Periods of Abelian Varieties

34 Pages·2003·0.25 MB·English

Periods of Abelian Varieties
J.S. Milne∗
November 8, 2002; May 18, 2003. Submitted version.
Abstract
Weprovevariouscharacterizationsoftheperiodtorsorofabelianvarieties.
Contents
Introduction 1
1 Preliminaries 4
2 Periodsofabelianvarietieswithcomplexmultiplication 13
3 Periodsofabelianvarieties 22
References 33
Introduction
For an abelian variety A over Q, H1(A(C),C) has two Q-structures, that provided by sin-
gularcohomologyH1(A(C),Q)andthatprovidedbydeRhamcohomologyH1(A ,Ω• ).
Zar A/k
The periods of A are the coefficients of the transition matrix from a Q-basis for one struc-
turetoaQ-basisfortheother. Itisknown(Deligne1982)thatHodgeclassesonAimpose
algebraic relations on the periods, and it is conjectured that these are the only such rela-
tions. Thus, there appears to be no hope of obtaining an explicit description of the periods
themselves, but one may still hope to characterize some of the objects attached to them.
The singular and de Rham cohomologies define fibre functors on the category of motives
based on the abelian varieties over Q, and the difference of these functors is measured by
theperiodtorsorPAV. Inthispaper,weobtainvariouscharacterizationsofPAV. Beyondits
intrinsic interest, the period torsor controls the rationality of automorphic vector bundles,
andthereforeofholomorphicautomorphicforms(seeMilne1988;Milne1990,III4).
∗PartiallysupportedbytheNationalScienceFoundation.
1
INTRODUCTION 2
ThefirstproblemonerunsintoisthatPAV isatorsorforanaffinegroupschemeGover
Q for the flat (more specifically, the f.p.q.c.) topology. Such torsors are classified by the
flat cohomology group H1(Q,G) rather than a more familiar Galois cohomology group.
For an algebraic quotient G of G, the two cohomology groups coincide. In §1 we prove
n
thatthecanonicalmap
H1(Q,G) → limH1(Q,G )
←− n
issurjective,andthatthefibreofthemapcontainingtheclassofaG-torsorP is
lim1(P ∧G G )(Q)
←− n
(nonabelian higher inverse limit). The limits are over the set of algebraic quotients G
n
of G. We show, moreover, that the full group H1(Q,G) can be interpreted as a Galois
cohomologygroup,butitisGaloiscohomologydefinedusingcochainsthatarecontinuous
relative to the inverse limit topology on limG (Qal) (discrete topology on each G (Qal)).
←− n n
It is important to note that these results depend crucially on the fact that the algebraic
quotientsofGformacountableset—Idonotevenknowhowtodefineanonabelianhigher
inverse limit except for countable coefficient sets. The remainder of §1 reviews results,
more-or-less known, concerning nonabelian higher inverse limits and the classification of
morphismsoftorsors.
In§2,wetakeuptheproblemofcharacterizingPCM,theperiodtorsorforthecategory
of motives based on abelian varieties of (potential) CM-type over Q. Since, as Deligne
has pointed out, the period torsor PArt attached to the subcategory of Artin motives can be
explicitlydescribed,itisnaturalrathertoconsiderthe“relative”problemofcharacterizing
the morphism PCM → PArt of torsors. Objects of this type are classified by the flat co-
homology group H1(Q, S) where S is a certain twist of the Serre group. As the Serre
f f
groupiscommutative,themainresultof§1simplifiestoanexactsequence
0 → lim1 S (Q) → H1(Q, S) → limH1(Q, S ) → 0.
←− f n f ←− f n
Blasius (unpublished) showed that limH1(Q, S ) satisfies a Hasse principle, and Win-
←− f n
tenberger (1990) showed that limH1(Q, S ) = 0. If lim1 S (Q) were also zero, then
←− f n ←− f n
PCM → PArt would be unique up to an isomorphism inducing the identity on PArt. Alas,
it is not zero — in fact, we show that lim1 S (Q) is uncountable. Our proof of this uses
←− f n
an old theorem of Scholz and Reichardt on the embedding problem for Galois groups of
numberfields. Itwouldbeinterestingtohavemoreinformationonlim1 S (Q).
