Tate Cycles on Abelian Varieties with Complex Multiplication
aa r X i v : . [ m a t h . N T ] A p r TATE CYCLES ON ABELIAN VARIETIES WITHCOMPLEX MULTIPLICATION
V. KUMAR MURTY AND VIJAY M. PATANKAR
Abstract.
We consider Tate cycles on an Abelian variety A definedover a sufficiently large number field K and having complex multiplica-tion. We show that there is an effective bound C = C ( A, K ) so that tocheck whether a given cohomology class is a Tate class on A , it sufficesto check the action of Frobenius elements at primes v of norm ≤ C . Wealso show that for a set of primes v of K of density 1, the space of Tatecycles on the special fibre A v of the Néron model of A is isomorphic tothe space of Tate cycles on A itself. Introduction and Statement of Results
Let K be a number field or a finite field. Let A be an Abelian varietydefined over K of dimension d , say. For a prime ℓ with ℓ char ( K ), and n ≥
1, we have the ℓ -adic Tate module T ℓ ( A ) = proj lim A [ ℓ n ] ≃ Z dℓ . In a natural way, it is a G K = Gal ( K/K )-module. Set also V ℓ ( A ) = T ℓ ( A ) ⊗ Z ℓ Q ℓ . Set ¯ A = A × K ¯ K . Then, we may identify V ℓ ( A ) with H ,ℓ ( A ) and thecohomology of A with the exterior algebra on the dual of V ℓ ( A ). As usual,for a G K -module W , we denote by W ( k ) the k -fold Tate twist of W .For any field F with K ⊆ F ⊆ ¯ K , set T kℓ ( A, F ) = H kℓ ( ¯ A )( k ) Gal ( ¯
K/F ) T ∗ ℓ ( A ) = M k X K ⊆ F ⊆ ¯ K T kℓ ( A, F )This is the space of Tate cycles on A . It is a finite dimensional Q ℓ vectorspace and we may in fact restrict the field F above to be of finite degreeover K . Thus, for some finite extension M of K , we have T ∗ ℓ ( A ) = ⊕ k H kℓ ( ¯ A )( k ) Gal ( ¯
K/M ) = ⊕ k T kℓ ( A, M ) Date : Sunday 28 th October, 2018.Research of VKM partially supported by an NSERC Discovery Grant. gain, for K ⊆ F ⊆ ¯ K , let Z k ( A, F ) denote the free Abelian group gener-ated by algebraic cycles of co-dimension k on A (modulo homological equiv-alence) with a representative defined over F . The ℓ -adic cycle class mapis c ℓ,k,F : Z k ( A, F ) ⊗ Z Q l → T kℓ ( A, F ) . Conjecture 1.1. [19] (Tate) The ℓ -adic cycle class map is surjective. This is known for k = 1 by Tate in the case K is a finite field and byFaltings in the case K is a number field. This conjecture is not known to beindependent of ℓ .There is a related conjecture involving L -functions associated to A . Conjecture 1.2. (Tate) If K is a number field, L ( H kℓ ( ¯ A ) , s ) has a mero-morphic continuation to Re ( s ) = 1 + k and ord s =1+ k L ( H kℓ ( ¯ A ) , s ) = − dim T kℓ ( A, K ) . If K is a finite field, ord s = k L ( H kℓ ( ¯ A ) , s ) = − dim T kℓ ( A, K ) . Our aim is to study (in the case K is a number field) the relationshipbetween T ∗ ℓ ( A ) and T ∗ ℓ ( A v ). Here v is a prime of K of good reduction for A , ℓ is assumed to be distinct from the characteristic of the residue field k v (of K at v ) and A v stands for the reduction of A modulo v .At a prime v of good reduction for A , we have the natural reduction map, A → A v , which induces a natural map (under pull-back) H ∗ ℓ ( A v ) ≃ H ∗ ℓ ( A ) . Let us set ι v : H ∗ ℓ ( A ) −→ H ∗ ℓ ( A v ) (1)be the inverse of the above isomorphism. We may view both sides as Q ℓ [ G v ]-modules where G v ⊂ G K is the decomposition group at v (subgroup of G K unique up to conjugation). Indeed, the left hand side H ∗ ℓ ( A ) is naturallya G v -module under restriction from G K . The right hand side is naturallya Q ℓ [ G k v ] module where G k v is the Galois group of the residue field k v at v . Since G k v is naturally a quotient of G v , we can consider H ∗ ℓ ( A v ) asa Q ℓ [ G v ]-module. Also, as v is a prime of good reduction, the criterionof Néron-Ogg-Shafarevich [16] implies that the inertia subgroup at v actstrivially on H ∗ ℓ ( A ). Thus, the two sides in (1) are isomorphic as Q ℓ [ G v ]-modules and so, restriction gives us a map ι v : T ∗ ℓ ( A ) → T ∗ ℓ ( A v ) (2)which is in fact injective. e begin with the following observation. A class ω ∈ H mℓ ( ¯ A )( m ) forwhich ι v ( ω ) is in T mℓ ( A v , k v ) at almost all places v of K (i.e. at all but finitelymany places of K ) is in T mℓ ( A, K ). Indeed, this follows from the Chebotarevdensity theorem. This observation can be strengthened as follows.
