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On Ext to the 2nd of motives over arithmetic curves.

0. Introduction. In this note we give yet another piece of evidence for the thesis that number theory is geometry. For a number field k consider the category [MM.sub.k] of mixed motives over k constructed by Jannsen [J] and Deligne [De1]. In [Sch] Scholl has defined a full subcategory [MM.sub.[O.sub.k]] of mixed motives over the ring of integers [O.sub.k] by imposing a local condition at every finite prime of k. In accordance with the Beilinson-Deligne conjectures on values of L-functions one expects that

(0.1) [Mathematical Expression Omitted]

for every mixed motive M over [O.sub.k].

Let us now turn to the compactification y = spec [O.sub.k] [union] {p [where] [infinity]} of spec [O.sub.k]. Scholl's local condition on a motive in [MM.sub.k] to be integral at a finite prime p can be reformulated in terms of the functor [F.sub.p] of [D, (3.1.1)]. Since an analogous functor [F.sub.p] exists for archimedian primes p as well we are led to a definition of integrality at p [where] [infinity] in terms of [F.sub.p]. In the appendix we show that it is equivalent to the following Hodge theoretic condition:

CONDITION 0.2. A mixed motive is integral at p [where] [infinity] if its real Betti realization at p is semisimple as an R-mixed Hodge structure over [k.sub.p].

That this is the analogue of Scholl's integrality condition at an infinite prime is also confirmed by the following example. Consider the natural injection (cf. [Sch, 2.7]):

[Mathematical Expression Omitted]

where K(x [cross product] 1) for x [element of] [k.sup.*] is the class of the 1-motive [[Mathematical Expression Omitted]]. Then for any place p of k the extension K(x [cross product] 1) is integral at p in the sense of Scholl resp. in the sense of Condition 0.2 if and only if [[absolute value of x].sub.p] = 1 as follows easily from loc. cit.

Condition 0.2 was earlier introduced by Fontaine and Perrin-Riou in [F-PR, III 2.1.5 ii] as the definition for good reduction of a motive at p [where] [infinity]. In essence it was also suggested to us independently by Flach.

Let [MM.sub.y] be the full subcategory of [MM.sub.[O.sub.k]] obtained by imposing integrality at all places of k.

The considerations in [D, [section]7] e.g. (7.22) suggest the following analogies:

[Mathematical Expression Omitted] for M in [Mathematical Expression Omitted] for an affine curve U/[F.sub.q] and a smooth [Q.sub.l]-sheaf M on U.

(0.3) [Mathematical Expression Omitted] for M in [Mathematical Expression Omitted] for a projective curve Y/[F.sub.q] and a smooth [Q.sub.l]-sheaf M on Y. Here [Mathematical Expression Omitted].

This gives a heuristic explanation to conjecture (0.1) on the vanishing of [Ext.sup.2] in [MM.sub.[O.sub.k]]. On the other hand we are led to expect that [Mathematical Expression Omitted] is nonzero. One result of our paper confirms this provided the following conjecture is true:

Conjecture 0.4. [Mathematical Expression Omitted] is an isomorphism.

Our approach was suggested by a reexamination of the trace map Tr for projective curves X over algebraically closed fields: Note that for n invertible on X, Tr is the composition:

(0.5) [Mathematical Expression Omitted].

We found a construction of the inverse [cl.sup.-1] of the cycle class map and hence of Tr on elements of [H.sup.2](X, Z/n(1)) viewed as 2-extensions of etale sheaves on X.

A generalization of this construction to an abstract categorial framework is contained in [section]1. In [section]2 among other things the trace map for curves over algebraically closed fields is recovered by specializing [section]1 suitably.

In [section]3 we apply the abstract formalism of [section]1 to motives: A subgroup [Mathematical Expression Omitted] of [Mathematical Expression Omitted], conjecturally the whole group, is defined by certain local conditions. For example if (0.1) holds then we have

[Mathematical Expression Omitted].

Assuming Conjecture 0.4 we obtain the "inverse of a cycle class map":

(0.6) [Mathematical Expression Omitted]

into the Arakelov Chow group [CH.sup.1](y) and hence by composition a trace map:

(0.7) [Mathematical Expression Omitted].

If we do not assume Conjecture 0.4 the formalism of [section]1 still leads to a map "[cl.sup.-1]" in (0.6) but into an a priori other group than [CH.sup.1](Y).

For simplicity of exposition let us now assume (0.1) in addition to Conjecture 0.4. Then the composition

(0.8) [Mathematical Expression Omitted]

is the motivic height pairing. The verification turns out to be no more than a reinterpretation of the work of Scholl in [Sch]. Let us also note that Fontaine and Perrin-Riou have constructed the motivic height pairing as such a duality but with an ad hoc definition of [Mathematical Expression Omitted], which is not in terms of 2-extensions [F-PR, III 3.2.9]. The first ideas in the direction of (0.8) seem to be due to Beilinson in [Bel, 0.3].

In section 3 we give a more involved comparison statement with known height pairings which assumes only Conjecture 0.4 but still implies that the map

[Mathematical Expression Omitted]

and hence the group [Mathematical Expression Omitted] are nonzero.

After we had found the map (0.6) the way described above, we noticed that the local constructions involved agree with Scholl's local height pairings. This made the comparison of (0.8) with the global height pairing straightforward.