←− f n
In §3, we take up the problem of characterizing PAV. Again, it is more natural to con-
sider the relative problem of characterizing PAV → PCM. Among other results, we prove
that the isomorphism class of PAV → PCM is uniquely determined by its classes over Q
l
(l = 2,3,5,...,∞),aboutwhichagreatdealisknown.
Blasius and Borovoi (1999) study the problem of characterizing PH where H $AV
is the category of motives based on a certain class of abelian varieties over Q whose
Mumford-Tate groups have simply connected derived group. Their main theorem (ibid.
1.5)statesthattheisomorphismclassofPH → PCM isdeterminedbytheGaloiscohomol-
ogy class of PH over R. Unfortunately, their proof of this is inadequate for two reasons.
First, they make the (false!) assumption that the flat cohomology groups coincide with
INTRODUCTION 3
the inverse-limit Galois cohomology groups — this amounts to setting all lim1s equal to
←−
zero. Second, they misidentify the cohomology class that must be proved trivial for their
theorem to hold1. It seems unlikely that the statement of their theorem is correct (see 3.26
below),butbycombining(1.14)ofthispaperwiththeirarguments,oneobtainsthefollow-
ing theorem (3.25): the isomorphism class of PH → PCM is uniquely determined by its
isomorphismclassoverR. Thisobservationbeganmyworkonthispaper.
Notationsandconventions
“Variety” means geometrically reduced scheme of finite type over a field. Semisimple al-
gebraicgroupsareconnectedand“simple”foranalgebraicgroupmeans“noncommutative
and having no proper closed connected normal subgroup 6= 1”. The identity component of
a group scheme G over a field is denoted by G◦. For a connected (pro)reductive group G
over a field, ZG is the centre of G, Gad is the adjoint group G/ZG of G, Gder is the de-
rived group of G, and Gab is the largest commutative quotient G/Gder of G. The universal
˜
coveringofasemisimplegroupGisdenotedG → G.
ThealgebraicclosureofQinCisdenotedbyQal,and(exceptin§1)Γ = Gal(Qal/Q).
We set Gal(C/R) = {1,ι}. A CM-field is any field E algebraic over Q admitting a non-
trivialinvolutionι suchthatι◦ρ = ρ◦ι forallρ: E → C.
E E
All categories of motives will be defined using absolute Hodge classes as the corre-
spondences(Deligne1982a;DeligneandMilne1982,§6).
Wesometimesuse[x]todenoteanequivalenceorisomorphismclasscontainingx. The
∼
notationX ≈ Y meansthatX andY areisomorphic,andX = Y meansthatX andY are
canonicallyisomorphic(orthataparticularisomorphismisgiven).
1Withtheirnotations,inordertoprovetheirtheorem,theywouldhavetoshowin5.1oftheirpaperthat
theclassofP → PCMinH1(Q,(G◦ )der)istrivial(see(1.10)below). Instead,theyproveonlytheweaker
DR
statement that the image of the class in H1(Q,G◦ ) is trivial, which, in fact, is all their hypotheses imply,
DR
evenwhenoneignoresthelim1terms.
←−
1 PRELIMINARIES 4
1 Preliminaries
Inverse limits
Foraninversesystemofgroupsindexedby(N,≤),
(A ,u ) = (A ← ··· ← A ←un A ← ···),
n n n∈N 0 n−1 n
define lim1A to be the set of orbits for the left action of the group Q A on the set
←− n n n
Q
A ,
n n
Q Q Q
A × A → A
n n n n n n
(...,a ,...) (...,x ,...) 7→ (...,a ·x ·u (a )−1,...).
n n n n n+1 n+1
Thisisaset,pointedbytheorbitof1 = (1,1,...). Notethat
Q
limA = {a ∈ A | a·1 = 1}.