Proposition 1.1.
Let S be a set of places of K with positive Dirichletdensity, say δ > . A class ω ∈ H mℓ ( ¯ A )( m ) for which ι v ( ω ) is in T mℓ ( A v , k v ) at all places v in S is in T mℓ ( A ) .Proof. Note that the ℓ -adic representation ρ ℓ : G K → Aut ( H mℓ ( ¯ A )( m ))is unramified at all but finitely many places v of K . Let L be the sub-extension of ¯ K fixed by Ker ( ρ ℓ ). Then L is a Galois extension of K andis unramified at all except finitely many places of K . At a finite prime v ,to be a Tate class means that it is fixed by the Frobenius automorphism in Gal ( k v /k v ) given by x x N v , where N v is the cardinality of k v (alsothe norm of v ). The Frobenius automorphism lifts to F rob v (at almost all v ) and thus, by (1) as above and the assumption, ω is fixed by F rob v ∈ Gal ( L/K ). Let G S be the group generated by { F rob v | v ∈ S } in Gal ( L/K )as a topological group. Let L G S be the sub-extenstion of L cut out by G S . The places of S split completely in L G S . This implies that L G S isa finite extension of K . Moreover, by the Chebotarev density theorem,[ Gal ( L/K ) : G S ] = [ L G S : K ] ≤ /δ . Thus, it follows that if for all v ∈ S , ι v ( ω ) is in T mℓ ( A v , k v ), then ω is in T mℓ ( A, L G S ), i.e. it is a Tate class. (cid:3) With more work, we will show that in the case of Abelian varieties withcomplex multiplication, the condition of the above result can be replacedwith a condition at a finite set of primes. Moroever, this can be done in aneffective manner. Here also, the role of split primes is crucial.We recall that for each prime ℓ , Serre and Tate [16] define a conductor of A (in terms of the Artin and the Swan character). It is an integer that theyprove ([16], pp. 499-500) is independent of ℓ . Theorem 1.1.
Let A be an Abelian variety with complex multiplication andlet K be a number field over which both A and its endomorphisms are defined.Denote by d the dimension of A and by N the conductor of A . Then, thereis an effective bound C = C ( A, K ) such that if ω ∈ H mℓ ( ¯ A )( m ) , for some prime ℓ , is a Tate class considered as an element of H mℓ ( ¯ A v )( m ) for all v of K with N K/ Q v ≤ C , then ω is Tate, i.e. ω ∈ T mℓ ( A, K ) . Several remarks are in order.
Remark 1.1.
The main point of the above theorem is that the bound C (defined by (24)) depends only on the Abelian variety A (in fact, only onthe conductor, the dimension and the discriminant of the field of complexmultiplication) and the number field K and does not depend on ℓ . However, f we fix a prime ℓ , it is possible to prove (using the Chebotarev densitytheorem and Nakayama’s lemma for finitely generated modules over Z ℓ ) theexistence of a constant (that would depend on A , K and ℓ ) so that similarstatement as in the theorem above is true. Remark 1.2.
The proof of Theorem 1.1 will produce an explicit expressionfor C (as defined by (24)) only in terms of the degree n K , the discriminant d K of K , the conductor N , the dimension d of A and the discriminant d F ofthe field F of multiplication of A . We have not made any effort to find anoptimal bound. However, we note that with the assumption of the RiemannHypothesis for Dedekind zeta functions, the estimate for C that the proofof Theorem 1.1 yields can essentially be replaced by (log C ) . It would bevery interesting to study to what extent it would be possible to get a boundwhich is uniform in each of the parameters n K , d K , N, d F and d . Remark 1.3.