We would like to mention that our note and in particular the map "[cl.sup.-1]" in (0.6) may be viewed as a modest first step towards linearizing Arakelov theory in terms of motivic sheaves.

It is a pleasure for us to thank the referee for his valuable suggestions, which have led to a number of improvements in content and exposition.

1. Remarks on Yoneda [Ext.sup.2]-groups in abelian categories. In this section we construct in an abstract categorial framework maps between certain subquotients of Ext-groups. In the concrete situations of sections 2 and 3 these homomorphisms give rise to trace maps.

Let us recall the definition of Yoneda [Ext.sup.2]-groups. Given an abelian category A assume that we have a commutative diagram in A with exact lines:

(1.1) [Mathematical Expression Omitted].

Then we say that the upper 2-extension is related to the lower one. Consider the equivalence relation on 2-extensions of [A.sub.1] by [A.sub.2] generated by this relation. The set of equivalence classes is denoted by [Mathematical Expression Omitted], it is naturally an abelian group.

(1.2) We now introduce our basic data obtained by generalizing the geometric situation from the beginning of section 2:

1) A diagram of exact functors between abelian categories:

together with a natural transformation sp: [r.sub.0] [approaches] [r.sub.(0)] [convolution] [r.sub.[Eta]]. We write [r.sub.[Eta]](L) = [L.sub.[Eta]], [r.sub.(0)](E) = [E.sub.(0)], ([r.sub.(0)] [convolution] [r.sub.[Eta]])(L) = [L.sub.[Eta](0)] = [L.sub.(0)], [r.sub.0](L) = [L.sub.0] and sp = [sp.sub.L] : [L.sub.0] [approaches] [L.sub.(0)] for the induced morphism.

2) An object Q of [Chi] such that

i) sp: [Q.sub.0] [approaches] [Q.sub.(0)] is an isomorphism in [[Chi].sub.0].

ii) The functor [Mathematical Expression Omitted] is exact on [Chi]. In particular for any extension 0 [approaches] L[prime] [approaches] L [approaches] Q [approaches] 0 in [Chi] the induced extension 0 [approaches] [L[prime].sub.0] [approaches] [L.sub.0] [approaches] [Q.sub.0] [approaches] 0 splits in [[Chi].sub.0].

(1.3) In the situation of (1.2) for any L in [Chi] set

[Mathematical Expression Omitted]


[Mathematical Expression Omitted]

the categories being omitted from the notation.

The assumptions will allow us to construct a canonical homomorphism:

[Mathematical Expression Omitted]

which is functorial in the following sense:

(1.5) Let ([X.sup.*], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], [Mathematical Expression Omitted], s[p.sup.*], [Q.sup.*]) be another setup as in (1.2). Any diagram with exact functors * and commuting squares

[Mathematical Expression Omitted]

such that *Q = [Q.sup.*] and [Mathematical Expression Omitted] holds for all L in X - a morphism of data - induces a commutative diagram:

[Mathematical Expression Omitted]

with [Mathematical Expression Omitted].

(1.7) Construction of [Phi]. Let m [element of] [Eta][Ext.sup.2](Q,L) be represented by the 2-extension:

0 [right arrow] L [right arrow] F[prime] [right arrow] F[double prime] [right arrow] Q [right arrow] 0.

Setting F = Im (F[prime] [right arrow] F[double prime]) we obtain two extensions:

[Mathematical Expression Omitted].

The exact sequence 0 [right arrow] [F.sub.[Eta]] [right arrow] [F[double prime].sub.[Eta]] [right arrow] [Q.sub.[Eta]] [right arrow] 0 gives rise to the exact sequence

[Mathematical Expression Omitted].

Because of [Delta][[F[prime].sub.[Eta]]] = [r.sub.[Eta]](m) = 0 there exists an element e of [Ext.sup.1]([F[double prime].sub.[Eta]], [L.sub.[Eta]]) which is mapped to [[F[prime].sub.[Eta]]]. It is unique up to elements coming from [Ext.sup.1]([Q.sub.[Eta]], [L.sub.[Eta]]). By assumption we have a commutative diagram

[Mathematical Expression Omitted]

in which the lower exact sequence admits a splitting [s.sub.0] : [Q.sub.0] [right arrow] [F[double prime].sub.0]. Hence the upper exact sequence is split by [s.sub.(0)] := sp [convolution] [s.sub.0] [convolution] s[p.sup.-1]. It gives rise to the split exact sequence:

[Mathematical Expression Omitted]

where setting [e.sub.(0)] = [r.sub.(0)](e) we have [Mathematical Expression Omitted].

We define [Phi](m) to be the image of [Mathematical Expression Omitted] under the natural projection

[Ext.sup.1]([Q.sub.(0)], [L.sub.(0)]) [right arrow] [Ext.sup.1][([Q.sub.(0)], [L.sub.(0)]).sub.[Eta]].

Let us check that this is well defined. If we choose a different splitting [s[prime].sub.0]: [Q.sub.0] [right arrow] [F[double prime].sub.0] of the lower exact sequence in (1.8) with associated upper splitting [s[prime].sub.(0)] := sp [convolution] [s[prime].sub.0] [convolution] s[p.sup.-1] we claim that [Mathematical Expression Omitted]. Since [s.sub.0] - [s[prime].sub.0] = [i.sub.0] [convolution] [v.sub.0] for some map [v.sub.0] : [Q.sub.0] [right arrow] [F.sub.0] we have [s.sub.(0)] - [s[prime].sub.(0)] = [i.sub.(0)] [convolution] [v.sub.(0)] where [v.sub.(0)] = sp [convolution] [v.sub.0] [convolution] s[p.sup.-1] and hence

[Mathematical Expression Omitted].