←− n n
Let(A ) → (B ) beaninversesystemsofinjectivehomomorphisms. From
n n∈N n n∈N
0 → (A ) → (B ) → (B /A ) → 0
n n∈N n n∈N n n n∈N
weobtainanexactsequence
1 → limA → limB → lim(B /A ) → lim1A → lim1B (1)
←− n ←− n ←− n n ←− n ←− n
ofgroupsandpointedsets. Exactnessatlim(B /A )meansthatthefibresoflim(B /A ) →
←− n n ←− n n
lim1A are the orbits for the natural action of limB on lim(B /A ). When each A is
←− n ←− n ←− n n n
normalinB ,sothatC = B /A isagroup,(1)canbeextendedtoanexactsequence
n n df n n
1 → limA → limB → limC → lim1A → lim1B → lim1C → 1. (2)
←− n ←− n ←− n ←− n ←− n ←− n
Exactness at lim1A means that the fibres of lim1A → lim1B are the orbits for the
←− n ←− n ←− n
naturalactionoflimC onlim1A .
←− n ←− n
Recallthataninversesystem(X ) ofsets(orgroups)issaidtosatisfythecondition
n n∈N
(ML) if, for each m, the decreasing chain in X of the images of the X for n ≥ m is
m n
eventuallyconstant.
PROPOSITION 1.1. Let(An,un)n∈N beaninversesystemofgroups.
(a) If(A ,u )satisfies(ML),thenlim1A = 0.
n n ←− n
(b) IftheA arecountableand(A ,u ) fails(ML),thenlim1A isuncountable.
n n n n∈N ←− n
PROOF. (a)Theaction2
(...,a ,...)·(...,x ,...) = (...,a ·x ·(ua )−1 ,...)
n n n n n+1
2Weusuallyomitthesubscriptontransitionmaps.
1 PRELIMINARIES 5
Q Q
of the group G = A on the set S = A is transitive, and the
N+1 df 0≤n≤N+1 n N df 0≤n≤N n
projection (a ) 7→ a gives an isomorphism from the stabilizer of any x ∈ S onto
n n N+1 N
Q
A . Letx,y ∈ A ,andlet
N+1 n∈N n
(cid:8) (cid:9)
P = g ∈ G | gxN = yN .
N N+1
where xN and yN are the images of x and y in S . We have to show that limP is
N ←− N
nonempty. TheobservationsinthefirstsentenceshowthatP isnonemptyandthatA
N N+1
acts simply transitively on it. It follows that the inverse system (P ) satisfies (ML). Let
N
Q = ∩ Im(P → P ). TheneachQ isnonempty,and(Q ) isaninversesystem
N i N+i N N N N∈N
with surjective transition maps. Hence, limQ is (obviously) nonempty, and any element
←− N
ofitisanelementoflimP .
←− N
(b)If(A )fails(ML),thenthereexistsanmsuchthatinfinitelymanyofthegroups
n
df
B = Im(A → A )
i m+i m
aredistinct. As
lim1A → lim1B
←− m+i ←− i
i i
is surjective (see (2)), it suffices to show that lim1B is uncountable. This is accomplished
←− i
bythenextlemma(appliedwithA = A ).
m
LEMMA 1.2. Let ··· ⊃ An ⊃ An+1 ⊃ ··· be a sequence of subgroups of a countable
groupA. IfinfinitelymanyoftheA aredistinct,thenlim1A isuncountable.
n ←− n
PROOF. From(1)appliedtotheinversesystem(An ,→ A)n∈N,weobtainabijection
A\(limA/A ) → lim1A .
←− n ←− n
Asamapofsets,A/A → A/A isisomorphictotheprojectionmap
n+1 n
A/A ×A /A → A/A ,
n n n+1 n
Q
andsolimA/A ≈ A /A (assets),whichisuncountable.
←− n n n+1
REMARK 1.3. The above statements apply to inverse systems indexed by any directed set
I containing an infinite countable cofinal set, because such an I will also contain a cofinal
setisomorphicto(N,≤).
NOTES. Thedefinitionoflim1 fornonabeliangroupsandthesequence(2)canbefoundin
←−
Bousfield and Kan 1972, IX §2. In the commutative case, statement (a) of Proposition 1.1
isprovedinAtiyah1961andstatement(b)inGray1966.
Torsors
LetEbeacategorywithfinitefibredproducts(inparticular,afinalobjectS)endowedwith
a topology in the sense of Grothendieck (see Bucur and Deleanu 1968, Chapter 2). Thus,
Eisasite. By“torsor”wemean“righttorsor”.
