On the Hasse principle for finite group schemes over global function fields
aa r X i v : . [ m a t h . N T ] J a n ON THE HASSE PRINCIPLE FOR FINITE GROUP SCHEMESOVER GLOBAL FUNCTION FIELDS
CRISTIAN D. GONZ ´ALEZ-AVIL´ES AND KI-SENG TAN
Abstract.
Let K be a global function field of characteristic p > M be a (commutative) finite and flat K -group scheme. We show that the kernelof the canonical localization map H ( K, M ) −→ Q all v H ( K v , M ) in flat (fppf)cohomology can be computed solely in terms of Galois cohomology. We then giveapplications to the case where M is the kernel of multiplication by p m on anabelian variety defined over K . Statement of the main theorem
Let K be a global field and let K be a fixed algebraic closure of K . Let K s be the separable closure of K in K and set G K = Gal( K s /K ). Further, foreach prime v of K , let K v be the completion of K at a fixed prime v of K ly-ing above v and let K s v denote the completion of K s at the prime of K s lyingbelow v . Set G v = Gal( K s v /K v ). If M is a commutative, finite and flat K -group scheme, let H i ( G K , M ) (respectively, H i ( G v , M )) denote the Galois coho-mology group H i ( G K , M ( K s )) (respectively, H i ( G v , M ( K s v ))). The validity of theHasse principle for M ( K s ), i.e., the injectivity of the canonical localization map H ( G K , M ) → Q all v H ( G v , M ) in Galois cohomology, has been discussed in [12, 6].See also [9], § I.9, pp. 117-120. However, if K is a global function field of charac-teristic p > M has p -power order, the injectivity of the canonical localizationmap β ( K, M ) : H ( K, M ) → Q all v H ( K v , M ) in flat (fppf) cohomology has notbeen discussed before (but see [7], Lemma 1, for some particular cases). In thispaper we investigate this problem and show that the injectivity of β ( K, M ) de-pends only on the finite G K -module M ( K s ), which may be regarded as the maximal´etale K -subgroup scheme of M . Indeed, let X ( K, M ) = Ker β ( K, M ) and set X ( G K , M ) = Ker[ H ( G K , M ) → Q all v H ( G v , M )]. Then the following holds. Main Theorem.
Let K be a global function field of characteristic p > and let M be a commutative, finite and flat K -group scheme. Let v be any prime of K . Thenthe inflation map H ( G K , M ) → H ( K, M ) induces an isomorphism Ker (cid:2) H ( G K , M ) → H ( G v , M ) (cid:3) ≃ Ker (cid:2) H ( K, M ) → H ( K v , M ) (cid:3) . Mathematics Subject Classification.
Primary 11G35; Secondary 14K15.C.G.-A. is partially supported by Fondecyt grant 1080025. K.-S. T. is partially supported bythe National Science Council of Taiwan, NSC99-2115-M-002-002-MY3.
In particular, X ( K, M ) ≃ X ( G K , M ) . Thus the Hasse principle holds for M , i.e., X ( K, M ) = 0, if, and only if, theHasse principle holds for the G K -module M ( K s ). An interesting example is thefollowing. Let A be an ordinary abelian variety over K such that the Kodaira-Spencer map has maximal rank. Then A p m ( K s ) = 0 for every integer m ≥ A p m The theorem is proved in the next Section. In Section 3, which concludes thepaper, we develop some applications.2.
Proof of the main theorem
We keep the notation introduced in the previous Section. In addition, we will write(Spec K ) fl for the flat site on Spec K as defined in [8], II.1, p.47, and H r ( K, M )will denote H r ((Spec K ) fl , M ). We will see presently that, in fact, H r ( K, M ) ≃ H r ((Spec K ) fppf , M ).By a theorem of M.Raynaud (see [14] or [1], Theorem 3.1.1, p. 110), there existabelian varieties A and B defined over K and an exact sequence of K -group schemes(1) 0 → M ι → A ψ → B → , where ι is a closed immersion. For r ≥
0, let ι ( r ) : H r ( K, M ) → H r ( K, A ) and ψ ( r ) : H r ( K, A ) → H r ( K, B ) be the maps induced by ι and ψ . The long exact flatcohomology sequence associated to (1) yields an exact sequence0 → Coker ψ ( r − → H r ( K, M ) → Ker ψ ( r ) → , where r ≥
1. Since the groups H r ( K, A ) and H r ( K, B ) coincide with the corre-sponding ´etale and fppf cohomology groups [8], Theorem III.3.9, p.114, and [5],Theorem 11.7, p.180, we conclude that H r ( K, M ) = H r ((Spec K ) fppf , M ). Lemma 2.1.
