The Poitou-Tate exact sequence

We keep notation from the previous sections. In particular, k is a number field, M_k is a 1-motive over k, and U is an open subset of SpecOk such that M_k ex-tends to a 1-motiveM overU. For each finite place vofk, we denote by Obv the

ring of integers of the completion ˆk_v. The groups Hⁱ(k,M_k) (−1≤i≤2) and Hⁱ(kˆ_v,M_k)fori≥1 (v∈Ω_k) are equipped with the discrete topology. Fori≥0, we define Pⁱ(M_k) as the restricted product over v∈Ω_k of the Hⁱ(kˆ_v,M_k), with respect to the images of Hⁱ(Obv,M) for finite placesv ofU. The groups Pⁱ(F_k) are defined similarly for any ´etale sheafF_k over Speck. Clearly all these groups are independent of the choice ofU. We equip them with their restricted prod-uct topology. Note thatP²(M_k)is the direct sum of the groupsH²(kˆ_v,M_k)for all placesv: indeed, for each finite place vofU, we have H²(Oˆv,M) =H²(F_v,M),e and we have already seen that this last group vanishes. We have natural restriction mapsβ_i: Hⁱ(k,M_k)→Pⁱ(M_k)fori=1,2; their kernels are precisely the groups Xⁱ(M_k).

In this section we establish a Poitou-Tate type exact sequence forM_k. A tech-nical complication for our considerations to come arises from the fact that one is forced to work with two kinds of completions: the first one is what we have used up till now, i.e. the inverse limitA^∧ of all open subgroups of finite index of a topological abelian groupA, and the second is the inverse limitA_∧ of the quo-tientsA/nfor alln>0. These are not the same in general (indeed, the latter is not necessarily profinite). But it is easy to see that(A_∧)^∧ is naturally isomorphic to A^∧; in particular, we have a natural mapA_∧→A^∧.

The natural mapH⁰(k,M_k)→P⁰(k,M_k)therefore gives rise to a commutative diagram between the two different kinds of completions:

H⁰(k,M_k)_∧ −−−→^θ0 P⁰(k,M_k)_∧



 y



 y

H⁰(k,M_k)^∧ −−−→^β0 P⁰(k,M_k)^∧.

(2.14)

Denote by X⁰∧(M_k) the kernel of the above map θ₀. We first prove the fol-lowing duality result analogous to [47], I.6.13 (b).

Proposition 2.6.1 Let M_k= [Y_k→G_k]be a 1-motive over k. Assume thatX¹(A_k) is finite. Then there is a perfect pairing

X⁰∧(M_k)×X²(M_k^∗)→Q/Z where the first group is compact and the second is discrete.

Remarks 2.6.2

1. The special caseM_k=Y_k[1]corresponds to the duality between the groups X¹(Y_k)andX²(T_k^∗), whereT_k^∗is the torus with character moduleY_k; com-pare [47], I.4.20 (a). Indeed, in this case the groupsH⁰(k,M_k)andP⁰(k,M_k) are finite, soX⁰∧(M_k)∼=X¹(Y_k).

2. The proof below will show that if one only assumes the finiteness of the `-primary torsion part ofX¹(A_k), then one gets a similar duality between the

`-primary torsion part ofX<sup>2</sup>(M<sub>k</sub><sup>∗</sup>)and a groupX<sup>0</sup><sub>(`)</sub>(M_k)defined similarly toX⁰∧(M_k)but taking only`-adic completions.

3. In the paper [30] we guessed thatX²(M_k)is finite, as it is finite forM_k=T_k and even trivial forM_k =A_k andM_k=Y_k[1]. This finiteness was proven in important mixed cases in the thesis of P. Jossen [34], among them 1-motives of the form[Y_k →A_k], withA_k a geometrically simple abelian variety. He also proved that in these cases there is a perfect pairing

X⁰(M_k)×X²(M_k^∗)→Q/Z

of finite groups (in fact, the finiteness of X⁰(M_k) can be proven to hold in general). However, he recently produced an example whereX²(M_k)is infinite!

For the proof of the proposition we first show a lemma.

Lemma 2.6.3 For any n>0, the natural map P⁰(M_k)/n→

∏

v

H⁰(kˆ_v,M_k)/n

is injective and its image is the restricted product of the groupsH⁰(kˆ_v,M_k)/n with respect to the subgroupsH⁰(Oˆv,M_k)/n.

