Table Of ContentA NOTE ON THE BLOCH-TAMAGAWA SPACE AND SELMER GROUPS
NIRANJANRAMACHANDRAN
ABSTRACT. ForanyabelianvarietyAoveranumberfield,weconstructanextensionoftheTate-
ShafarevichgroupbytheBloch-TamagawaspaceusingtherecentworkofLichtenbaumandFlach.
5 ThisgivesanewexampleofaZagiersequencefortheSelmergroupofA.
1
0
2
n Introduction. Let A be an abelian variety over a number field F and A∨ its dual. B. Birch and
a
P. Swinnerton-Dyer, interested in defining the Tamagawa number τ(A) of A, were led to their
J
celebrated conjecture [2, Conjecture 0.2] for the L-function L(A,s) (of A overF) which predicts
4
both its order r of vanishing and its leading term c at s = 1. The difficulty in defining τ(A)
A
]
T directlyisthattheadelicquotient A(AF) isHausdorffonlywhenr = 0,i.e.,A(F)isfinite. S.Bloch
A(F)
N
[2] has introduced a semiabelian variety G overF with quotientA such that G(F) is discrete and
h. cocompactinG(AF)[2,Theorem1.10]andfamouslyproved[2,Theorem1.17]thattheTamagawa
t number conjecture - recalled briefly below, see (5) - for G is equivalent to the Birch-Swinnerton-
a
m DyerconjectureforAoverF. ObservethatGisnotalinearalgebraicgroup. TheBloch-Tamagawa
[ spaceX = G(AF) ofA/F is compactand Hausdorff.
A G(F)
1 Theaim ofthisshortnoteis toindicateafunctorial constructionofalocallycompact groupY
A
v
0 (1) 0 → X → Y → Ш(A/F) → 0,
A A
4
6 an extension of the Tate-Shafarevich group Ш(A/F) by XA. The compactness of YA is clearly
0 equivalenttothefinitenessofШ(A/F). ThisconstructionwouldbestraightforwardifG(L)were
0
discreteinG(A ) forall finiteextensionsL ofF. But thisisnottrue(Lemma4): thequotient
. L
1
0 G(A )
L
5
G(L)
1
: isnot Hausdorff,ingeneral.
v
i The very simple idea for the construction of Y is: Yoneda’s lemma. Namely, we consider the
X A
categoryoftopologicalG-modulesasasubcategoryoftheclassifyingtoposBGofG(naturalfrom
r
a the context of the continuous cohomology of a topological group G, as in S. Lichtenbaum [10],
M.Flach [5])and constructY viatheclassifyingtoposoftheGaloisgroupofF.
A
D. Zagier [18] has pointed out that the Selmer groups Sel (A/F) (6) can be obtained from
m
certain two-extensions (7) of Ш(A/F) by A(F); these we call Zagier sequences. We show how
Y providesanewnaturalZagiersequence. Inparticular,thisshowsthatY isnotasplitsequence.
A A
Bloch’s construction has been generalized to one-motives; it led to the Bloch-Kato conjecture
onTamagawanumbersofmotives[3];itiscloseinspirittoScholl’smethodofrelatingnon-critical
valuesofL-functionsofpuremotivestocriticalvaluesofL-functionsofmixedmotives[9,p.252]
[13, 14].
Notations. We write A = A × R for the ring of adeles over Q; here A = Zˆ ⊗ Q is the ring
f f Z
of finite adeles. For any number field K, we let O be the ring of integers, A denote the ring of
K K
adeles A⊗ K over K; write I for the ideles. Let F¯ be a fixed algebraic closure of F and write
Q K
1
Γ = Gal(F¯/F) for the Galois group of F. For any abelian group P and any integer m > 0, we
writeP forthem-torsionsubgroupofP. A topologicalabeliangroup isHausdorff.
m
Construction of Y . This will use the continuous cohomology of Γ via classifying spaces as in
A
[10, 5] towhichwerefer foradetailed exposition.
For each field L with F ⊂ L ⊂ F¯, the group G(A ) is a locally compact group. If L/F is
L
Galois,then
G(A )Gal(L/F) = G(A ).
L F
So
E = limG(A ),
L
→
the direct limit of locally compact abelian groups, is equipped with a continuous action of Γ. The
natural map
(2) E := G(F¯) ֒→ E
isΓ-equivariant. ThoughthesubgroupG(F) ⊂ G(A )is discrete,thesubgroup
F
E ⊂ E
fails to be discrete; this failure happens at finite level (see Lemma 4 below). The non-Hausdorff
natureofthequotient
E/E
directs ustoconsidertheclassifyingspace/topos.