The exact sequence 0 [right arrow] L [right arrow] F[prime] [right arrow] F [right arrow] 0 gives rise to the following commutative diagram

[Mathematical Expression Omitted]

where c is surjective by assumption (1.2) ii) and [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Since by general nonsense [Mathematical Expression Omitted] and by definition b([v.sub.0]) = [v.sub.(0)] it follows [Mathematical Expression Omitted] and hence that [Mathematical Expression Omitted]. A short calculation shows that the image of [Mathematical Expression Omitted] in [Ext.sup.1][([Q.sub.(0)], [L.sub.(0)]).sub.[Eta]] is independent of the choice of e. Finally one checks that [Phi](m) is independent of the choice of a representing 2-extension for m. In fact by writing down the obvious commutative diagrams one sees that any two 2-extensions in m that are related (1.1) lead to the same [Phi](m). We omit the verification that [Phi] is actually a homomorphism. Finally the naturality (1.5) of the maps [Phi] is clear by construction.

Remark 1.9. If in (1.2) for any F in X the map

1) [Ext.sup.1](F,L) [right arrow] [Ext.sup.1]([F.sub.(0)], [L.sub.(0)]) is trivial then [[Phi].sup.Q,L] is independent of [r.sub.0] and sp.

2) [Ext.sup.1]([F.sub.[Eta]], [L.sub.[Eta]]) [right arrow] [Ext.sup.1]([F.sub.(0)], [L.sub.(0)]) is trivial then [[Phi].sup.Q,L] = 0.

This is obvious since under assumption 1) the upper sequence in (1.8) always splits and since under 2) we have [e.sub.(0)] = 0.

In applications we will need the following criterion for injectivity of [Phi].

LEMMA 1.10. If in (1.2) for any F in X the sequence

[Mathematical Expression Omitted]

is exact then [[Phi].sup.Q,L] is injective and independent of [r.sub.0] and sp.

Proof. Assume that for m in [Eta][Ext.sup.2](Q,L) we have [Phi](m) = 0. Then there exists an element e of [Ext.sup.1]([F[double prime].sup.[Eta]], [L.sub.[Eta]]) which maps to [[F[prime].sub.[Eta]]] = [r.sub.[Eta]][F[prime]] and satisfies [Mathematical Expression Omitted] in [Ext.sup.1]([Q.sub.(0)], [L.sub.(0)]). Since [Mathematical Expression Omitted] by assumption, it follows that [e.sub.(0)] = 0. There is thus an element [Mathematical Expression Omitted] in [Ext.sup.1](F[double prime], L) with [Mathematical Expression Omitted]. Injectivity of [r.sub.[Eta]] implies that [Mathematical Expression Omitted] maps to [F[prime]] in the exact sequence:

[Mathematical Expression Omitted].

Hence m = 0.

(1.11) In the applications we will have a collection of data (X, [X.sub.[Eta]], [X.sub.x], [r.sub.[Eta]], [r.sub.x], [r.sub.(x)], s[p.sub.x], Q) for x [element of] X satisfying the conditions in (1.2). Applying our construction of [Phi] to the datum (X, [X.sub.[Eta]], [X.sub.0], . . .) with [Mathematical Expression Omitted] etc. we obtain a natural homomorphism for any L in X

[Mathematical Expression Omitted].

Remark 1.13. If in addition for any F in X the sequence

[Mathematical Expression Omitted]

is exact then Lemma 1.10 implies that [Phi] = [[Phi].sup.Q,L] in (1.12) is injective and independent of the choices of [r.sub.x] and s[p.sub.x].

(1.14) The following additional consideration is necessary for technical reasons when dealing with motives. Suppose that in (1.11) there is a subset [X.sub.f] [subset] X such that for every x in [X.sub.f] we are given a morphism of data

[Mathematical Expression Omitted] etc.

such that for a fixed L and any F in X the map

[Mathematical Expression Omitted]

is trivial. Then the map [Phi] = [[Phi].sup.Q,L] of (1.12) restricts to a map

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted]

and [Mathematical Expression Omitted]

and [Mathematical Expression Omitted].

To prove this note that by (1.5) we have a commutative diagram for x [element of] [X.sub.f]:

[Mathematical Expression Omitted].

By (1.9) 2) we know that [[Phi].sub.X], [X.sub.[Eta]x], [X.sub.x] = 0. Hence restricted to [Mathematical Expression Omitted] the "x-component" of [Phi] in (1.12) maps to zero in [Mathematical Expression Omitted] and hence lies in [E.sub.x]. Here we have used naturality (1.5) again to see that the "x-component" of [Phi] is just [[Phi].sub.X], [X.sub.[Eta]], [X.sub.x]].

Remark 1.16. If the condition in (1.13) is satisfied the map [Phi] of (1.15) is injective.