Let v be any prime of K . If A is an abelian variety defined over K ,then A ( K s ) = A (cid:0) K (cid:1) ∩ A ( K s v ) , where the intersection takes place inside A (cid:0) K v (cid:1) .Proof. Let
F/K be a finite subextension of K s /K and let F v denote the completionof F at the prime of F lying below v . Choose an element t ∈ F such that F v = k (( t )),where k is the field of constants of F , and let m ≥ F p − m v = k (( t p − m )) = P p m i =1 ( t p − m ) i F v ⊂ F p − m F v , whence F p − m v = F p − m F v . Now let a ∈ K be inseparable over K . Then there exists an integer m ≥ F/K as above such that K ( a ) = F p − m . Consequently a ∈ K ( a ) · F v = F p − m F v = F p − m v ,whence a is also inseparable over K v . This shows that K s = K ∩ K s v . Now let V ⊂ A nK be an affine K -variety and let P = ( x , ..., x n ) ∈ V (cid:0) K (cid:1) ∩ V ( K s v ). Theneach x i ∈ K ∩ K s v = K s , whence P ∈ V ( K s ). Thus V ( K s ) = V (cid:0) K (cid:1) ∩ V ( K s v ), andthe lemma is now clear since A is covered by affine K -varieties. (cid:3) HE HASSE PRINCIPLE OVER FUNCTION FIELDS 3 If v is a prime of K , we will write ψ v = ψ × Spec K Spec K v . Since H ( K s , A ) = H ( K s v , A ) = 0, the exact sequence (1) yields a commutative diagram(2) B ( K s ) /ψ ( A ( K s )) (cid:15) (cid:15) ∼ / / H ( K s , M ) (cid:15) (cid:15) B ( K s v ) /ψ v ( A ( K s v )) ∼ / / H ( K s v , M ) . Lemma 2.2.
Let v be a prime of K . Then the canonical map B ( K s ) /ψ ( A ( K s )) → B ( K s v ) /ψ v ( A ( K s v )) is injective.Proof. Write M = Spec R , where R is a finite K -algebra, and identify M (cid:0) K v (cid:1) with Hom K v ( K v ⊗ K R, K v ). If s ∈ M (cid:0) K v (cid:1) , then the image of the composition˜ f : R → K v ⊗ K R s → K v is a finite K -algebra and so, in fact, a finite field extensionof K . Consequently, ˜ f factors through some f ∈ Hom K (cid:0) R, K (cid:1) = M (cid:0) K (cid:1) . Thisimplies that M (cid:0) K v (cid:1) = M (cid:0) K (cid:1) . Now let P ∈ B ( K s ) ∩ ψ v ( A ( K s v )) ⊆ B ( K s v ) and let Q ∈ A ( K s v ) be such that P = ψ v ( Q ). Since A (cid:0) K (cid:1) ψ → B (cid:0) K (cid:1) is surjective, thereexists an R ∈ A (cid:0) K (cid:1) such that ψ ( R ) = P . Then R − Q ∈ M (cid:0) K v (cid:1) = M (cid:0) K (cid:1) .This shows that Q ∈ A (cid:0) K (cid:1) ∩ A ( K s v ) = A ( K s ), by the previous lemma. Thus P = ψ ( Q ) ∈ ψ ( A ( K s )), as desired. (cid:3) The above lemma and diagram (2) show that the localization map H ( K s , M ) → H ( K s v , M ) is injective. The main theorem is now immediate from the exact com-mutative diagram0 / / H ( G K , M ) / / (cid:15) (cid:15) H ( K, M ) / / (cid:15) (cid:15) H ( K s , M ) (cid:127) _ (cid:15) (cid:15) / / H ( G v , M ) / / H ( K v , M ) / / H ( K s v , M ) , whose rows are the inflation-restriction exact sequences in flat cohomology [16],p.422, line -12. 3. Applications