Tate [20] (section 3, pp.76-77) discusses the notion of an“almost algebraic” cycle. A Tate cycle is said to be almost algebraic if allbut finitely many of its specializations are algebraic. According to Tate,this notion of almost algebraic cycle seems to be part of the folklore. Tateadds: “it is mentioned explicitly in [12], 5.2 that Künneth components of the“diagonal” are almost algebraic (by part (1) of Theorem 2 of [1])”. Thus these“Künneth components of the diagonal” are examples of almost algebraiccycles that are not known to be algebraic. Tate further mentions a weakerconjecture than Conjecture (1.1): The space of Tate cycles is spanned byalmost algebraic cycles.
Remark 1.4.
Let us assume the Tate conjecture for Abelian varieties overfinite fields. Suppose ω is a class in H mℓ ( A )( m ) with the property that ι v ( ω )is algebraic for each prime v of K of norm ≤ C . Then ι v ( ω ) is algebraicfor all but finitely many v . Indeed, by the Theorem, ω is a Tate class, andhence so is its reduction ι v ( ω ). Now by our assumption, it follows that ι v ( ω )is algebraic. Hence ω is almost algebraic.Our next result is about the field of definition of Tate cycles on an Abelianvariety over a finite field obtained by reduction of Tate cycle on a fixedAbelian variety over a number field. Theorem 1.2.
Let A be an Abelian variety defined over a number field K .Then, there is a bound D = D ( A ) so that for all v of good reduction, allTate cycles on A v are defined over an extension of the residue field k v ofdegree ≤ D . Remark 1.5.
In fact our proof of this theorem can be suitably adapted toshow that all the Tate cycles on an Abelian variety A over a number field K with complex multiplication by F are defined over a ‘specific’ extension of K that depends on d = dim( A ), the conductor of A and the normal closureof F . s is well known (and as implied by (2)) the dimension of the space ofTate cycles does not decrease under reduction modulo v . Our next resultshows that in the CM case, for a set of primes of Dirichlet density 1, it doesnot increase either. The proof of this theorem uses Thereom (1.2). Theorem 1.3.
Let A be an Abelian variety of CM type. Let K be sufficientlylarge so that all the Tate cycles on A and all the endomorphisms of A aredefined over K . Then, for a set of primes v of K of Dirichlet density , T ∗ ℓ ( A ) ≃ T ∗ ℓ ( A v ) . In particular, the Tate conjecture for A implies the Tate conjecture for A v for a set of primes v of Dirichlet density . Remark 1.6.
The condition that K be sufficiently large is necessary asthe following example illustrates: Let E be an elliptic curve over Q withcomplex multiplication by an imaginary quadratic field, say K = Q ( √− d ),with d square-free and positive integer. Let A = E × E . Then, Z ( A, K ) isgenerated by E × { } , { } × E , the diagonal ∆ := { ( x, x ) | x ∈ E } and thegraph of complex multiplication∆ ′ := { ( x, [ √− d ] x ) | x ∈ E } , where [ √− d ] denotes the endomorphism of E whose square is [ − d ].We remark that E has super-singular reduction at a prime p of Q whichremains inert in K and of good reduction for E . Let A p denote the reductionof A modulo p over F p . Then, Z ( A p , ¯ F p ) is generated by E p × { } , { } × E p ,∆ p (the reduction of ∆ modulo p ) , ∆ ′ p (the reduction of ∆ ′ modulo p ),∆ φ := { ( x, φ ( x )) | x ∈ E p } and ∆ ψ := { ( x, ψ ( x )) | x ∈ E p } , where φ and ψ arethe other two generators of End ¯ F p ( E p ). This proves that for such primes p ,the rank of T ℓ ( A ) is 4 and the rank of T ℓ ( A p ) is 6. Thus, the reduction mapas in (2) is strictly injective for a set of primes p of Q of Dirichlet density ! However, this is not a problem as the set of primes of Q that are inert in K , has Dirichet density 0 considered as a set of primes of K . Remark 1.7.
