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A note on the Bloch-Tamagawa space and Selmer groups.

Introduction. Let A be an abelian variety over a number field F and [A.sup.[disjunction]] its dual. Birch and Swinnerton-Dyer, interested in defining the Tamagawa number [tau](A) of A, were led to their celebrated conjecture [2, Conjecture 0.2] for the L-function L(A, s) (of A over F) which predicts both its order r of vanishing and its leading term ca at s = 1. The difficulty in defining [tau](A) directly is that the adelic quotient A([A.sub.F])/A(F) is Hausdorff only when r = 0, i.e., A(F) is finite. Bloch [2] has introduced a semiabelian variety G over F such that G(F) is discrete and cocompact in G([A.sub.F]) [2, Theorem 1.10] and famously proved [2, Theorem 1.17] that the Tamagawa number conjecture (4) for G is equivalent to the Birch-Swinnerton-Dyer conjecture for A over F. Observe that G is not a linear algebraic group.

The aim of this short note is to indicate a functorial construction of a locally compact group [Y.sub.A] given as

(1) 0 [right arrow] [X.sub.A] [right arrow] [Y.sub.A] [right arrow] III(A/F) [right arrow] 0,

an extension of m(A/F) by [X.sub.A]. The compactness of [Y.sub.A] is clearly equivalent to the finiteness of m(A/F). This would be straightforward if G(L) were discrete in G([A.sub.L]) for all finite extensions L of F. But this is not true (Lemma 4):

G([A.sub.L])/G(L)

is not Hausdorff in general.

The very simple idea for the construction of [Y.sub.A] is: Yoneda's lemma. Namely, consider the category of topological G-modules as a subcategory of the classifying topos BG of G (continuous cohomology of a topological group G, as in S. Lichtenbaum [10], M. Flach [5]).

D. Zagier [18] has pointed out that the Selmer groups [Sel.sub.m](A/F) (5) can be obtained from certain two-extensions (6) of III(A/F) by A(F); these we call Zagier sequences. We show how [Y.sub.A] provides a new natural Zagier sequence. In particular, this shows that [Y.sub.A] is not a split sequence.

Bloch's construction has been generalized to one-motives; it led to the Bloch-Kato conjecture on Tamagawa numbers of motives [3]; it is also closely related to Scholl's method of relating non-critical values of L-functions of pure motives to critical values of L-functions of mixed motives [9, p. 252] [13, 14].

Notations. We write A = [A.sub.f] x R for the ring of adeles over Q; here [A.sub.f] = [??] [[cross product].sub.z] Q is the ring of finite adeles. For any number field K, we let [O.sub.K] be the ring of integers, [A.sub.K] denote the ring of adeles A [[cross product].sub.Q] K over K; write [I.sub.K] for the ideles. Let [??] be a fixed algebraic closure of F and write [GAMMA] = Gal(F/F) for the Galois group of F. For any abelian group P and any integer m > 0, we write [P.sub.m] for the m-torsion subgroup of P. A topological abelian group is Hausdorff.

Construction of [Y.sub.A]. This will use the continuous cohomology of [GAMMA] via classifying spaces [10, 5] to which we refer for a detailed exposition.

For each field L with F [subset] L [subset] [??], the group G([A.sub.L]) is a locally compact group. If L/F is Galois, then

G[([A.sub.L]).sup.Gal(L/F)] = G([A.sub.F]).

So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

the direct limit of locally compact abelian groups, is equipped with a continuous action of [GAMMA] = [[GAMMA].sub.F]. The natural map

E := G([??])[??] E

is [[GAMMA].sub.F]-equivariant. Though the subgroup G(F) [subset] G([A.sub.F]) is discrete, the subgroup

E [subset] E

fails to be discrete; this failure happens at finite level (see Lemma 4 below). The non-Hausdorff nature of the quotient

E/E

directs us to consider the classifying space/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, [section]2]). Any locally compact abelian group M defines a sheaf yM of abelian groups on Top; this (Yoneda) provides a fully faithful embedding of the (additive, but not abelian) category Tab of locally compact abelian groups into the (abelian) category Tab of sheaves of abelian groups on Top. Write Top for the category of sheaves of sets on Top and let y : Top [right arrow] Top be the Yoneda embedding. A generalized topology on a given set S is an object F of Top with F(*) = S.