Let K and M be as in the previous Section. We will write K ( M ) for the subfieldof K s fixed by Ker (cid:2) G K → Aut (cid:0) M ( K s ) (cid:1)(cid:3) . We note that the Hasse principle is knownto hold for M ( K s ) under any of the following hypotheses:(a) Gal( K ( M ) /K ) ⊆ Aut (cid:0) M ( K s ) (cid:1) is cyclic. See [9], Lemma I.9.3, p.118. CRISTIAN D. GONZ ´ALEZ-AVIL´ES AND KI-SENG TAN (b) M ( K s ) is a simple G K -module such that pM ( K s ) = 0 and Gal( K ( M ) /K )is a p -solvable group, i.e., Gal( K ( M ) /K ) has a composition series whosefactors of order divisible by p are cyclic. See [9], Theorem I.9.2(a), p.117.(c) There exists a set T of primes of K , containing the set S of all primes of K which split completely in K ( M ), such that T \ S has Dirichlet density zeroand [ K ( M ) : K ] = l . c . m . { [ K ( M ) v : K v ] : v ∈ T } . See [11], Theorem 9.1.9(iii),p.528.In this Section we focus on case (a) above when M = A p m is the p m -torsionsubgroup scheme of an abelian variety A defined over K . More precisely, we areinterested in the class of abelian varieties A such that A p m ( K s ) is cyclic, for thenGal( K ( A p m ) /K ) ֒ → Aut (cid:0) A p m ( K s ) (cid:1) is cyclic as well if p is odd or m ≤ A such that theassociated Kodaira-Spencer map has maximal rank since, as noted in Section 1, A p m ( K s ) is in fact zero. To find more examples, recall that A p m (cid:0) K (cid:1) ≃ ( Z /p m Z ) f for some integer f (called the p -rank of A ) such that 0 ≤ f ≤ dim A . Thus, if f ≤ A p m ( K s ) is cyclic. Clearly, the condition f ≤ A is an elliptic curve,but there exist higher-dimensional abelian varieties A having f ≤
1. See [13], § Remark . Clearly, Gal( K ( A p m ) /K ) may be cyclic even if A p m ( K s ) is not. Forexample, let k be the (finite) field of constants of K , let A be an abelian varietydefined over k and let A = A × Spec k Spec K be the constant abelian variety over K defined by A . Then A ( K s ) tors = A (cid:0) k (cid:1) , and it follows that Gal( K ( A p m ) /K ) ≃ Gal( k ′ /k ) for some finite extension k ′ of k . Consequently Gal( K ( A p m ) /K ) is cyclicand the Hasse principle holds for A p m ( K s ).We will write A { p } for the p -divisible group attached to A , i.e., A { p } = lim −→ m A p m .If B is an abelian group, B ( p ) = S m B p m is the p -primary component of its torsionsubgroup, B ∧ = lim ←− m B/p m is the adic completion of B and T p B = lim ←− m B p m isthe p -adic Tate module of B . Further, if B is a topological abelian group, B D willdenote Hom cont. ( B, Q / Z ) endowed with the compact-open topology, where Q / Z isgiven the discrete topology.Let X denote the unique smooth, projective and irreducible curve over the fieldof constants of K having function field K . If A is an abelian variety over K , we willwrite A for the N´eron model of A over X .The following statement is immediate from the main theorem and the above re-marks. Proposition 3.2.
Let A be an abelian variety defined over K and let m be a positiveinteger. Assume that A p m ( K s ) is cyclic. Assume, in addition, that m ≤ if p = 2 .Then the localization map in flat cohomology H ( K, A p m ) → Y all v H ( K v , A p m ) HE HASSE PRINCIPLE OVER FUNCTION FIELDS 5 is injective. (cid:3)
The next lemma confirms a long-standing and widely-held expectation.
Lemma 3.3. H ( K, A ) = 0 .Proof.
