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A 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 REFERENCES [1] AravindAsok,BrentDoran,andFrancesKirwan.Yang-MillstheoryandTamagawanumbers:thefascinationof unexpectedlinksinmathematics.Bull.Lond.Math.Soc.,40(4):533–567,2008. [2] S.Bloch.Anoteonheightpairings,Tamagawanumbers,andtheBirchandSwinnerton-Dyerconjecture.Invent. Math.,58(1):65–76,1980. [3] Spencer Bloch and Kazuya Kato. L-functions and Tamagawa numbers of motives. In The Grothendieck Festschrift,Vol.I,volume86ofProgr.Math.,pages333–400.BirkhäuserBoston,Boston,MA,1990. [4] PierreDeligne.ThéoriedeHodge.III.Inst.HautesÉtudesSci.Publ.Math.,44:5–77,1974. [5] M.Flach.CohomologyoftopologicalgroupswithapplicationstotheWeilgroup.Compos.Math.,144(3):633– 656,2008. [6] KenkichiIwasawa.Ontheringsofvaluationvectors.Ann.ofMath.(2),57:331–356,1953. 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