For any (locally compact) topological group G, its classifying topos BG is the category of objects F of Top together with an action yG x F [right arrow] F. An abelian group object F of BG is a sheaf on Top, together with actions yG(U) x F(U) [right arrow] F(U), functorial in U; we write [H.sup.i](G, F) (objects of Tab) for the continuous/topological group cohomology of G with coefficients in F. These arise from the global section functor

BG [right arrow] Tab, F [??] [F.sup.yG].

Details for the following facts can be found in [5, [section]3] and [10].

(a) (Yoneda) Any topological G-module M provides an (abelian group) object yM of BG; see [10, Proposition 1.1].

(b) If 0 [right arrow] M [right arrow] N is a map of topological G-modules with M homeomorphic to its image in N, then the induced map yM [right arrow] yN is injective [5, Lemma 4].

(c) Applying Propositions 5.1 and 9.4 of [5] to the profinite group T and any continuous [GAMMA]-module M provide an isomorphism

[H.sup.i]([GAMMA], yM) [equivalent] [H.sup.i.sub.cts]([GAMMA], M)

between this topological group cohomology and the continuous cohomology (computed via continuous cochains). This is also proved in [10, Corollary 2.4].

For any map f : M [right arrow] N of topological abelian groups, the cokernel of yf : yM [right arrow] yN is well-defined in Tab even if the cokernel of f does not exist in Tab. If f is a map of topological G-modules, then the cokernel of the induced map yf : yM [right arrow] yN, a well-defined abelian group object of BG, need not be of the form yP.

The pair of topological [GAMMA]-modules E [??] E gives rise to a pair yE [??] yE of objects of B[GAMMA]. Write [gamma] for the quotient object yE/yE. As E/E is not Hausdorff (Lemma 4), [gamma] is not yN for any topological [GAMMA]-module N.

Definition 1. We set [gamma]A = [H.sup.0]([GAMMA], [gamma]) [member of] Tab. Our main result is the

Theorem 2.

(i) [[gamma].sub.A] is the Yoneda image y[Y.sub.A] of a Hausdorff locally compact topological abelian group [Y.sub.A].

(ii) [X.sub.A] is an open subgroup of [Y.sub.A].

(iii) The group [Y.sub.A] is compact if and only if III(A/F) is finite. If [Y.sub.A] is compact, then the index of [X.sub.A] in [Y.sub.A] is equal to #III(A/F).

As III(A/F) is a torsion discrete group, the topology of [Y.sub.A] is determined by that of [X.sub.A].

Proof of Theorem 2. The basic point is the proof of (iii). From the exact sequence

0 [right arrow] yE [right arrow] yE [right arrow] [gamma] [right arrow] 0

of abelian objects in B[GAMMA], we get a long exact sequence (in Tab)

0 [right arrow] [H.sup.0] ([GAMMA], yE) [right arrow] [H.sup.0] ([GAMMA], yE) [right arrow] [right arrow] [H.sup.0]([GAMMA], [gamma]) [right arrow] [H.sup.1] ([GAMMA], yE) [??] [H.sup.1]([GAMMA], yE) [right arrow]....