Proof: Take an element x= (x_v)∈P⁰(M_k)and assume x_v is innH⁰(kˆ_v,M)for allv. For almost allv, it also comes fromH⁰(Oˆv,M), hence in fact it comes from nH⁰(Oˆv,M), for in the exact commutative diagram

H⁰(Obv,M) −−−→ⁿ H⁰(Obv,M) −−−→ H¹(Obv,T_Z/nZ(M))



 y



 y



 y

H⁰(kˆ_v,M_k) −−−→ⁿ H⁰(kˆ_v,M_k) −−−→ H¹(kˆ_v,T_Z/nZ(M_k))

the third vertical map is injective,H_v¹(Obv,T_Z/nZ(M))being trivial by the same argument as in the proof of Lemma 2.5.5. But this means x∈nP⁰(M_k). The second statement is obvious.

Proof of Proposition 2.6.1: Using exact sequence (2.11) we get a commutative diagram with exact rows for eachn>0:

0 −−−→ H⁰(k,M_k)/n −−−→ H¹(k,T_Z/nZ(M_k)) −−−→ H¹(k,M_k)[n] −−−→ 0 and a similar exact diagram holds with Obv instead of ˆk_v. Taking restricted products and using the above lemma, we get a commutative exact diagram:

0 −−−→ H⁰(k,M_k)/n −−−→ H¹(k,T_Z/nZ(M_k)) −−−→ H¹(k,M_k)[n] −−−→ 0 If we pass to the inverse limit over alln, the two lines remain left exact, whence a commutative exact diagram implies the finiteness ofX¹(M_k)as in Corollary 2.5.8).

Therefore we obtain thatX⁰∧(M_k) =Kerθ0is isomorphic to Kerθ. By Poitou-Tate duality for finite modules ([49], VIII.6.8), there is a perfect pairing between the latter group andX²(T(M_k^∗)_tors).

We conclude by observing that X²(T(M_k^∗)_tors) is also X²(M_k^∗)_tors by the same argument as in [47], I.6.8 (use the analogue of the exact sequence (2.11) for M_k over Speckwithi=2 and pass to the limit).

We now return to diagram (2.14) and prove:

Proposition 2.6.4 Keep the finiteness assumption onX¹(A_k). Then, with nota-tions as in diagram (2.14), the natural mapkerθ0→kerβ0is an isomorphism.

By virtue of the proposition, we may employ the notationX⁰∧(M_k)for kerβ₀ as well and use the resulting duality. The finiteness assumption on X¹(A_k) is presumably superfluous here but we did not succeed in removing it (and use it elsewhere anyway).

For the proof we need a lemma about abelian groups.

Lemma 2.6.5 Let A be a discrete abelian group of finite exponent n. Then the intersection of the finite index subroups of A is trivial.

Proof: Consider the profinite groupA^D=Hom(A,Z/n). AsA^DD=A, the state-ment is equivalent to saying that any character ofA^D vanishing on all finite sub-groups ofA^Dis trivial. This holds because all finitely generated subgroups ofA^D are finite.

Proof of Proposition 2.6.4: We begin by showing that the vertical maps in dia-gram (2.14) above are injective, whence the injectivity of the map kerθ₀→kerβ₀. For the left one, note that the topology on H⁰(k,M) being discrete, injectivity means that any element of H⁰(k,M) which is nontrivial modulo n for some n gives a nonzero element in some finite quotient – this holds by the lemma above.

For injectivity of the map P⁰(k,M)_∧ →P⁰(k,M)^∧, take an element x= (x_v) in P⁰(M_k) not lying in nP⁰(M_k). We have to find an open subgroup of finite index avoiding x. By Lemma 2.6.3, there is a local component x_v not lying in nH⁰(kˆ_v,M). We thus get that for some v, our x is not contained in the inverse image inP⁰(M_k)of the subgroupnH⁰(kˆ_v,M)⊂H⁰(kˆ_v,M), which is open of finite index, by definition of the topology onH⁰(kˆ_v,M).

For the surjectivity of the map kerθ₀ →kerβ₀, remark first that kerθ₀ is a profinite group, being dual to the torsion groupX²(M_k)by the previous proposi-tion.