The theorem is not true for Abelian varieties without complexmultiplication as the following example illustrates : Let E be an ellipticcurve over Q without complex multiplication. Let A = E × E be as above.Then, Z ( A, K ) is generated by E × { } , { } × E and the diagonal ∆ forany number field K . Let v be a finite place of K of good reduction for E and let k v be the finite residue field at v . Then, E always acquires theextra endomorphism (the Frobenius ) and Z ( A v , k v ) has rank at least 4.This implies that T ℓ ( A ) strictly injects into T ℓ ( A v ) for all but finitely manyplaces of K . Remark 1.8.
The above result is, in some sense, related to our previouswork [7]. There, we studied the problem of when A v stays simple for a set We thank the referee for suggesting us to bring this to the attention of the reader. f primes of positive density, given that A is simple (or absolutely simple).If A v splits, this gives rise to ‘extra’ classes in the Néron-Severi group of A v (and hence also extra Tate cycles on A v ).In section 2, we recall some basic properties of a compatible family of λ -adic representations. In section 3, we recall some basic properties of thetheory of complex multiplication. In section 4, we develop an analytic es-timate that will be crucial in the proofs of our main results. In section 5,we present the Main Lemma and in the following three sections, we give theproofs of Theorems 1.1, 1.2 and 1.3.2. Brief background on compatible family of λ -adicrepresentations We recall (from [10]) some basics about compatible families of λ -adic Ga-lois representations.Let K be a number field. Let K be a separable algebraic closure of K .Let G K denote the Galois group of K over K . Let E λ be a non-archimedeanlocal field (finite extension of some p -adic field). Let V be a vector spaceover E λ . Let Aut( V ) be the general linear group of V with topology inducedby that on End ( V ): If n = dim( V ), we have Aut( V ) ≃ GL( n, E λ ). Definition 2.1. A λ -adic representation of G K ( or of K ) is a continuoushomomorphism ρ : G K → Aut( V ), where V is vector space over a non-archimedean local field E λ , a finite extension of Q p .For any number field F , by Σ F we denote the set of finite places of F .Recall that a λ -adic representation ρ is said to be rational (respectively, integral ) if there exists a finite subset S of Σ K such that(a) For any place v in Σ K − S , ρ is unramified at v .(b) For v / ∈ S , the coefficients of P v,ρ ( T ) belong to Q (respectively, Z ).Let E be a number field. For each finite prime λ of E , let ρ λ be a rational λ -adic representation of K . For any given finite prime λ of E , let us denoteby ℓ λ the prime of Q that lies below the prime λ of E . The system { ρ λ } issaid to be strictly compatible if there exists a finite subset S of Σ K , calledthe exceptional set, such that:(a) For S λ := { v | v lies over ℓ λ } and every v / ∈ S ∪ S λ , ρ λ is unramified at v and P v,ρ λ ( T ) has rational integral coefficients.(b) P v,ρ λ ( T ) = P v,ρ λ ′ ( T ) if v / ∈ S ∪ S λ ∪ S λ ′ . Definition 2.2.