We have the following identities of topological groups: [H.sup.0]([GAMMA], yE) = yG(F) and [H.sup.0]([GAMMA], yE) = yG([A.sub.F]). Thus, it suffices to identify Ker(j) as yIII(A/F). Let [E.sup.[delta]] denote E endowed with the discrete topology; the identity map on the underlying set provides a continuous [GAMMA]-equivariant map [E.sup.[delta]] [right arrow] E. Since E is a discrete [GAMMA]-module, the inclusion E [right arrow] E factorizes via [E.sup.[delta]]. By item (c) above, Ker(j) is isomorphic to the Yoneda image of the kernel of the composite map

[H.sup.1.sub.cts]([GAMMA], E) [??] [H.sup.1.sub.cts]([GAMMA], [E.sup.[delta]]) [??] [H.sup.1.sub.cts]([GAMMA], E).

Since E and [E.sup.[delta]] are discrete r-modules, the map j' identifies with the map of ordinary Galois cohomology groups

[H.sup.1]([GAMMA], E) [??] [H.sup.1]([GAMMA], [E.sup.[delta]]).

The traditional definition [2, Lemma 1.16] of III(G/F) is as Ker(j"). As

III(A/F) [equivalent] III(G/F)

[2, Lemma 1.16], to prove Theorem 2, all that remains is the injectivity of k. This is straightforward from the standard description of [H.sup.1] in terms of crossed homomorphisms: if f : [GAMMA] [right arrow] [E.sup.[delta]] is a crossed homomorphism with kf principal, then there exists [alpha] [member of] E with f : [GAMMA] [right arrow] E satisfies

f([gamma]) = [gamma]([alpha]) - [alpha] [gamma] [member of] [GAMMA].

This identity clearly holds in both E and [E.sup.[delta]]. Since the r-orbit of any element of E is finite, the left hand side is a continuous map from r to [E.sup.[delta]]. Thus, f is already a principal crossed (continuous) homomorphism. So k is injective, finishing the proof of Theorem 2.

Remark 3. The proof above shows: If every element of a topological [GAMMA]-module N has open stabilizer, then the natural map [H.sup.1]([GAMMA], [N.sup.[delta]]) [right arrow] [H.sup.1]([GAMMA], N) is injective.

Bloch's construction of G [2,11]. Write [A.sup.[disjunction]](F) = B x finite. By the Weil-Barsotti formula,

[Ext.sup.1.sub.F](A, [G.sub.m]) [equivalent] [A.sup.[disjunction]](F).

Every point P [member of] [A.sup.[disjunction]](F) determines a semi-abelian variety [G.sub.P] which is an extension of A by [G.sub.m]. Let G be the semiabelian variety determined by B:

(2) 0 [right arrow] T [right arrow] G [right arrow] A [right arrow] 0,

an extension of A by the torus T = Hom(B, [G.sub.m]). The semiabelian variety G is the Cartier dual [4, [section]10] of the one-motive

[B [right arrow] [A.sup.[disjunction]]].

The sequence (2) provides (via Hilbert Theorem 90) [2, (1.4)] the following exact sequence

(3) 0 [right arrow] T([A.sub.F])/T(F) [right arrow] G([A.sub.F])/G(F) [right arrow] A([A.sub.F])/A(F) [right arrow] 0.

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 Hausdorff group!

Lemma 4. For any field L with F [subset] L [subset] [??], the group G(L) is a discrete subgroup of G([A.sub.L]) if and only if A(K) [subset] A(L) is of finite index.

Proof. Pick a subgroup C [equivalent] [Z.sup.s] of [A.sup.[disjunction]](L) such that B x C has finite index in [A.sup.[disjunction]](L). The Bloch semiabelian variety G' over L determined by B x C is an extension of A by T' = Hom(B x C, [G.sub.m]). One has an exact sequence 0 [right arrow] T" [right arrow] G' [right arrow] G [right arrow] 0 defined over L where T" = Hom(C, [G.sub.m]) is a split torus of dimension s. Consider the commutative diagram with exact rows and columns

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof of surjectivity in the columns follows Hilbert Theorem 90 applied to T" [2, (1.4]). The Bloch-Tamagawa space [X'.sub.A] = G'([A.sub.L])/G'(L) for A over L is compact and Hausdorff; its quotient by

T"([A.sub.L])/T"(L) = [([I.sub.L]/[L.sup.*]).sup.s]

is G([A.sub.L])/G(L). The quotient is Hausdorff if and only if s = 0.