1 PRELIMINARIES 6
1.4. ForasheafofgroupsAonE,arightA-sheafX,andaleftA-sheafY,X∧AY denotes
thecontractedproductofX andY,i.e.,thequotientsheafofX×Y bythediagonalaction
of A, (x,y)a = (xa,a−1y). When A → B is a homomorphism of sheaves of groups,
X ∧A B is the B-sheaf obtained from X by extension of the structure group. In this last
case,ifX isanA-torsor,thenX ∧A B isaB-torsor.
1.5. ForanA-torsorP andaleftA-sheafX,define
PX = P ∧A X.
When X is a sheaf of groups and A acts by group homomorphisms, PX is a sheaf of
groups. For example, when we let A act on itself by inner automorphisms, PA is the inner
form of A defined by P. There is a natural left action of PA on P, which makes P into a
leftPA-torsorandinducesanisomorphism
PA → Aut (P). (3)
A
LetP(A)denotethecategoryofA-torsorsandH1(S,A)thesetofisomorphismclassesof
objectsinP(A)(pointedbytheclassofthetrivialtorsorA ).
A
1.6. Letv: B → C beahomomorphismofsheavesofgroupsonE,andletQbeaC-torsor.
DefineP(B → C;Q)tobethecategorywhoseobjectsarethev-morphismsoftorsorsP →
Q and whose morphisms Hom(P → Q,P0 → Q) are the B-morphisms P → P0 giving
a commutative triangle. Let H1(S,B → C;Q) denote the set of isomorphism classes in
P(B → C;Q). When Q = C , we drop it from the notation; then H1(S,B → C) is
C
pointed by the class of B → C . The category P(B → 0) is canonically equivalent with
B C
thecategoryofB-torsors,andso
H1(S,B → 0) ∼= H1(S,B).
Let A = Ker(B → C). Then A is stable under the action of B on itself by inner automor-
phisms,andforanyobjectP → QofP(B → C;Q),
Aut(P → Q) ∼= P ∧B A.
1.7. Let v: B → C be a surjective homomorphism with kernel A. To give a B-torsor P
with vP trivialised by e ∈ (vP)(S) amounts to giving the A-torsor f−1(e) where f is the
mapP → vP: thenaturalfunctor
P(A → 0) → P(B → C)
isanequivalence.
1.8. Let v: B → C be a homomorphism of sheaves of groups on E. A B-torsor P allows
ustotwistv:
Pv: PB → PC, PC =df P ∧B C.
HerealocalsectionbofB actsonC byc 7→ (vb)c(vb)−1. LetQ = vP. ThenPC ∼= QC.
1 PRELIMINARIES 7
1.9. Let v: B → C be a homomorphism of sheaves of groups on E. A B-torsor P can be
regardedasa(PB,B)-bitorsor(see(3)). Thereisafunctor
P(PB → QC) → P(B → C;Q) (4)
sending P0 → Q0 to P0 ∧PB P → Q0 ∧QB Q. In particular, the neutral object of P(PB →
QC) is sent to the object P → Q of P(B → C;Q). Let Popp denote the (B,PB)-bitorsor
withthesameunderlyingsheafasP butwithlocalsectionsbandb0 ofB andPB actingas
(b,b0)·p = b0−1 ·p·b−1. Thefunctor
P(B → C;Q) → P(PB → QC) (5)
sending P0 → Q to (P0 → Q)∧B Popp is a quasi-inverse to the functor in (4). Therefore,
bothfunctorsareequivalencesofcategories.
PROPOSITION 1.10. Let
v
1 → A → B → C → 0 (6)
beanexactsequenceofsheavesofgroupsonE,andletP → Qbeav-morphismoftorsors.
Thereisanaturalbijection
H1(S,PA) → H1(S,B → C;Q)
sendingtheneutralelementofH1(S,PA)totheelement[P → Q]ofH1(S,B → C;Q).
PROOF. WecanuseP totwistthesequence(6):
1 → PA → PB → QC → 1, PA = P ∧B A.
Nowcombine
H1(S,PA) →1.7 H1(S,PB → QC) →(4) H1(S,B → C;Q).