Let ρ (= ρ λ ) be a λ -adic representation of G K . Then, wesay that ρ is pure of weight w ∈ Z if there is a finite set S of finite places of K such that, for each finite place v / ∈ S ∪ S λ , ρ ( F rob v ) is unramified, andthe eigenvalues of ρ ( F rob v ) are algebraic integers whose complex conjugateshave complex absolute value q w/ v , where q v is the cardinality of the residue eld of K at v . We say that a compatible family { ρ λ } is of weight w ∈ Z iffor each finite place λ of E , ρ λ is of pure weight w . As usual, we denote by χ ℓ the cyclotomic character giving the action ofGal( K/K ) on ℓ -power roots of unity. The { χ ℓ } form a compatible family of ℓ -adic representations.3. Abelian Varieties with Complex Multiplication andShimura-Taniyama theorem
We now briefly recall a few basic facts about Abelian varieties of
Com-plex Multiplication type (for short, Abelian varieties with CM). The mainreferences are [3], [8], [16], [17].Let A be an Abelian variety defined over a number field K of dimension d := dim( A ). Let F be a number field of degree 2 d . Then, A is said tohave Complex Multiplication by F (or for short, CM by F ) if there existsan embedding ι : F → End Q ( A ) := End Q ( A ) ⊗ Z Q . It is a fact that such an F must be a totally imaginary extension of a totallyreal number field. Let E be the Galois closure of F . To such an Abelianvariety, the Shimura-Taniyama theory associates Hecke characters ψ i,λ ofthe following type. For each 1 ≤ i ≤ d and for each finite place λ of E (lying over a prime ℓ ), we have a family of continuous characters of theGalois group G K := Gal ( K/K ): ψ i,λ : G K → E × λ with the property that for all v not dividing ℓN , ψ i,λ ( F rob v ) ∈ F × . Thus,the field generated by ψ i,λ ( F rob v ) for all v over Q is contained in E . Wehave the following well-known result ([8], Theorem 2 ′ , page 171): Theorem 3.1. (Shimura-Taniyama) (With notations and definitions asabove) Let A be an Abelian variety defined over a number field with Com-plex Multiplication F . Let H kℓ ( A ) be the k -th ℓ -adic étale cohomology of A .Then, H kℓ ( A ) ⊗ Q ℓ Q ℓ = ∧ k H ℓ ( A ) ⊗ Q ℓ Q ℓ = ⊕ I H I , where the direct sum is over subsets I ⊆ { ψ ,λ , . . . , ψ d,λ } of cardinality k and where each H I is a one dimensional G K invariant Q ℓ subspace of H kℓ ( A ) ⊗ Q ℓ Q ℓ . Further more, for any finite place v of K away from ℓ andfrom places of bad reduction for A , and for all x ∈ H I ( F rob v ) − ( x ) = ( Y ψ i,λ ∈ I ψ i,λ (Frob v )) · x In fact, the Shimura-Taniyama theory implies that for 1 ≤ i ≤ d , ( ψ i,λ ) λ forms a compatible family of 1-dimensional continuous λ -adic representa-tions of G K , where λ runs over finite places of E . We denote these compat-ible families by ψ i . Let Ψ := { ψ , . . . , ψ d } and Ψ λ := { ψ ,λ , . . . , ψ d,λ } . heorem 3.2. (With notations as above) Let λ be a prime of E lying over ℓ . Then, H kℓ ( A ) ⊗ Q ℓ E λ = ⊕ I H I,λ (3) where I ⊆ Ψ λ of size k and H I,λ is a 1 dimensional G K invariant E λ -subspace such that for any finite place v of K away from ℓ and places of badreduction for A , ( F rob v ) − ( x ) = ( Y ψ i,λ ∈ I ψ i,λ ( F rob v )) · x for x ∈ H I,λ . We separate this as a corollary:
Corollary 3.1.
With notations and definitions as above, { H I,λ } is a strictly-compatible family of λ -adic representations of G K as λ varies over finiteplaces of E . The least prime that does not split completely
Let L and K be number fields with K ⊆ L . In the proof of our results, itwill be necessary to have an estimate for the norm of a prime v of K thatdoes not split completely in L . Such an estimate is given in [6], Theorem 1assuming the Generalized Riemann Hypothesis for the Dedekind zeta func-tion ζ L ( s ) of s and assuming that L/K is Galois. In the case that K = Q ,an unconditional estimate is given by X. Li in [4], Theorem 1 and also byVaaler and Voloch in [21]. However, we need an (unconditional) estimatefor a general K . Such an estimate is remarked after the proof of Theorem1 in [6]. What we prove is slightly weaker, though more than sufficient forour purposes.Let us denote by n L , n K the degrees of L and K over Q respectively. Let d L , d K denote the discriminants of L/ Q and K/ Q , respectively. We also notethe effective prime number theorem in K as given by Lagarias and Odlyzko(see [2] and [14], Théorème 2). This theorem tells us that the number π K ( x )of primes of K of norm ≤ x satisfies | π K ( x ) − Li x + Li ( x β ) | ≤ c x exp( − c s log xn K )) (4)provided x ≥ c n K (log | d K | ) . Here, c , c , c > β is the possible exceptional zero of the Dedekind zeta function ζ K ( s ) . If it exists, it is real and satisfies1 −
14 log | d K | < β < . Let us set f ( K ) = ( n K if ζ K ( s ) does not have an exceptional zeromax( n K ! log | d K | , | d K | /n K ) + n K otherwise . (5) hus, for example f ( Q ) = 1. Theorem 4.1.