2. The trace homomorphism for curves via 2-extensions. In this section we explain how the trace map on the second cohomology of a curve over an algebraically closed field k can be obtained as a special case of the construction in section 1. Thus let X be a projective curve over k which for simplicity we assume to be smooth. For any closed point x of X we define a datum (X, [X.sub.[Eta]], [X.sub.x], [r.sub.[Eta]], [r.sub.x], [r.sub.(x)], s[p.sub.x], Q) as follows. Choose a separable closure [Mathematical Expression Omitted] of the function field K of X and set [Eta] = spec K, [Mathematical Expression Omitted]. For any x fix an embedding [Mathematical Expression Omitted] of the (strict) Henselization of [O.sub.x] in x. It defines a morphism [Mathematical Expression Omitted]. Let [Mathematical Expression Omitted] be the quotient field of [Mathematical Expression Omitted] set [Mathematical Expression Omitted] and let

[Mathematical Expression Omitted]

be the induced morphisms. For any scheme Z denote by [Mathematical Expression Omitted] the category of sheaves of abelian groups on the small etale site of Z. Define:

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[r.sub.x](F) = [F.sub.x] viewed as a trivial [I.sub.x]-module.

The map [Alpha] above induces the specialization map

[Mathematical Expression Omitted]

[Mi. II(3.2(a))] which gives a natural transformation [Mathematical Expression Omitted]. Finally for Q we take the constant sheaf Z on X. It is then immediately verified that the assumptions of (1.2) are satisfied. For any torsion sheaf L we have:

[Mathematical Expression Omitted]

since by Tsen's theorem the cohomological dimension of K is [less than or equal to] 1. Moreover

[Mathematical Expression Omitted]


[Mathematical Expression Omitted]

The map [Phi] of (1.12) gives a homomorphism:

[Mathematical Expression Omitted].

Since [H.sup.2](U,L) = 0 for any affine open subscheme U of X and because of functoriality we could write [[symmetry].sub.x[element of][absolute value of X]] and in fact [[symmetry].sub.x[element of][absolute value of X\U]] instead of [[Pi].sub.x[element of][absolute value of X]].

If in particular L = [[Mu].sub.n] for some n prime to char (k) then Kummer theory provides us with canonical homomorphisms

[Mathematical Expression Omitted]

via the discrete valuation [ord.sub.x] on [K.sub.x] and isomorphisms

[H.sup.1]([G.sub.K], [[Mu].sub.n]) = [K.sup.*]/[([K.sup.*]).sup.n] = [K.sup.*] [cross product] Z/n.

Thus (2.1) becomes a map

[Mathematical Expression Omitted].

Note that [Mathematical Expression Omitted] is n-divisible.

THEOREM 2.3. The map of (2.2)

[Mathematical Expression Omitted]

is the trace map.

Proof. Let D = [Sigma] [n.sub.x] [center dot] x be any divisor of X. The class of -D in [Mathematical Expression Omitted] is the extension obtained by pullback from the divisor sequence on X:

[Mathematical Expression Omitted].

where Di[v.sub.x] = [[symmetry].sub.x[element of][absolute value of X]] [i.sub.x*]Z cf. [D1 (cycle)] (1.1).

The cycle class map sends -D to the 2-extension

[Mathematical Expression Omitted]

obtained by cup-product with the Kummer sequence loc. cit. (2.1). We have canonical identifications

[Mathematical Expression Omitted]

[Mathematical Expression Omitted].

Choose [Mathematical Expression Omitted] as a trivializing module fitting into the commutative diagram

[Mathematical Expression Omitted].

For any x [element of] X a section [s.sub.x] of [G.sub.x] [right arrow] Z [right arrow] 0 has the form [s.sub.x](l) = (f, 1) with [Mathematical Expression Omitted] and [ord.sub.x] (f) = [n.sub.x]. The induced section [s.sub.(x)] of [Mathematical Expression Omitted] is given by [s.sub.(x)](1) = (f, 1) as well. Under the pullback

[Mathematical Expression Omitted]

we have [Mathematical Expression Omitted] as follows from a short calculation.

For general sheaves L the homomorphism [Phi] in (2.1) turns out to be the inverse of a map [Psi] constructed as follows. For any open U [subset] X we have the relative exact sequence

[Mathematical Expression Omitted].

By excision we have:

[Mathematical Expression Omitted].

Taking the inductive limit over shrinking U's gives the exact sequence

[Mathematical Expression Omitted]

Using the relative exact sequence for ([X.sub.x], [[Eta].sub.x]) and taking into account that because [Mathematical Expression Omitted] is Henselian and L is torsion we have [H.sup.i]([X.sub.x], L) = [H.sup.i](x, L) = 0 for i [greater than] 0 we get:

[Mathematical Expression Omitted].

The exact sequence therefore gives an isomorphism

[Mathematical Expression Omitted].

THEOREM 2.4. [Phi] in (2.1) and [Psi] are inverse to each other.

We omit the proof since the result is not needed in the sequel. The direct verification is somewhat lengthy. However as the referee points out (2.4) also follows from (2.3) by standard reduction techniques.