Let Top be the site defined by the category of (locally compact) Hausdorff topological spaces
with the open covering Grothendieck topology (as in the "gros topos" of [5, §2]). Any locally
compact abelian group M defines a sheaf yM of abelian groups on Top; this (Yoneda) provides a
fullyfaithfulembeddingofthe(additive,butnotabelian)categoryTaboflocallycompactabelian
groups into the (abelian) category Tab of sheaves of abelian groups on Top. Write Top for the
categoryofsheavesofsetsonTopandlety : Top → TopbetheYonedaembedding. Ageneralized
topologyon agivenset S is an objectF ofTop withF(∗) = S.
Forany(locallycompact)topologicalgroupG,itsclassifyingtoposBGisthecategoryofobjects
F of Top together with an action yG× F → F. An abelian group object F of BG is a sheaf on
Top, together with actions yG(U) ×F(U) → F(U), functorial in U; we write Hi(G,F) (objects
of Tab) for thecontinuous/topologicalgroup cohomologyofG with coefficients in F. These arise
from theglobalsectionfunctor
BG → Tab, F 7→ FyG.
Detailsforthefollowingfacts can befoundin [5,§3]and [10].
(a) (Yoneda)AnytopologicalG-moduleM providesan(abeliangroup)objectyM ofBG; see
[10, Proposition1.1].
(b) If 0 → M → N isa mapoftopologicalG-moduleswithM homeomorphicto itsimagein
N, then theinducedmapyM → yN isinjective[5, Lemma4].
(c) Applying Propositions 5.1 and 9.4 of [5] to the profinite group Γ and any continuous Γ-
moduleM providean isomorphism
Hi(Γ,yM) ≃ yHi (Γ,M)
cts
between this topological group cohomology and the continuous cohomology (computed
viacontinuouscochains). This isalsoprovedin[10, Corollary 2.4].
2
For any continuoushomomorphismf : M → N of topological abelian groups, the cokernel of
yf : yM → yN iswell-definedinTabevenifthecokerneloff doesnotexistinTab. Iff isamap
of topological G-modules, then the cokernel of the induced map yf : yM → yN, a well-defined
abelian groupobjectofBG, need not beoftheform yP.
By(a)and(b)above,thepairoftopologicalΓ-modulesE ֒→ E(2)givesrisetoapairyE ֒→ yE
of objects of BΓ. Write Y for the quotient object yE. As E/E is not Hausdorff (Lemma 4), Y is
yE
notyN foranytopologicalΓ-moduleN.
Definition 1. Weset Y = H0(Γ,Y) ∈ Tab.
A
Ourmain resultisthe
Theorem 2. (i) Y is the Yoneda image yY of a Hausdorff locally compact topological abelian
A A
groupY .
A
(ii)X is anopen subgroupof Y .
A A
(iii) The group Y is compact if and only if Ш(A/F) is finite. If Y is compact, then the index
A A
ofX in Y is equalto#Ш(A/F).
A A
AsШ(A/F)is atorsiondiscretegroup,thetopologyofY isdeterminedby thatofX .
A A
Proof. (ofTheorem 2)Thebasicpointistheproofof(iii). From theexact sequence
0 → yE → yE → Y → 0
ofabelian objectsinBΓ, weget alongexactsequence (inTab)
0 → H0(Γ,yE) → H0(Γ,yE) →
→ H0(Γ,Y) → H1(Γ,yE) −→j H1(Γ,yE) → ··· .
Wehavethefollowingidentitiesoftopologicalgroups: H0(Γ,yE) = yG(F)and H0(Γ,yE) =
yG(A ), and by [5, Lemma 4], yG(AF) ≃ yX . This exhibits yX as a sub-object of Y and
F yG(F) A A A
providestheexact sequence
0 → yX → Y → Ker(j) → 0.
A A
If Ker(j) = yШ(A/F), then Y = yY for a unique topological abelian group Y because
A A A
Ш(A/F) is a torsion discrete group. Thus, it suffices to identify Ker(j) as yШ(A/F). Let Eδ
denote E endowed with the discrete topology; the identity map on the underlying set provides a
continuous Γ-equivariant map Eδ → E. Since E is a discrete Γ-module, the inclusion E → E
factorizes via Eδ. By item (c) above, Ker (j) is isomorphic to the Yoneda image of the kernel of
thecompositemap
H1 (Γ,E) −→j′ H1 (Γ,Eδ) −→k H1 (Γ,E).
cts cts cts
Since E and Eδ are discrete Γ-modules, the map j′ identifies with the map of ordinary Galois
cohomologygroups
H1(Γ,E) −j→” H1(Γ,Eδ).