Since H ( K v , A ) = 0 for every v [9], Theorem III.7.8, p.285, it suffices tocheck that X ( A ) = 0. For any integer n , there exists a canonical exact sequenceof flat cohomology groups0 → H ( K, A ) /n → H ( K, A n ) → H ( K, A ) n → . Since the Galois cohomology groups H i ( K, A ) are torsion in degrees i ≥ Q / Z is divisible, the direct limit over n of the above exact sequences yields a canonicalisomorphism H ( K, A ) = lim −→ n H ( K, A n ). An analogous isomorphism exists over K v for each prime v of K , and we conclude that X ( A ) is canonically isomorphicto lim −→ n X ( A n ). Now, by Poitou-Tate duality [9], Theorem I.4.10(a), p.57, and [4],Theorem 4.8, the latter group is canonically isomorphic to the Pontryagin dual oflim ←− n X ( A tn ), where A t is the dual abelian variety of A . Thus, it suffices to showthat lim ←− n X ( A tn ) = 0. Let U be the largest open subscheme of X such that A tn extends to a finite and flat U -group scheme A tn . For each closed point v of U , let O v denote the completion of the local ring of U at v . Now let V be any nonempty opensubscheme of U . By the computations at the beginning of [9], III.7, p.280, and thelocalization sequence [8], Proposition III.1.25, p.92, there exists an exact sequence0 → H ( U, A tn ) → H ( V, A tn ) → M v ∈ U \ V H ( K v , A tn ) /H ( O v , A tn ) . Taking the direct limit over V in the above sequence and using [4], Lemma 2.3, weobtain an exact sequence0 → H ( U, A tn ) → H ( K, A tn ) → Y v ∈ U H ( K v , A tn ) /H ( O v , A tn ) , where the product extends over all closed points of U . The exactness of the lastsequence shows the injectivity of the right-hand vertical map in the diagram below0 / / H ( U, A tn ) / / (cid:15) (cid:15) H ( K, A tn ) / / (cid:15) (cid:15) H ( K, A tn ) /H ( U, A tn ) (cid:127) _ (cid:15) (cid:15) / / Y v ∈ U H ( O v , A tn ) / / Y v ∈ U H ( K v , A tn ) / / Y v ∈ U H ( K v , A tn ) /H ( O v , A tn ) . We conclude that X U ( A tn ) := Ker[ H ( K, A tn ) → Q v ∈ U H ( K v , A tn )] equals H ( U, A tn ) := Ker " H ( U, A tn ) → Y v ∈ U H ( O v , A tn ) . CRISTIAN D. GONZ ´ALEZ-AVIL´ES AND KI-SENG TAN
Now, it is shown in [10], Propositions 5 and 6, that lim ←− n H ( U, A tn ) = 0, whencelim ←− n X U ( A tn ) = 0. Now the exact sequence0 → X ( A tn ) → X U ( A tn ) → Y v / ∈ U H ( K v , A tn )shows that lim ←− n X ( A tn ) = 0, as desired. (cid:3) Remark . With the notation of the above proof, the kernel-cokernel exact se-quence [9], Proposition I.0.24, p.16, for the pair of maps H ( K, A tn ) → M all v H ( K v , A tn ) → M all v H ( K v , A t )yields an exact sequence0 → X ( A tn ) → Sel( A t ) n → M all v H ( K v , A t ) /n, where Sel( A t ) n := Ker[ H ( K, A tn ) → L all v H ( K v , A t )]. Since lim ←− n X ( A tn ) = 0 asshown above, the inverse limit over n of the preceding sequences yields an injectionlim ←− n Sel( A t ) n ֒ → Y all v lim ←− n H ( K v , A t ) /n. This injectivity was claimed in [3], p.300, line -8, but the “proof” given there isinadequate and should be replaced by the above one.
Proposition 3.5.
Let A be an abelian variety over K and let A t be the correspondingdual abelian variety. Assume that A tp m ( K s ) is cyclic, where m is a positive integersuch that m ≤ if p = 2 . Then the localization maps H ( K, A p m ) → M all v H ( K v , A p m ) and H ( K, A ) /p m → M all v H ( K v , A ) /p m are injective.Proof. The injectivity of the first map is immediate from Proposition 3.2 and Poitou-Tate duality [4], Theorem 4.8. On the other hand, the lemma and the long exact flatcohomology sequence associated to 0 → A p m → A p m → A → K and over K v foreach v identifies the second map of the statement with the first, thereby completingthe proof. (cid:3) Remark . When A is an elliptic curve over a number field and p is any prime,the injectivity of the second map in the above proposition was first established in[2], Lemma 6.1, p.107. HE HASSE PRINCIPLE OVER FUNCTION FIELDS 7
Let Q ′ all v H ( K v , A p m ) denote the restricted product of the groups H ( K v , A p m )with respect to the subgroups H ( O v , A p m ). Proposition 3.7.
Let A be an abelian variety over K and let m be a positive integersuch that m ≤ if p = 2 . Assume that both A p m ( K s ) and A tp m ( K s ) are cyclic. Thenthere exist exact sequences → A p m ( K ) → Y all v A p m ( K v ) → H ( K, A tp m ) D → , → H ( K, A p m ) → Y all v ′ H ( K v , A p m ) → H ( K, A tp m ) D → and → H ( K, A p m ) → M all v H ( K v , A p m ) → A tp m ( K ) → . Proof.