Example 2.5. Let us review the map [Phi] of (1.4) in the setting of locally constant sheaves. Define X = category of locally constant finite Z/n-sheaves and let [X.sub.[Eta]], [X.sub.x], [r.sub.[Eta]], [r.sub.(x)] be as before. In this situation we can take [r.sub.x] = [r.sub.(x)] [convolution] [r.sub.[Eta]] and s[p.sub.x] = id. Setting Q = Z/n the conditions of (1.2) are verified for each x [element of] [absolute value of X]. In addition condition (1.13) holds true in this context:

For any F, L in X the sequence

[Mathematical Expression Omitted]

is exact. Let us check for example that an extension

[Mathematical Expression Omitted]

of [G.sub.K]-modules that is mapped to zero by [Pi] [r.sub.(x)] comes from [Mathematical Expression Omitted]. Since [r.sub.(x)]([M]) = 0 there exists an [I.sub.x]-invariant splitting of (2.6). Hence [Mathematical Expression Omitted] as [I.sub.x]-modules and therefore [I.sub.x] acts trivially on M. Since [[Pi].sub.1] (X) is the quotient of [G.sub.K] by the closed subgroup generated by all the [I.sub.x] for x [element of] [absolute value of X] it follows that M is a [[Pi].sub.1](X)-module and thus can be viewed as an object of X.

Using (1.13) we get that:

[Mathematical Expression Omitted]

is injective. By functoriality (1.5) we obtain for L = [[Mu].sub.n] the commutative diagram:

[Mathematical Expression Omitted].

Remark. It is known that [Mathematical Expression Omitted], if X is not isomorphic to [P.sub.1] so in this case the upper [Phi] is also an isomorphism.

3. The trace map on 2-extensions of motives over y. In this section we first introduce the notion of a motive over an arithmetic curve Y = [Y.sub.k] = spec [O.sub.k] [union] {p [where] [infinity]}. Then we specialize the abstract framework of section 1 suitably to obtain a trace map on 2-extensions of such motives. Finally we interpret the motivic height pairing of [Sch] as a Yoneda pairing followed by this trace map.

We first recall a general notion

Definition 3.1. Let H be an object in an abelian category endowed with an increasing filtration [W.sub.i]H for i [element of] Z. The filtration is split, if there exists an isomorphism

[Mathematical Expression Omitted] such that [Mathematical Expression Omitted].

In this case there also is a [Tau] with [Mathematical Expression Omitted].

As yet there is no entirely satisfactory definition of a category M[M.sub.k] of mixed motives with coefficients in Q over a number field k. However using realizations Deligne [De1] and Jannsen [J] have proposed manageable candidates for M[M.sub.k]. The semisimple objects in M[M.sub.k] form the full subcategory [M.sub.k] of (direct sums of) pure motives for absolute Hodge cycles earlier introduced by Deligne. We refer to [Sch, [section] 1] for a short review of their theory.

For a finite place p of k choose an inertia group [I.sub.p] [subset] [G.sub.k] at p. Following Scholl [Sch, 1.7] we call a motive M in M[M.sub.k] integral at p if the weight filtration on the l-adic realization [M.sub.l] of M splits over [I.sub.p] for every l prime to p.

Let M[H.sub.C] (resp. M[H.sub.R]) denote the category of mixed Hodge structures with coefficients in R (and equipped with the action of an infinite Frobenius [F.sub.[infinity]] which respects the weight filtration and maps [F.sup.i] to [Mathematical Expression Omitted]).

For an infinite prime p of k the considerations of the appendix show that an analogue of Scholl's condition of integrality at p is the following: M in M[M.sub.k] is integral at p [where] [infinity] if the real Betti realization [M.sub.p,R] corresponding to an embedding p : k [right arrow] C is semisimple in M[H.sub.[k.sub.p]].

For any place p the motives integral at p form a full Tannakian subcategory M[M.sub.k,p] of M[M.sub.k] containing [M.sub.k]. The same is true for the categories M[M.sub.[O.sub.k]] resp. M[M.sub.y] of motives which are integral at every finite resp. at every place of k.

For every finite prime p of k and prime number l with [Mathematical Expression Omitted] we denote by [X.sub.p,l] the category of finite dimensional [Q.sub.l]-vector spaces with continuous action of [I.sub.p]. Set

[Mathematical Expression Omitted].

For p [where] [infinity] we set [X.sub.p] = M[H.sub.[k.sub.p]]. If M is in M[M.sub.k] we write again M for the image under the realization functor [r.sub.(p)] : M[M.sub.k] [right arrow] [X.sub.p]. Then we have:

LEMMA 3.2. For every place p of k and any motives F, L in M[M.sub.k,p] the following sequence is exact:

[Mathematical Expression Omitted].

In particular the functor

L [right arrow] [Hom.sub.[X.sub.p]] (Q(0), L)

is exact on M[M.sub.k,p].

Proof. Since [Ext.sup.1](A,B) = [Ext.sup.1](Q(0), [A.sup.*] [cross product] B) in M[M.sub.k,p], M[M.sub.k] and [X.sub.p] we are reduced to the case F = Q(0). First we show that any extension

[Mathematical Expression Omitted]

splits in [X.sub.p] after applying [r.sub.(p)]. For p [where] [infinity] this is clear so let p be a finite place. Applying Gr. = [Gr..sup.W] we obtain the exact sequence

[Mathematical Expression Omitted]

with Q(0) in degree zero. Since [M.sub.k] is semisimple this sequence is split by a graded map s: Q(0) [right arrow] Gr. E. Hence for the l-adic realization [s.sub.l] : [Q.sub.l] [right arrow] Gr. [E.sub.l] we have Gr. [[Psi].sub.l] [convolution] [s.sub.l] = id and thus there exists an element a in [([Gr.sub.0] [E.sub.l]).sup.[G.sub.k]] such that ([Gr.sub.0] [[Psi].sub.l])(a) = 1. Since E is integral at p there exists an [I.sub.p]-equivariant isomorphism

[Mathematical Expression Omitted].