There is an effective and absolute constant c > with thefollowing property. Let K be a number field and let L/K be a finite non-trivial Galois extension of degree n . Then, there exists a prime ideal ℘ of K such that • ℘ is of degree 1 over Q and unramified in L . • ℘ does not split completely in L and N K/ Q ℘ < max(55 , e cf ( K ) | d L | / n − ) (6) Proof.
From [18], Lemma 3, we have for σ >
1, the inequality − ζ ′ L ζ L ( σ ) < σ + 1 σ − (cid:18) | d L | r π n L (cid:19) + r ′ Γ ( σ/
2) + r Γ ′ Γ ( σ ) . (7)On the other hand, we have the Dirichlet series expansion − ζ ′ L ζ L ( σ ) = X p ,m log N p ( N p ) mσ where the sum ranges over primes p of L and 1 ≤ m ∈ Z . If all primes ℘ of K which are of degree 1 over Q and of norm less than y (say) ramify orsplit completely in L , then we see that with n = [ L : K ] = n L /n K , − ζ ′ L ζ L ( σ ) ≥ n X N ℘ ≤ y ′ log N ℘ ( N ℘ ) σ where the prime on the summation indicates that we range over primes ℘ of K that are of degree 1 over Q and which are unramified over L . Thus, − ζ ′ L ζ L ( σ ) ≥ n X N ℘ ≤ y log N ℘ ( N ℘ ) σ − nS − nS where S = X N ℘ ≤ yr log N ℘ ( N ℘ ) σ and the sum is taken over primes ℘ of K that ramify in L , and S = X N ℘ ≤ y ℘ of degree ≥ log N ℘ ( N ℘ ) σ . For each prime ℘ counted in this sum, let p be the rational prime that itdivides. Then (log N ℘ ) / ( N ℘ ) σ ≤ p ) /p σ . Moreover, given a rationalprime p , there are at most n K primes ℘ of K dividing p and so S ≤ n K X p log pp σ ≤ n K . e are using here the fact that for the Riemann zeta function, the inequality(7) gives X p log pp σ < − ζ ′ ζ (2 σ ) < σ >
1. For S , we have ([6], p. 558) S ≤ X ℘ | d L/K log N ℘ ≤ n log d L . Thus, − ζ ′ L ζ L ( σ ) ≥ n X N ℘ ≤ y log N ℘ ( N ℘ ) σ − d L − n L . (8)To estimate the sum on the right, we use (4). Let us assume that y ≥ y = c n K (log | d K | ) . By partial summation, we have X N ℘ ≤ y log N ℘ ( N ℘ ) σ = π K ( y )(log y ) y − σ + Z y π K ( t ) t − − σ ( σ log t + 1) dt. (9)The first term on the right is bounded by using the estimate π K ( x ) ≤ n K π ( x ) . Thus, we see that as σ > π K ( y )(log y ) y − σ ≤ c n K . Similarly, we see that Z y π K ( t ) t − − σ ( σ log t + 1) dt ≤ c n K log y The estimate (4) implies that replacing π K ( t ) with Li ( x ) − Li ( x β ) in theintegral in (9) results in an error of at most ≤ Z log y exp( − c q u/n K ) udu ≤ c n K . The term coming from the possible exceptional zero contributes to the inte-gral an amount which is easily seen to be < ( σ − β ) − < (1 − β ) − . (10)To estimate this, we note that by [18], equations (27) and (28), we have β < max(1 − (4 n K ! log | d K | ) − , − ( c | d K | /n K ) − ) . This implies that(1 − β ) − < max(4 n K ! log | d K | , c | d K | /n K ) . Let us denote the right hand side by c K . Finally, it remains to estimate Z yy (Li t ) t − − σ ( σ log t + 1) dt. nd this is easily seen to be σ Z y t − σ dt + O ( Z y t − σ (log t ) − ) dt. Taking σ = 1 + (log y ) − , (11)this is easily seen to be ≥ log y − c log log y. Putting all this together into (8), we deduce that − ζ ′ L ζ L ( σ ) ≥ n log y − n ( c K + c (log | d K | ) + c log log y ) − c n L log y − d L − n L . On the other hand, we can combine this with the upper bound given above(7). As mentioned in [18], p. 142, we have Γ ′ ( σ/ < ′ ( σ ) < < σ < /
