid:2)(cid:2) H1 (G) (cid:2)(cid:2) H1 (T) (cid:2)(cid:2) 0 dR dR dR (2.4) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:2) (cid:1)(cid:1)(cid:1) (cid:2) 0 (cid:2)(cid:2) H1†(B) (cid:2)(cid:2) H1† G† (cid:2)(cid:2) H1† T† (cid:2)(cid:2) 0. We know the top sequence is exact, and we can check that the bottom sequence is as well. Now the outer vertical arrows are isomorphisms. The first is well known, and the last one is easy to check since T is essentially a product of Gms (or one can use [BC]). Thus H1 (G) is isomorphic to H1†(G†). Now by P. Monsky and G. dR Washnitzer[MW],theobjectsinthebottomrowhavecompatibleactionsofFrobenius overK.Thatis,wehaveendomorphisms(cid:10)B for(cid:10)G and(cid:10)T ofH1†(B),H1†(G†), andH1†(T†),suchthattheobviousdiagramscommute.Wenowidentifytheobjects onthetoprowof(2.4)withtheobjectsdirectlybeneaththem.Asweshallsee,(cid:10)Gis thepoweroftheFrobeniusoperator(tensorK)thatweseek.Tomakethisoperator, all we have to do is split the exact sequence (2.4) since the outer members of this sequence have W(k)-structures with σ-linear Frobenius operators. Suppose q =|k|. Then(cid:10)T −q annihilatesH1 (T).ItfollowsfromtheRiemannhypothesisforB that dR thekernelM of(cid:10)G−q inH1 (G)mapsisomorphicallyontoH1 (T).Thisgivesus dR dR thedesiredsplitting. Wesawabovethat(cid:12) mapsintoG∗. Proposition2.2. (cid:6)omK(G∗,Ga)∼=(cid:6)om((cid:12),K)underthenaturalmap. Proof. Considerthecommutativediagram 0 (cid:2)(cid:2) (cid:6)om((cid:12),K) (cid:2)(cid:2) H1 (A) (cid:2)(cid:2) H1 (G) (cid:2)(cid:2) 0 dR dR (cid:1)(cid:1) (cid:1)(cid:1) 0 (cid:2)(cid:2) (cid:6)omK(G∗,Ga) (cid:2)(cid:2) InvK(G∗) (cid:2)(cid:2) H1 (G∗) (cid:2)(cid:2) 0 dR 180 COLEMAN AND IOVITA inwhichtherowsareexactandtheverticalarrowsareisomorphisms.Itfollowsthat (cid:6)omK(G∗,Ga)∼=(cid:6)om((cid:12),K).Theassertionthatthismapisthenaturalonefollows bychasingthediagram. 3. Equality of the monodromy operators. Now suppose X is a curve over K withsemistablemodel(cid:1)asaboveandJ istheJacobianofX.ThenJ hassemistable reduction.SinceH1 (X)iscanonicallyisomorphictoH1 (J),wemayconsiderNJ dR dR andN(cid:1) asoperatorsonthesamegroup.WenowshowthatNJ =N(cid:1). FirstletAbeanAbelianvarietyoverKwithsemistablereduction.Letthefollowing betheuniformizationcrossesofAandAˆ: T T(cid:29) (cid:1)(cid:1) (cid:1)(cid:1) (cid:12) (cid:2)(cid:2) G π (cid:2)(cid:2) A and (cid:12)(cid:29) (cid:2)(cid:2) G(cid:29) π (cid:2)(cid:2) Aˆ. (cid:1)(cid:1) (cid:1)(cid:1) B Bˆ Then (cid:12)(cid:29) = (cid:6)om(T,Gm), T(cid:29) = (cid:6)om((cid:12),Gm), and Bˆ is the dual of B. We have a canonical pairing (cid:12)×(cid:12)(cid:29) →Z, called the monodromy pairing, which we denote by (, )Mon.Now(cid:12)(cid:29) iscanonicallyisomorphicto(cid:6)om(T,Gm),whichinjectsontoaZ- latticeofH1 (T)viathemapthattakesh∈(cid:6)om(T,Gm)totheclassofh∗(dT/T). dR Wealsohaveamapof(cid:12)ontoalatticeinH1 (T(cid:29)).Thus,byextensionofscalars,we dR obtainapairingH1 (T)×H1 (T(cid:29))withvaluesinK.Pullingbackviatheprojections dR dR H1 (A)→H1 (T) and H1 (Aˆ)→H1 (T(cid:29)), dR dR dR dR weobtainapairing (cid:1) (cid:2) H1 (A)×H1 Aˆ →K, dR dR which we also call ( , ) . Let ( , ) be the cup-product (Poincaré) pairing on Mon Poin H1 (A)×H1 (Aˆ). dR dR Theorem3.1. Supposeα∈H1 (A)andβ∈H1 (Aˆ).Then dR dR (cid:1) (cid:2) α,N β =(α,β) . Aˆ Poin Mon We also saw that H1 (T) ∼= (cid:6)om(T,Gm)⊗K = (cid:12)(cid:29)⊗K. Let f be the natural dR mapfromH1 (A)into(cid:12)(cid:29)⊗K andg bethemapfrom(cid:6)om((cid:12)(cid:29),K)intoH1 (Aˆ),as dR dR describedinSection2.1.Thetheoremwillfollowfromresultof[Co7],whichasserts thatthepairingofRaynaudisthesameas(, ) and Mon