For b = [[Tau].sup.-1](a) [element of] [([W.sub.0][E.sub.l]).sup.[I.sub.p]] and any representative a [element of] [W.sub.0][E.sub.l] for a we then have b [equivalent to] a mod [W.sub.-1][E.sub.l]. Since [W.sub.-1]Q(0) = 0 it follows that [[Psi].sub.l](b) = [[Psi].sub.l](a) = 1. Hence for all l with [Mathematical Expression Omitted] the sequence

0 [right arrow] [L.sub.l] [right arrow] [E.sub.l] [right arrow] [Q.sub.l] [right arrow] 0

is split as an extension of [I.sub.p]-modules.

It remains to show that for any extension in M[M.sub.k]

[Mathematical Expression Omitted]

which splits in [X.sub.p] we have E [element of] M[M.sub.k,p]. Again this is clear for p [where] [infinity] so let p be a finite prime. By assumption there is an element [Mathematical Expression Omitted] with [[Psi].sub.l](a[prime]) = 1 for any l with p [Mathematical Expression Omitted]. Since [Psi]([W.sub.-1]E) = 0 we have a[prime] [element of] [([W.sub.n][E.sub.l]).sup.[I.sub.p]] for some n [greater than or equal to] 0. Assume that n [greater than] 0. Since L is in M[M.sub.k,p] the natural projection [Mathematical Expression Omitted] remains surjective upon passing to invariants under inertia. A consideration of

[Mathematical Expression Omitted]

shows that there are elements c [element of] [([W.sub.n][L.sub.l]).sup.[I.sub.p]] and a[double prime] [element of] [W.sub.n-1][E.sub.l] such that a[prime] = c + a[double prime].

Thus a[double prime] [element of] [([W.sub.n-1][E.sub.l]).sup.[I.sub.p]] satisfies [[Psi].sub.l](a[double prime]) = 1. By induction them is thus an element a [element of] [([W.sub.0][E.sub.l]).sup.[I.sub.p]] with [[Psi].sub.l](a) = 1. It defines an [I.sub.p]-equivariant splitting [Mathematical Expression Omitted] that induces isomorphisms [Mathematical Expression Omitted]. Since L, Q(0) are in M[M.sub.k,p] it follows that E is in M[M.sub.k,p] as well.

We can now apply the abstract considerations of section 1 to the motivic situation. Let X be the set of places of k and define for any p [element of] X a datum (X, [X.sub.[Eta]], [X.sub.p], [r.sub.[Eta]], [r.sub.p], [r.sub.(p)], s[p.sub.p], Q) as follows: X = M[M.sub.y], [X.sub.[Eta]] = M[M.sub.k], [X.sub.p], [r.sub.(p)] as above, Q = Q(0), [r.sub.[Eta]] is the inclusion, [r.sub.p] = [r.sub.(p)] [convolution] [r.sub.[Eta]], s[p.sub.p] = id. It follows from (3.2) that the conditions of (1.2) and (1.13) are satisfied. For finite p Kummer theory gives an isomorphism:

[Mathematical Expression Omitted]

where [v.sub.p] is induced by the valuation on the maximal unramified extension [Mathematical Expression Omitted] of [k.sub.p]. The composition

[Mathematical Expression Omitted]

and the map K of Conjecture 0.4 fit into a commutative diagram

[Mathematical Expression Omitted].

For p [where] [infinity] there is a canonical isomorphism

[Mathematical Expression Omitted]

such that the composition

[Mathematical Expression Omitted]

fits into the commutative diagram

[Mathematical Expression Omitted]

where [v.sub.p](f) = -log [absolute value of [f.sup.[Sigma]]] for f [element of] [k.sup.*] and [Mathematical Expression Omitted] is an embedding in the class of p.

From (1.11) and (1.13) we obtain an injective homomorphism which is independent of the choices of [r.sub.p] and [sp.sub.p]:

[Mathematical Expression Omitted].


[Mathematical Expression Omitted].

Note that we could write [[symmetry].sub.p] here instead of [[Pi].sub.p] as in (1.12) because of (3.2), (1.5) and the fact that every motive over k is integral at almost every place cf. [Sch, 3.16].

We can replace the spaces [Q.sup.p] by much smaller ones using the considerations of (1.14). Let [X.sub.f] be the set of finite places of k take [X.sub.[[Eta].sub.p]] = M[M.sub.k,p] for p [element of] [X.sub.f] and extend this in an obvious way to morphisms of data as in (1.14). We then have the following result:

THEOREM 3.5. The map [Phi] above restricts to an injection:

[Mathematical Expression Omitted]


[Mathematical Expression Omitted]


[Mathematical Expression Omitted]

[E.sub.p] = R for p [where] [infinity].

Note that if e.g. [Mathematical Expression Omitted] as expected, then

[Mathematical Expression Omitted].

The existence of the monomorphism [Phi] in (3.6) is about all that we can say on the structure of [Mathematical Expression Omitted] without assuming any unproved hypotheses. We can go much further if we assume Conjecture 0.4.

Recall the Arakelov Chow group of y:

[Mathematical Expression Omitted]

and its degree map deg: C[H.sup.1] (y) [approaches] R given by

[Mathematical Expression Omitted].