A more general form of Lemma 4 is implicit in [2]: For any one-motive [N [??] [A.sup.[disjunction]]] over F, write V for its Cartier dual (a semiabelian variety), and put

X = V([A.sub.F])/V(F).

Then X is compact if and only if Ker([phi]) is finite; X is Hausdorff if and only if the image of [phi] has finite index in [A.sup.[disjunction]](F).

Tamagawa numbers. Let H be a semisimple algebraic group over F. Since H(F) embeds discretely in H([A.sub.F]), the adelic space [X.sub.H] = H([A.sub.F])/H(F) is Hausdorff. The Tamagawa number [tau](H) is the volume of [X.sub.H] relative to a canonical (Tamagawa) measure [15]. The Tamagawa number theorem [8,1] (which was formerly a conjecture) states

(4) [tau](H) = #Pic[(H).sub.torsion]/#III(H)

where Pic(H) is the Picard group and III(H) the Tate-Shafarevich set of H/F (which measures the failure of the Hasse principle). Taking H = S[L.sub.2] over Q in (4) recovers Euler's result

[zeta](2) = [[pi].sup.2]/6.

The above formulation (4) of the Tamagawa number theorem is due to T. Ono [12,17] whose study of the behavior of [tau] under an isogeny explains the presence of Pic(H), and reduces the semisimple case to the simply connected case. The original form of the theorem (due to A. Weil) is that [tau] (H) = 1 for split simply connected H. The Tamagawa number theorem (4) is valid, more generally, for any connected linear algebraic group H over F. The case H = [G.sub.m] becomes the Tate-Iwasawa [16,7] version of the analytic class number formula: the residue of the zeta function [zeta](F, s) is the volume of the (compact) unit idele class group [J.sup.1.sub.F] of F.

Zagier extensions [18]. The m-Selmer group [Sel.sub.m] (A/F) (for m > 0) fits into an exact sequence

(5) 0 [right arrow] A(F)/mA(F) [right arrow] [Sel.sub.m](A/F) [right arrow] III[(A/F).sub.m] [right arrow] 0.

D. Zagier [18, [section]4] has pointed out that while the m-Selmer sequences (5) (for all m > 1) cannot be induced by a sequence (an extension of III(A/F) by A(F))

0 [right arrow] A(F) [right arrow] ? [right arrow] III(A/F) [right arrow] 0,

they can be induced by an exact sequence of the form

(6) 0 [right arrow] A(F) [right arrow] A [right arrow] S [right arrow] III(A/F) [right arrow] 0

and gave examples of such (Zagier) sequences. Combining (1) and (3) above provides the following natural Zagier sequence

0 [right arrow] A(F) [right arrow] A([A.sub.F]) [right arrow] [Y.sub.A]/T([A.sub.F]) [right arrow] III(A/F) [right arrow] 0.

Write A([A.sub.[??]]) for the direct limit of the groups A([A.sub.L]) over all finite subextensions F [subset] L [subset] [??]. The previous sequence discretized (neglect the topology) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 5. (i) For an elliptic curve E over F, Flach has indicated how to extract a canonical Zagier sequence via [sup.[tau][greater than or equal to]1[tau][greater than or equal to]2]R[GAMMA]([S.sub.et], [G.sub.m]) from any regular arithmetic surface S [right arrow] Spec [O.sub.F] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Spec F.

(ii) It is well known that the class group Pic([O.sub.F]) is analogous to m(A/F) and the unit group [O.sup.x.sub.F] is analogous to A(F). Iwasawa [6,p. 354] proved that the compactness of [J.sup.1.sub.F] is equivalent to the two basic finiteness results of algebraic number theory: (i) Pic([O.sub.F]) is finite; (ii) [O.sup.x.sub.F] is finitely 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.sub.A] uses the Mordell-Weil theorem (A(F) is finitely generated) and the non-degeneracy of the Neron-Tate pairing on A(F) x [A.sup.[disjunction]](F) (modulo torsion).