Therefore(kerθ₀)^∧=kerθ₀, so by completing the exact sequence

0→kerθ₀→H⁰(k,M_k)_∧→imθ₀→0 (2.16) we get an exact sequence kerθ0→H⁰(k,M_k)^∧→(imθ0)^∧→0.

To conclude, it is thus enough to show the injectivity of the natural map (imθ0)^∧→P⁰(M_k)^∧. By the above considerations,P⁰(M_k)_∧ injects into its com-pletion P⁰(M_k)^∧, so the completion of the subgroup imθ₀ is simply its closure in P⁰(M_k)^∧, whence the claim. But there is a subtle point here: in this ar-gument, imθ0 is equipped with the subspace topology inherited from P⁰(M_k)_∧, whereas in exact sequence (2.16) it carried the quotient topology fromH⁰(M_k)_∧; we have to check that the two topologies are the same, or in other words that the morphism θ0 is strict. For every n>0, the groups H⁰(k,M_k)/n and P⁰(M_k)/n

are locally compact (the latter thanks to Lemma 2.6.3). Moreover, the mor-phism f_n:H⁰(k,M_k)/n→P⁰(M_k)/n is strict for every n>0. Indeed, the mor-phisms H⁰(k,M_k)/n→ H¹(k,T_Z/nZ(M_k)) and P⁰(M_k)/n→ P¹(T_Z/nZ(M_k)) are strict (thanks to [33], Theorem 5.29 and to diagram (2.15)), hence the image of H⁰(k,M_k)/n in P⁰(M_k)/n identifies with a subspace of the image of the group H¹(k,T_Z/nZ(M_k))inP¹(T_Z/nZ(M_k)), which is discrete. Using the Poitou-Tate ex-act sequence for finite modules ([49], VIII.6.13) we get that this image is a closed subset ofP¹(T_Z/nZ(M_k)), hence the morphismH¹(k,T_Z/nZ(M_k))→P¹(T_Z/nZ(M_k)) is strict (again by [33], Theorem 5.29). Nowθ₀is obtained as the projective limit of the strict morphisms f_n with kerf_n finite and H⁰(k,M_k)/n discrete; it is not difficult to check that this implies thatθ0is strict.

We can now state the main result of this section.

Theorem 2.6.6 (Poitou-Tate exact sequence) Let M_k be a 1-motive over k. As-sume thatX¹(A_k)andX¹(A^∗_k)are finite, where A_kis the abelian variety corre-sponding to M_k. Then there is a twelve term exact sequence of topological groups

0 −−−→ H⁻¹(k,M_k)^{∧ γ2}

D

−−−→ ∏v∈Ω_kH²(kˆ_v,M_k^∗)^{D β2}

D

−−−→ H²(k,M_k^∗)^D



 y

H¹(k,M_k^∗)^D ←−−−^γ0 P⁰(M_k)^∧ ←−−−^β0 H⁰(k,M_k)^∧



 y

H¹(k,M_k) −−−→^β1 P¹(M_k)_tors −−−→^γ1 (H⁰(k,M_k^∗)^D)_tors



 y

0 ←−−− H⁻¹(k,M_k^∗)^D ←−−−^{γ2 L}_v∈ΩkH²(kˆ_v,M_k) ←−−−^β2 H²(k,M_k) (2.17) where the mapsβ_i are the restriction maps defined at the beginning of this sec-tion, the mapsγi are induced by the local duality theorem of Section 2, and the unnamed maps come from the global duality results of Proposition 2.6.1 (com-pleted by Proposition 2.6.4) and Corollary 2.5.8.

Remarks 2.6.7

1. In the above sequence the group P¹(M_k)_tors is equipped with the discrete topology, andnotthe subspace topology fromP¹(M_k).

2. The sequence (2.17) is completely symmetric in the sense that if we replace M_k byM_k^∗and dualize, we obtain exactly the same sequence.

3. If we only assume the finiteness ofX¹(A_k){`} andX<sup>1</sup>(A<sup>∗</sup><sub>k</sub>){`}for some prime number`, then the analogue of the exact sequence (2.17) still holds, with profinite completions replaced by`-adic completions and the torsion groups involved by their`-primary part.