We then have the following result:

THEOREM 3.7. Assume Conjecture 0.4. Then (3.6) becomes an invective map:

[Mathematical Expression Omitted]

and we have

[Mathematical Expression Omitted].

The trace map Tr obtained by composition

[Mathematical Expression Omitted]

is nonzero. In particular [Mathematical Expression Omitted].

Remarks. Concerning (3.7.1) compare the closely related statement in [Sch, Prop. (6.1)]. The normalization of the trace map is chosen in analogy with (2.3).

Proof. The nontriviality of the trace map will follow from the relation with the height pairing in theorem (3.12) below. For the proof of (3.7.1) we now recall the definition of the map [Phi] in the situation of the theorem i.e. assuming that K is an isomorphism. This will also ease the comparison with the local height pairings of Scholl. The remaining assertion of the theorem is clear.

(3.8) Let [Mathematical Expression Omitted] be represented by the 2-extension

0 [approaches] Q(1) [approaches] f[prime] [approaches] f[double prime] [approaches] Q(0) [approaches] 0.

Setting F = Im (F[prime] [approaches] F[double prime]) we obtain two extensions in MMy

[Mathematical Expression Omitted].

In the exact sequence

[Mathematical Expression Omitted].

Hence there is an [Mathematical Expression Omitted] with [i.sup.*](e) = [F[prime]]. Let [e.sub.p] be the image of e in [Mathematical Expression Omitted]. By lemma (3.2) the exact sequence

[Mathematical Expression Omitted]

is split in [X.sub.p] and hence gives rise to an exact sequence

[Mathematical Expression Omitted].

Again by lemma (3.2) we have

[i.sup.*][e.sub.p] = [F[prime] in [X.sub.p]] = 0

and hence there is a class

[Mathematical Expression Omitted].

Under the natural isomorphisms (3.3), (3.4) of [Mathematical Expression Omitted] with [Q.sup.p] for [Mathematical Expression Omitted] and R for p [where] [infinity] we have [f.sub.p] [element of] Q for [Mathematical Expression Omitted] by Conjecture 0.4 since m goes to zero in [Mathematical Expression Omitted] and [f.sub.p] [element of] R for p [where] [infinity]. Then:

[Phi](m) = class of [summation over p] [f.sub.p] [center dot] p in C[H.sup.1](y) [cross product] Q.

Note that because of lemma (3.2) we have

[Mathematical Expression Omitted].

Given e there exists z [element of] [k.sup.*] [cross product] Q such that:

[Mathematical Expression Omitted].


[Mathematical Expression Omitted]

we have

[Mathematical Expression Omitted]

and hence

[Mathematical Expression Omitted].

Since [Mathematical Expression Omitted] in the exact sequence

[Mathematical Expression Omitted]

it follows that

[Mathematical Expression Omitted].

Under Conjecture 0.4 we therefore have:

[Mathematical Expression Omitted]

as claimed in (3.7).

(3.9) Consider the Yoneda pairing

[Mathematical Expression Omitted]

obtained by splicing extensions and denote by

[Mathematical Expression Omitted]

the inverse image of [Mathematical Expression Omitted] under [Union]. Under the condition of the theorem we obtain a map <,>M by composition:

[Mathematical Expression Omitted].

In the situation of (3.8) the construction of [Sch, [section]3] attaches to every extension

0 [approaches] Q(1) [approaches] E [approaches] F[double prime] [approaches] 0 in M[M.sub.k]

with [E] = e and every place p of k a map

[Mathematical Expression Omitted].

It is not difficult to see that [f.sub.p,e] = [b.sub.p,E](1) if p is finite and that

[f.sub.p,e] = -[[[k.sub.p] : R].sup.-1] [b.sub.p,E](1)

if p is infinite. It follows that (3.10) agrees with the global motivic height pairing of [Sch, [section]6] whenever the latter is defined.

(3.11) We now state the relation of (3.10) with the geometric height pairing of Beilinson [Be2]. Let X be a smooth, projective variety over k = Q of equidimension N. Choose integers a, b [greater than or equal to] 1 with a + b = N + 1 and consider M = [H.sup.2a-1](X)(a) in [M.sub.Q]. Let C[H.sup.n][(X).sup.0] be the subspace in C[H.sup.n](X) [cross product] Q of cycles homologically equivalent to zero. There are canonical Abel-Jacobi maps

[Mathematical Expression Omitted] and [Mathematical Expression Omitted]

defined by pulling back suitable relative exact cohomology sequences, see [Sch, [section]7]. For technical reasons we will have to consider certain subspaces of C[H.sup.*][(X).sup.0]. Assume that X extends to a regular scheme X proper and flat over Z. Then we write C[H.sup.n][(X).sup.00] for the image in C[H.sup.n][(X).sup.0] of

[Mathematical Expression Omitted]

where [Z.sup.n](X) denotes the free abelian group generated by the codimension n cycles on X. According to conjecture 2.2.5 of [Be2] we should have

C[H.sup.n][(X).sup.00] = C[H.sup.n][(X).sup.0].

This is known for curves for example [Be2, 2.2.6].

The next result is an immediate consequence of [Sch, Th. 7.7] combined with Cor. 2.3(i), 2.5, 3.4-3.6 of loc. cit. and lemma (3.2) above:

THEOREM 3.12. Let X be as above and assume that M satisfies Deligne's conjecture on the purity of the monodromy filtration. Then for y = [y.sub.Q] we have:

[Mathematical Expression Omitted]


[Mathematical Expression Omitted].

