the frobenius and monodromy operators for curves and abelian varieties

45 Pages·1999·0.28 MB·English

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