4. Some special cases of the theorem are worth noting. For M_k = [0→A_k] we get the ten-term exact sequence of [47], I.6.14. ForM_k= [0→T_k]one can show (see proof of Proposition 2.6.9 below) thatβ₂^Dis an isomorphism, hence we obtain a classical (?) nine-term exact sequence; the case M_k = Y_k[1]is symmetric (compare [47], I.4.20).

The proof will use the following lemma.

Lemma 2.6.8 Let M_kbe a 1-motive over k. ThenH⁰(k,M_k)_torsis finite.

Proof: LetM_k⁰ =M_k/W₋₂(M_k) = [Y_k→A_k]. From the exact sequence H⁰(k,Y_k)→H⁰(k,A_k)→H⁰(k,M_k⁰)→H¹(k,Y_k)

we deduce that H⁰(k,M_k⁰) is of finite type because H⁰(k,A_k) is of finite type (Mordell-Weil theorem) andH¹(k,Y_k)is finite (becauseY_k(k)¯ is a lattice). There is also an exact sequence

H⁻¹(k,M_k⁰)→H⁰(k,T_k)→H⁰(k,M_k)→H⁰(k,M_k⁰)

whereT_k is the torus corresponding to M_k. It is therefore sufficient to show that the torsion subgroup of the group B :=T_k(k)/imH⁻¹(k,M_k⁰) is finite. Choose generators a₁, ...,a_r of the lattice H⁻¹(k,M_k⁰), and let b₁, ...,b_r be their images in T_k(k). We can find an open subset U of SpecOk such that T_k extends to a torus T over U, and b_i ∈H⁰(U,T) for any i∈ {1, ...,r}. In this way, each x∈B_tors is the image inBof an element y∈T_k(k) for which yⁿ ∈H⁰(U,T) for some n> 0. LetV/U be an ´etale covering such that T splits over V, i.e. it becomes isomorphic to some power G^N_m. If L denotes the fraction field of V, we have T_k(L)/H⁰(V,T)∼= (L^×/H⁰(V,G_m))^N which is naturally a subgroup of the free abelian group Div(V)^N; in particular, it has no torsion. Therefore, since yⁿ∈H⁰(V,T)via the inclusionH⁰(U,T)→H⁰(V,T)we get thaty∈H⁰(V,T) = H⁰(V,G_m)^N. Let H be the subgroup of H⁰(V,T) generated by the y’s; since H⁰(V,G_m) is of finite type by Dirichlet’s Unit Theorem, so is H. We thus get a surjection from the finitely generated groupHto the torsion groupB_tors, whence the claim.

Proof of Theorem 2.6.6:The first line is dual to the last one, so for its exactness it is enough to show the exactness of the latter. We proceed as in [47], I.6.13 (b).

For eachn>0, we have an exact commutative diagram (using the exact sequence (2.11) and the Poitou-Tate sequence for the finite moduleT_Z/nZ(M_k)):

0

By Lemma 2.6.8, the full Tate moduleT(H⁰(k,M_k^∗))is trivial. Therefore taking the inductive limit over alln, we obtain the exact commutative diagram, where the right vertical map is an isomorphism: We remark that the first two vertical maps are also isomorphisms by the same argument as at the end of the proof of Proposition 2.6.1. Since the first row is exact by the Poitou-Tate sequence for finite modules ([49], VIII.6.13), so is the second one. Therefore the exactness of the last line of (2.17) (and hence that of the first) follows noting that the dual of the profinite completion of the latticeH⁻¹(k,M_k^∗) is the same as the dual ofH⁻¹(k,M_k^∗)itself.

To prove exactness of the second line, note first that it is none but the profinite completion of the sequence

H⁰(k,M_k)_∧→^θ0 P⁰(M_k)_∧ ^γ

00

→H¹(k,M_k^∗)^D, (2.18) where the mapγ₀⁰ is induced by local duality, taking Lemma 2.6.3 into account.