REMARK 1.11. If in the proposition A is commutative, then the action of B on A factors
throughanactionofC onA,andso
PA =df P ∧B A ∼= P ∧B C ∧C A ∼= Q∧C A =df QA.
NOTES. The basic definitions 1.4–1.5 are from Giraud 1971. The remaining statements
can be found, or are hinted at, in Deligne 1979a, 2.4.3–2.4.4. See also Breen 1990. (The
mainideasgobacktoDedeckerandGrothendieckinthe1950s.)
1 PRELIMINARIES 8
Cohomology and inverse limits
WenowfixanaffineschemeS andletEbethecategoryofaffineschemesoverS endowed
withthefpqctopology(thatforwhichthecoveringfamiliesarethefinitesurjectivefamilies
offlatmorphisms).
Throughout this subsection, (G ,u ) is an inverse system, indexed by (N,≤), of flat
n n n
affine group schemes of finite type over S with faithfully flat transition maps, and G =
limG . Thus,GisaflataffinegroupschemeoverS.
←− n
PROPOSITION 1.12. Themap[P] 7→ ([P ∧G Gn])n≥0
H1(S,G) → limH1(S,G ) (7)
←− n
n
is surjective. For a G-torsor P, the fibre of the map containing [P] is lim1G0 (S) where
←− n
G0 istheinnerformP ∧G G ofG .
n n n
PROOF. Aclasscinl←im−H1(S,Gn)isrepresentedbyaninversesystem
P ← P ← ··· ← P ← ···
0 1 n
withP aG -torsor. TheinverselimitofthissystemisaG-torsormappingtoc.
n n
Let P0 and P be G-torsors such that P0 ≈ P for all n, and choose isomorphisms
n n
a : P0 → P . Consider
n n n
P0 −a−n−+→1 P
n+1 n+1
 
 
yv0 yv
P0 −−a−n→ P .
n n
There is a unique isomorphism b : P0 → P for which the diagram commutes, i.e., such
n n n
that
b ◦v0 = v ◦a .
n n+1
Lete betheelementofAut(P )suchthate ◦b = a ;then
n n n n n
e ◦v ◦a = a ◦v0.
n n+1 n
Replacing(a ) with(c ◦a ) replaces(e ) with(c ·e ·u0c−1 ) whereu0 is
n n≥0 n n n≥0 n n≥0 n n n+1 n≥0
the transition map Aut(P ) → Aut(P ). Thus, the class of (e ) in lim1Aut(P ) is
n+1 n n n≥0 ←− n
independentofthechoiceofthea . Similarly,itdependsonlyontheisomorphismclassof
n
P0. Therefore, we have a well-defined map from the fibre containing [P] to lim1Aut(P ),
←− n
and it is straightforward to check that it is a bijection. Finally, (3) allows us to replace
Aut(P )withG0 (S).
n n
COROLLARY 1.13. WhentheGn arecommutative,thereisanexactsequence
0 → lim1G (S) → H1(S,G) → limH1(S,G ) → 0.
←− n ←− n
PROOF. Inthiscase,G0n = Gn.
1 PRELIMINARIES 9
COROLLARY 1.14. Foranycountablefamily(Gi)i∈I offlataffinegroupschemes,
H1(S,QG ) = QH1(S,G ).
i i
PROOF. This is certainly true for finite families. Thus, we may assume that I is infinite,
and equals N. Let A = Q G . For any Q G -torsor P, the projection maps
n 0≤i≤n i i≥0 i
PA (S) → PA (S) admit sections, and so are surjective. Therefore lim1PA (S) = 0
n n−1 ←− n
(1.1a),anditfollowsthat
H1(S,G) 1∼=.12 limH1(S,A ) ∼= lim Q H1(S,G ) ∼= QH1(S,G ).
←− n ←− i i
n n 0≤i≤n i≥0
REMARK 1.15. Let S = Spec(Q). Although the maps un: Gn → Gn−1 are surjective,
typically the maps G (Q) → G (Q) will not be. In fact, typically, the inverse sys-
n n−1
tem (G (Q)) will not satisfy (ML) and so lim1G (Q) will be uncountable (1.1b). For
n n ←− n
example,consideratowerofdistinctsubfieldsofQal,
Q ⊂ F ⊂ ··· ⊂ F ⊂ F ⊂ ··· , [F : Q] < ∞.