Thetraditionaldefinition[2, Lemma1.16]ofШ(G/F)is as Ker(j”). As
Ш(A/F) ≃ Ш(G/F)
3
[2,Lemma1.16],toproveTheorem2,allthatremainsistheinjectivityofk. Thisisstraightforward
from the standard description of H1 in terms of crossed homomorphisms: if f : Γ → Eδ is a
crossed homomorphismwithkf principal,then thereexistsα ∈ Ewithf : Γ → Esatisfies
f(γ) = γ(α)−α γ ∈ Γ.
ThisidentityclearlyholdsinbothEandEδ. SincetheΓ-orbitofanyelementofEisfinite,theleft
hand side is a continuous map from Γ to Eδ. Thus, f is already a principal crossed (continuous)
homomorphism. So k isinjective,finishingtheproofofTheorem 2. (cid:3)
Remark 3. The proof above shows: If the stabilizer of every element of a topological Γ-module
N is openin Γ, thenthenaturalmap H1(Γ,Nδ) → H1(Γ,N) is injective.
Bloch’s semi-abelianvarietyG. [2, 11]
WriteA∨(F) = B ×finite. By theWeil-Barsottiformula,
Ext1(A,G ) ≃ A∨(F).
F m
EverypointP ∈ A∨(F)determinesa semi-abelianvarietyG which isan extensionofAby G .
P m
Let G bethesemiabelianvarietydeterminedby B:
(3) 0 → T → G → A → 0,
an extension of A by the torus T = Hom(B,G ). The semiabelian variety G is the Cartier dual
m
[4, §10]oftheone-motive
[B → A∨].
Thesequence(3)provides(viaHilbertTheorem 90)[2,(1.4)]thefollowingexact sequence
T(A ) G(A ) A(A )
(4) 0 → F → F → F → 0.
T(F) G(F) A(F)
It is worthwhile to contemplate this mysterious sequence: the first term is a Hausdorff, non-
compact group and the last is a compact non-Hausdorff group, but the middle term is a compact
Hausdorffgroup!
Lemma 4. ConsiderafieldLwithF ⊂ L ⊂ F¯. ThegroupG(L) isadiscretesubgroupofG(A )
L
ifand onlyifA(K) ⊂ A(L) isoffiniteindex.
Proof. Pick a subgroup C ≃ Zs of A∨(L) such that B ×C has finite index in A∨(L). The Bloch
semiabelianvarietyG′overLdeterminedbyB×C isanextensionofAbyT′ = Hom(B×C,G ).
m
Onehas anexactsequence0 → T” → G′ → G → 0 definedoverLwhereT” = Hom(C,G ) is
m
asplittorusofdimensions. Considerthecommutativediagramwithexact rowsand columns
4
0 0
y y
T”(AL) T”(AL)
T”(L) T”(L)
y y
0 −−−→ T′(AL) −−−→ G′(AL) −−−→ A(AL)
T′(L) G′(L) A(L)
(cid:13)
(cid:13)
y y (cid:13)
0 −−−→ T(AL) −−−→ G(AL) −−−→ A(AL)
T(L) G(L) A(L)
y y y
0 0 0.
The proof of surjectivityin thecolumns followsHilbert Theorem 90 applied to T” [2, (1.4]). The
Bloch-TamagawaspaceX′ = G′(AL) forAoverLis compactand Hausdorff;itsquotientby
A G′(L)
T”(A ) I
L = ( L)s
T”(L) L∗
is G(AL).ThequotientisHausdorffifand onlyifs = 0. (cid:3)
G(L)
A more general form of Lemma 4 is implicit in [2]: For any one-motive [N −→φ A∨] over F,
writeV foritsCartier dual(a semiabelianvariety), andput
V(A )
F
X = .
V(F)
We assumethat the Γ-action on N is trivial. Then X is compact if and only if Ker(φ) is finite; X
isHausdorffifand onlyiftheimageofφ hasfiniteindexinA∨(F).