First we show that this latter sequence is a complex. The map γ₀⁰ has the fol-lowing concrete description at a finite level: forα = (α_v)∈P⁰(M_k)/nand β ∈ H¹(k,M_k^∗)[n], denote by β_v the image ofβ inH¹(kˆ_v,M_k^∗)[n]. Then [γ₀⁰(α)](β) is the sum over allv of the elements j_v(α_v∪βv), where (α_v∪βv)∈H²(kˆ_v,µn) via

the pairingM×M^∗→G_m[1], and j_v is the local invariant. (The sum is finite by virtue of the propertyH²(Obv,µn) =0 because the elementsαvandβvare unrami-fied for almost allv.) Thenγ₀⁰◦θ₀=0 follows (after passing to the limit) from the reciprocity law of global class field theory, according to which the sequence

0→H²(k,µn)→ ^M

v∈Ω_k

H²(kˆ_v,µn)^∑→^jvZ/n→0 (2.19) is a complex.

For the exactness of the sequence (2.18) recall that, as remarked at the end of the proof of Proposition 2.6.4, diagram (2.15) and the Poitou-Tate sequence for finite modules imply that imθ₀is the kernel of the composed map

P⁰(M_k)_∧→P¹((T(M_k))→H¹(k,T(M_k^∗)_tors)^D. The claimed exactness then follows from the commutative diagram

P⁰(M_k)_∧ −−−→ P¹(T(M_k))

whose commutativity arises from the compatibility of the duality pairings for 1-motives and their “n-adic” realizations via the Kummer map.

Now we show that the profinite completion of sequence (2.18), i.e. the second row of diagram (2.17) remains exact. This follows by an argument similar to the one at the end of the proof of Proposition 2.6.4, once having checked that imγ₀⁰ is closed inH¹(k,M_k^∗)^Dand(imγ₀⁰)^∧=imγ₀⁰. To see this, note that by applying the snake lemma to diagram (2.15) we get that Cokerθ0(with the quotient topology) injects as a closed subgroup into Cokerθ. But one sees using the Poitou-Tate sequence for finite modules that the latter group is profinite, hence so is imγ₀⁰; in particular, it is compact and hence closed inH¹(k,M_k^∗)^D.

Next, remark that by definition of the restricted product topology and Theo-rem 2.3.10, the dual of the group P⁰(M_k) (equipped with the restricted product topology) isP¹(M_k^∗). Thus the dual ofP⁰(M_k)^∧ isP¹(M_k^∗)_tors. Therefore we ob-tain the third line by dualizing the second one (and exchanging the roles ofMand M^∗), which consists of profinite groups.

Finally, the exactness of the sequence (2.17) at the “corners” follows imme-diately, in the first two rows, from the dualitiesX⁰∧(M_k)∼=X²(M_k^∗)^D (Proposi-tion 2.6.1 combined with Proposi(Proposi-tion 2.6.4) and X¹(M_k)∼=X¹(M_k^∗)^D (Corol-lary 2.5.8), respectively; the remaining corner is dual to the first one.

We conclude with the following complement.

Proposition 2.6.9 Let M_kbe a 1-motive over k. Then the natural mapHⁱ(k,M_k)→ L

v∈Ω_∞Hⁱ(k_v,M_k)is an isomorphism for i≥3.

Proof: When M= [0→G], this follows immediately by devissage from [47], I.4.21, and I.6.13 (c). To deal with the general case, it is sufficient to show that the map f_i:Hⁱ(k,Y_k)→^L_v∈ΩRHⁱ(k_v,Y_k)is an isomorphism fori≥3. Using the exact sequence

0→Hⁱ(k,Y_k)/n→Hⁱ(k,Y_k/nY_k)→Hⁱ⁺¹(k,Y_k)[n]→0

for eachn>0, we reduce to the casei=3. The last line of the Poitou-Tate exact sequence forM_k=Y_k[1]yields the surjectivity of f₃because ˆk_vis of strict cohomo-logical dimension 2 forvfinite. On the other hand, we remark thatH³(k,Z) =0 ([47], I.4.17), hence H³(k,Y_k) = H³(k,Y_k)[n] for some n >0 by a restriction-corestriction argument. In particular the divisible subgroup ofH³(k,Y_k) is zero.

The injectivity of f₃ now follows from Proposition 2.6.1 applied to M_k =T_k^∗, whereT_k^∗is the torus with module of charactersY_k.

In document 1–MOTIVES AND ALBANESE MAPS IN ARITHMETIC GEOMETRY Tam´as Szamuely A doctoral dissertation submitted to the Hungarian Academy of Sciences Budapest, 2011 (Pldal 63-72)