1 n−1 n n
There is an inverse system (G ,u ) with surjective transition maps for which G is the
n n n
Q-torusobtainedfromG byrestrictionofscalarsandu isthenormmap. Then
m/Fn n
(G (Q),u (Q)) = (F×,Nm ) ,
n n n∈N n Fn/Fn−1 n∈N
whichfails(ML),3 andsolim1G (Q)isuncountable.
←− n
Comparison with Galois cohomology
We now let S be the spectrum of a field k, and we let H1(k,−) denote H1(S,−). Choose
aseparableclosureksep ofk,andletΓ = Gal(ksep/k).
PROPOSITION 1.16. For any smooth algebraic group N over k, there is a canonical iso-
morphism
H1(k,N) → H1(Γ,N(ksep)).
PROOF. An N-torsor P is represented by an algebraic variety over k, and hence acquires
apointpoversomesubfieldofksep offinitedegreeoverk. Theformula
τp = p·a (8)
τ
definesacontinuouscrossedhomomorphisma : Γ → N(ksep)whosecohomologyclassis
τ
independent of the choice of p and depends only on the isomorphism class of P. Thus, we
have a well-defined map H1(k,N) → H1(Γ,N(ksep)), and it follows from descent theory
thatthisisanisomorphism.
3Toseethis,usethat,forafiniteextensionE/F ofnumberfieldsandafiniteprimevofF,ord (NmE×)
v
istheidealinZgeneratedbytheresidueclassdegreesoftheprimesofE lyingoverv.
1 PRELIMINARIES 10
1.17. Let N be a smooth algebraic group over k, and let f: Γ → Aut(N)(ksep) be a
continuous crossed homomorphism (discrete topology on Aut(N)(ksep)). The “twist” of
N by f is a smooth algebraic group N over k such that N(ksep) = N(ksep) but with
f f
τ ∈ Γactingaccordingtotherule
τ ∗x = f(τ)·τx.
When we let N act on itself by inner automorphisms, a crossed homomorphism f: Γ →
N(ksep)definesatwist N ofN withτ ∈ Γactingon N(ksep)by
f f
τ ∗x = f(τ)·τx·f(τ)−1.
1.18. LetG = lim(G ,u )beasintheprecedingsubsectionbutwithG nowasmoothal-
←− n n n
gebraicgroupoverk,anddefineH1 (Γ,G)bethecohomologysetcomputedusingcrossed
cts
homomorphisms Γ → G(ksep) that are continuous for the profinite topology on Γ and the
inverse limit topology on G(ksep) = limG (ksep) (discrete topology on G (ksep)). Thus,
←− n n
giving a continuous crossed homomorphism f: Γ → G(ksep) amounts to giving a compat-
iblefamilyofcontinuouscrossedhomomorphismsf : Γ → G (ksep).
n n
PROPOSITION 1.19. Themap
H1 (Γ,G) → limH1(Γ,G )
cts ←− n
sending[f]to([f ]) issurjective. Thefibreofthemapcontaining[f]equalslim1G0 (k)
n n≥0 ←− n
whereG0 = G .
n f n
PROOF. Each class c in l←im−H1(S,Gn) is represented by a family (fn)n≥0 of crossed ho-
momorphisms, which can be chosen so that f = u ◦ f . The f define a continuous
n−1 n n n
crossedhomomorphismf: Γ → G(ksep)mappingtoc.
Let f0 and f be continuous crossed homomorphisms such that f0 ∼ f for all n, and
n n
choosea ∈ G (ksep)insuchawaythat
n n
f0(τ) = a−1 ·f (τ)·τa . (9)
n n n n
Definee ∈ G (ksep)bytheequation
n n
e ·ua = a .
n n+1 n
Onapplyingutotheequation(9) ,weobtaintheequation
n+1
f0(τ) = (ua )−1 ·f (τ)·τ(ua ),
n n+1 n n+1
or,
f0(τ) = a−1 ·e ·f (τ)·τe−1 ·τa .
n n n n n n
Oncomparingthiswith(9),wefindthat
e = f (τ)·τe ·f (τ)−1,
n n n n