Question 6. Can one define directly a space attached to A/F whose compactness implies the Mordell-Weil theorem for A and the finiteness of III(A/F)?

doi: 10.3792/pjaa.91.61

Acknowledgements. I thank Profs. C. Deninger, M. Flach, S. Lichtenbaum, J. Milne, J. Parson, J. Rosenberg, 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, Prof. T. Ono (via B. Wieland [17]) on Tamagawa numbers, and Prof. D. Zagier on the Tate-Shafarevich group [18]. I am indebted to Ran Cui for alerting me to Wieland's ideas [17].

References

[1] A. Asok, B. Doran and F. Kirwan, Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics, Bull. Lond. Math. Soc. 40 (2008), no. 4, 533-567.

[2] S. Bloch, A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Invent. Math. 58 (1980), no. 1, 65-76.

[3] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progr. Math., 86, Birkhauser, Boston, MA, 1990, pp. 333-400.

[4] P. Deligne, Theorie de Hodge. III, Inst. Hautes Etudes Sci. Publ. Math. 44 (1974), 5-77.

[5] M. Flach, Cohomology of topological groups with applications to the Weil group, Compos. Math. 144 (2008), no. 3, 633-656.

[6] K. Iwasawa, On the rings of valuation vectors, Ann. of Math. (2) 57 (1953), 331-356.

[7] K. Iwasawa, Letter to J. Dieudonne, in Zeta functions in geometry (Tokyo, 1990), 445 450, Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo, 1992.

[8] R. E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), no. 3, 629-646.

[9] S. Lichtenbaum, Euler characteristics and special values of zeta-functions, in Motives and algebraic cycles, Fields Inst. Commun., 56, Amer. Math. Soc., Providence, RI, 2009, pp. 249-255.

[10] S. Lichtenbaum, The Weil-etale topology for number rings, Ann. of Math. (2) 170 (2009), no. 2, 657-683.

[11] J. Oesterle, Construction de hauteurs archimediennes et p-adiques suivant la methode de Bloch, in Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), 175-192, Progr. Math., 22, Birkhauser, Boston, MA, 1982.

[12] T. Ono, On Tamagawa numbers, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 122-132, Amer. Math. Soc., Providence, RI, 1996.

[13] A. J. Scholl, Extensions of motives, higher Chow groups and special values of L-functions, in Seminaire de Theorie des Nombres, Paris, 1991-92, Progr. Math., 116, Birkhauser, Boston, MA, 1993, pp. 279-292.

[14] A. J. Scholl, Height pairings and special values of L-functions, in Motives (Seattle, WA, 1991), 571-598, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

[15] T. Tamagawa, Adeles, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), 113-121, Amer. Math. Soc., Providence, RI, 1966.

[16] J. T. Tate, Fourier analysis in number fields, and Hecke's zeta-functions, in Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), 305-347, Thompson, Washington, DC, 1967.

[17] B. Wieland (http://mathoverflow.net/users/ 4639/ben-wieland), Why are tamagawa numbers equal to Pic/Sha? MathOverflow. URL: http://mathoverflow.net/q/44360 (version: 2010-10-31).

[18] D. Zagier, The Birch-Swinnerton-Dyer conjecture from a naive point of view, in Arithmetic algebraic geometry (Texel, 1989), 377-389, Progr. Math., 89, Birkhauser, Boston, MA, 1991.

By Niranjan Ramachandran

Department of Mathematics, University of Maryland, College Park, MD 20742, USA

(Communicated by Shigefumi MORI, M.J.A., April 13, 2015)

2010 Mathematics Subject Classification. Primary 11G40, 14H52, 20G30, 22E41.
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