This is immediate from Propositions 3.2 and 3.5 and the Poitou-Tate exactsequence in flat-cohomology [4], Theorem 4.11. (cid:3)
Finally, set Sel( A ) p m = Ker " H ( K, A p m ) → M all v H ( K v , A ) and define T p Sel( A ) = lim ←− m Sel( A ) p m . Further, recall the group X ( A ) = Ker " H ( K, A ) → M all v H ( K v , A ) . Now define H (cid:0) K, T p A t (cid:1) = lim ←− m H ( K, A tp m ). Corollary 3.8.
Under the hypotheses of the proposition, there exist canonical exactsequences → X ( A )( p ) → Q all v A ( K v ) ⊗ Q p / Z p A ( K ) ⊗ Q p / Z p → H ( K, T p A t ) D → ( T p Sel( A t )) D → , → T p X ( A ) → ( Q all v A ( K v ) ∧ ) /A ( K ) ∧ → H ( K, A t { p } ) D and → T p Sel( A ) → Y all v A ( K v ) ∧ → H ( K, A t { p } ) D . CRISTIAN D. GONZ ´ALEZ-AVIL´ES AND KI-SENG TAN
Proof.
Let m ≥ / / A ( K ) /p m / / (cid:127) _ (cid:15) (cid:15) H ( K, A p m ) / / (cid:127) _ (cid:15) (cid:15) H ( K, A ) p m (cid:15) (cid:15) / / / / Y all v A ( K v ) /p m / / Y all v ′ H ( K v , A p m ) / / M all v H ( K v , A ) p m / / , yields an exact sequence of profinite abelian groups(3) 0 → X ( A ) p m → Q all v A ( K v ) /p m A ( K ) /p m → H ( K, A tp m ) D → B m ( A ) → , where B m ( A ) = Coker [ H ( K, A ) p m → L all v H ( K v , A ) p m ]. By the main theoremof [3], lim −→ m B m ( A ) ≃ ( T p Sel( A t )) D and the first exact sequence of the statementfollows by taking the direct limit over m in (3). On the other hand, since the inverselimit functor is exact on the category of profinite groups [15], Proposition 2.2.4,p.32, the inverse limit over m of the sequences (3) is the second exact sequenceof the statement. The third exact sequence follows from the second and the exactcommutative diagram0 / / A ( K ) ∧ / / T p Sel( A ) / / (cid:127) _ (cid:15) (cid:15) T p X ( A ) (cid:127) _ (cid:15) (cid:15) / / / / A ( K ) ∧ / / Y all v A ( K v ) ∧ / / ( Q all v A ( K v ) ∧ ) /A ( K ) ∧ / / . (cid:3) Concluding remarks
Let K and M be as in Section 1 and assume that M has p -power order. Further,let M ∗ be the Cartier dual of M . Since X ( K, M ∗ ) ≃ X ( G K , M ∗ ) and there existsa perfect pairing of finite groups(4) X ( K, M ∗ ) × X ( K, M ) → Q / Z [4], Theorem 4.8, it is natural to expect a Galois-cohomological description of X ( K, M ). Note, however, that the natural guess X ( K, M ) ≃ X ( G K , M )is incorrect, since the latter group is zero (because the p -cohomological dimen-sion of G K is ≤ H i ( K s , A ) = H i ( K s , B ) = 0 for all i ≥
1, the exact sequence (1) shows that H i ( K s , M ) = 0 for all i ≥
2. On the other hand, H i ( G K , M ) = 0 for all i ≥ p ( G K ) ≤
1. Now the exact sequence of terms of low degree belonging to
HE HASSE PRINCIPLE OVER FUNCTION FIELDS 9 the Hochschild-Serre spectral sequence H i ( G K , H j ( K s , M )) ⇒ H i + j ( K, M ) yields acanonical isomorphism H ( K, M ) ≃ H ( G K , H ( K s , M )) ≃ H ( G K , B ( K s ) /ψ ( A ( K s ))) . Analogous isomorphisms exist over K v for every prime v of K , and we conclude that X ( K, M ) ≃ X ( G K , B/ψ ( A )) . For example, if M = A p m for some abelian variety A over K , then X ( K, A p m ) ≃ X ( G K , A/p m ) and the pairing (4) takes the form X ( G K , A tp m ) × X ( G K , A/p m ) → Q / Z . In particular, if A tp m ( K s ) is cyclic with m ≤ p = 2, then Proposition 3.2 appliedto A t and the perfectness of the above pairing yield X ( G K , A/p m ) = 0, i.e., theHasse principle holds for the G K -module A ( K s ) /p m . References [1] Berthelot, P., Breen, L. and Messing, W.:
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Departamento de Matem´aticas, Universidad de La Serena, Chile
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