Tamagawa numbers. Let H be a semisimple algebraic group over F. Since H(F) embeds dis-
cretely inH(A ),theadelicspaceX = H(AF) isHausdorff. TheTamagawanumberτ(H)isthe
F H H(F)
volumeof X relativeto a canonical (Tamagawa)measure [15]. The Tamagawanumber theorem
H
[8, 1](whichwas formerly aconjecture)states
#Pic(H)
(5) τ(H) = torsion
#Ш(H)
where Pic(H) is the Picard group and Ш(H) the Tate-Shafarevich set of H/F (which measures
thefailureoftheHasseprinciple). TakingH = SL overQ in(5)recovers Euler’sresult
2
π2
ζ(2) = .
6
The above formulation (5) of the Tamagawa number theorem is due to T. Ono [12, 17] whose
study of the behavior of τ under an isogeny explains the presence of Pic(H), and reduces the
semisimplecase to the simply connected case. The original form of the theorem (due to A. Weil)
is that τ(H) = 1 for splitsimplyconnected H. TheTamagawanumbertheorem (5)is valid,more
5
generally, for any connected linear algebraic group H over F. The case H = G becomes the
m
Tate-Iwasawa[16,7]versionoftheanalyticclassnumberformula: theresidueats = 1ofthezeta
functionζ(F,s)isthevolumeofthe(compact)unitideleclassgroupJ1 = Ker(|−| : IF → R )
F F∗ >0
ofF. Here |−| istheabsolutevalueornormmap onI .
F
Zagierextensions. [18]Them-Selmer groupSel (A/F)(form > 0)fits intoan exactsequence
m
A(F)
(6) 0 → → Sel (A/F) → Ш(A/F) → 0.
m m
mA(F)
D.Zagier[18,§4]haspointedoutthatwhilethem-Selmersequences(6)(forallm > 1)cannot
beinducedby asequence(an extensionofШ(A/F)by A(F))
0 → A(F) →? → Ш(A/F) → 0,
theycan beinducedby an exact sequenceoftheform
(7) 0 → A(F) → A → S → Ш(A/F) → 0
and gaveexamplesofsuch (Zagier) sequences. Combining(1) and (4) aboveprovidesthefollow-
ingnatural Zagiersequence
Y
0 → A(F) → A(A ) → A → Ш(A/F) → 0.
F T(A )
F
Write A(A ) for the direct limit of the groups A(A ) over all finite subextensions F ⊂ L ⊂ F¯.
F¯ L
Theprevioussequencediscretized (neglectthetopology)becomes
A(A )
0 → A(F) → A(A ) → ( F¯ )Γ → Ш(A/F) → 0.
F ¯
A(F)
Remark 5. (i) For an elliptic curve E over F, Flach has indicated how to extract a canonical
Zagier sequence via τ τ RΓ(S ,G ) from any regular arithmetic surface S → Spec O with
≥1 ≤2 et m F
E = S × Spec F.
Spec OF
(ii) It is well known that the class group Pic(O ) is analogous to Ш(A/F) and the unit group
F
O× is analogous to A(F). Iwasawa [6, p. 354] proved that the compactness of J1 is equivalentto
F F
thetwo basicfiniteness results of algebraic numbertheory: (i)Pic(O ) is finite; (ii)O× is finitely
F F
generated. His result provided a beautiful new proof of these two finiteness theorems. Bloch’s
result [2, Theorem 1.10] on the compactness of X uses the Mordell-Weil theorem (the group
A
A(F) is finitely generated) and the non-degeneracy of the Néron-Tate pairing on A(F)×A∨(F)
(modulotorsion).
Question 6. Can one define directly a space attached to A/F whose compactness implies the
Mordell-Weiltheorem forAandthefinitenessof Ш(A/F)?
Acknowledgements. IthankC.Deninger,M.Flach,S.Lichtenbaum,J.Milne,J.Parson,J.Rosen-
berg, L. Washington, and B. Wieland for interest and encouragement and the referee for helping
correct some inaccuracies in an earlier version of this note. I have been inspired by the ideas of
Bloch,T.Ono(viaB.Wieland[17])onTamagawanumbers,andD.ZagierontheTate-Shafarevich
group[18]. Iam indebtedtoRan Cui foralertingmetoWieland’sideas[17].
6
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DEPARTMENTOF MATHEMATICS,UNIVERSITY OFMARYLAND, COLLEGEPARK, MD 20742USA.
E-mailaddress:[email protected]
URL:http://www2.math.umd.edu/~atma
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