Under Conjecture 0.4 there is a commutative diagram:

[Mathematical Expression Omitted]

where <,>X is Beilinson's global geometric height pairing [Be2], [Sch, 4.12].

Applying this to elliptic curves E/Q of nonzero rank it follows that <,>M and hence Tr is nonzero thus completing the proof of (3.7) in the case k = Q.

For arbitrary k note that [Pi]: spec k [approaches] spec Q induces exact base extension functors [J, 2.16]:

[[Pi].sup.*] : M[M.sub.[y.sub.Q]] [approaches] M[M.sub.[y.sub.k]] and M[M.sub.Z] [approaches] M[M.sub.[O.sub.k]].

The nontriviality of the trace map Tr over [Y.sub.k] now follows from the commutative diagram:

[Mathematical Expression Omitted].

4. Appendix. Here we translate Scholl's integrality condition to the infinite places using the functors [F.sub.p] from mixed motives into D-modules of [D]. According to Scholl the motive M in M[M.sub.k] is integral at [Mathematical Expression Omitted] if for all l prime to p there is an [I.sub.p]-equivariant isomorphism

[Mathematical Expression Omitted]

respecting the weight filtrations. This is equivalent to the condition that for some (any) embedding [Mathematical Expression Omitted] there exists such an isomorphism after tensoring with C via [Iota]. The equivalence can be seen as follows (we are indebted to the referee for the proof): Scholl's condition is equivalent to the [I.sub.p]-splitting of all sequences

[Mathematical Expression Omitted].

Equivalent is the splitting of the sequence

[Mathematical Expression Omitted]

obtained by tensoring with [Mathematical Expression Omitted] and pulling back via the map

[Mathematical Expression Omitted]

mapping 1 to the identity map. Now this sequence splits if and only if [E.sup.[I.sub.p]] [not equal to] 0. Now tensor everything with C via [Mathematical Expression Omitted] and note that

[Mathematical Expression Omitted].

In terms of the functor

[F.sub.p] = [F.sub.p,l,[Iota]] : M[M.sub.k] [approaches] [D.sub.p]

of [D, (3.1.1)] which is the composition of [Mathematical Expression Omitted] with a fully faithful functor Scholl's condition can thus be reformulated as follows: There exist

(4.1) [Phi] [element of] [F.sub.p]([W.sub.0]Hom(M, M[prime])), [Psi] [element of] [F.sub.p]([W.sub.0]Hom(M[prime], M))

such that [Phi] [convolution] [Psi] = id, [Psi] [convolution] [Phi] = id under the natural pairings.

In [D, (6.3)] an analogous functor

[Mathematical Expression Omitted]

was introduced for every p [where] [infinity]. It is the composition of [Mathematical Expression Omitted] with a fully faithful functor. Here as usual [().sup.+] = () if p is complex and [().sup.+] is the fixed module under [Mathematical Expression Omitted] if p is real.


[([F.sup.0][W.sub.0]Hom[(M, M[prime]).sub.B,C] [intersection] [W.sub.0]Hom[(M, M[prime]).sub.B,R]).sup.+] = [Hom.sub.[M[H.sub.[k.sub.p]]]] ([M.sub.B,R], [M[prime].sub.B,R])

the condition (4.1) applied to [Mathematical Expression Omitted] thus becomes:

There exist

[Phi] [element of] Hom ([M.sub.B,R], [M[prime].sub.B,R]), [Psi] [element of] Hom([M[prime].sub.B,R], [M.sub.B,R])

such that [Phi] [convolution] [Psi] = id, [Psi] [convolution] [Phi] = id i.e. [M.sub.B,R] is isomorphic to [Mathematical Expression Omitted]. [M.sub.B,R] as a real mixed Hodge structure over [k.sub.p] i.e. [M.sub.B,R] is semisimple (0.2).




[Be1] A. A. Beilinson, Notes on absolute Hodge cohomology, Contemp. Math. 55 (1986), 35-68.

[Be2] -----, Height pairings between algebraic cycles, K-Theory, Arithmetic and Geometry (Y. I. Manin, ed.), Lecture Notes in Math., vol. 1289, Springer-Verlag, New York, 1987, 1-26.

[De1] P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois Groups Over Q, Springer-Verlag, New York, 1989, pp. 79-297.

[De2] P. Deligne et al., Cohomologie Etale, Lecture Notes in Math., vol. 569, Springer-Verlag, New York, 1977.

[D] C. Deninger, Motivic L-functions and regularized determinants, Proc. Sympos. PureMath. (to appear).

[F-PR] J.-M. Fontaine and B. Perrin-Riou, Autour des conjectures de Bloch et Kato, Cohomologie Galoisienne et Valeurs de Fonctions L, Proc. Sympos. Pure Math. (to appear).

[J] U. Jannsen, Mixed Motives and Algebraic K-theory, Lecture Notes in Math., vol. 1400, Springer-Verlag, New York, 1990.

[Mi] J. S. Milne, Etale Cohomology, Princeton University Press, Princeton, NJ, 1980.

[Sch] A. J. Scholl, Height pairings and special values of L-functions, Proc. Sympos. Pure Math. (to appear).
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Author:Deninger, Christopher; Nart, Enric
Publication:American Journal of Mathematics
Date:Jun 1, 1995
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