4. Moreover, Γ( x ) > x > y > e ∼ .
6, there is an absolute and effective constant c > n log y < y + 52 log | d L | + c nf ( K )with f ( K ) given by (5). It follows that there is a c > y ≤ cf ( K ) + 52( n −
1) log | d L | . Thus, if y > e cf ( K ) | d L | / n − , we get a contradiction and this proves theresult. (cid:3) We record here the following estimate of Hensel (see for example, pp.44-45 of [5]) which we shall also need. We havelog d L ≤ ( n L − X p ∈ P ( L/ Q ) log p + n L (log n L ) | P ( L/ Q ) | (12)where P ( L/ Q ) is the set of rational primes p that ramify in L . In fact, when L is Galois over K , the following stronger estimate holds:log d L ≤ ( n L − n K ) X p ∈ P ( L/K ) log p + n L (log n L − log n K ) + n L n K log d K . (13)In particular, if L/K is a Galois extension unramified outside of primesdividing M then log d L ≪ n L log( M n L n K ) + n L n K log d K . (14) . Proof of the Main Lemma
Lemma 5.1.
Suppose ω ∈ H ∗ l ( A ) is a (simultaneous) eigenvector for aconjugacy set C ⊂ G K . Then so is every element of Q l [ G K ]( ω ) , the G K -module generated by ω inside H ∗ l ( A ) .Proof. We need to show that for any g ∈ G K , g ( ω ) is an eigenvector for any σ ∈ C . We have σ ( g ( ω )) = g (( g − σg )( ω ))= g ( τ ( ω )) , for some τ ∈ C as C is a conjugacy set= g ( λ τ · ω ) for some λ τ ∈ Q ℓ by assumption= λ τ · g ( ω ) , the last step follows from the Q ℓ linearity of the G K action. (cid:3) For integers
N, m, d ≥ K , let us set B = B ( N, K, m, d ) = e f ( K ) N mn K d ( f ( K ) + n K log N ) mn K d +1 . (15)Here f ( K ) = ( n K if ζ K ( s ) does not have an exceptional zeromax( n K ! log | d K | , | d K | /n K ) + n K otherwise . (16)as defined by (5), and n K = [ K : Q ].Let G K denote the absolute Galois group of a number field K . Let E be some number field and O E , the ring of integers of E . For a finite place λ of E , we denote by E λ , the completion of E at λ , and by O λ , the ringof integers of E λ . Let { M λ } be a family of continuous O λ [ G K ]-modulessuch that { M λ ⊗ O λ E λ } is a strictly compatible family of continuous semi-simple λ -adic integral representations of G K of weight 2 w , conductor N anddimension d , where λ varies over all finite places of E . Let S ′ be the set ofexceptional places of K for the system { M λ } . Lemma 5.2. Main Lemma
There are absolute and effective constants c , c > so that if for some finite place λ of E , the set S = { F rob v : N v ≤ c B ( N wd log d E , K, n E , d ) c } acts as scalars on M λ , then G K acts as scalars on M λ and hence on any M λ where λ is a finite place of E .Proof. By abuse of notation, below, we will use M λ to denote M λ as amodule over O E and as well as the associated vector space M λ ⊗ O E E λ over E λ .First suppose that F rob v acts as a scalar µ on M λ . Then ( N v ) wd = µ d ,since the characteristic polynomial of F rob v has coefficients in Z . So infact, µ = ǫ ( N v ) w with ǫ d = 1. Since { ρ λ } is a compatible family of rationalintegral representations, the trace dǫ ( N v ) w of F rob v , lies in Z and so ǫ = ± f d = 1 then there is nothing to prove, so we may assume without loss that d >
1. Suppose that G K does not act as scalars on M λ . By compatibility, G K does not act as scalars on the family { M λ } . We shall show that S doesnot act as scalars either.Let S ′ be the exceptional set (of places of K ) for the system { M λ } . Fora finite place λ of E , let ℓ λ denote the prime of Q that lies below λ . Let S λ := { v | v lies over ℓ λ } , the set of finite places of K that divide ℓ λ . For v / ∈ S ′ ∪ S λ , let P v,λ ( T ) := det( T − ρ λ ( F rob v )). By compatibility, P v,λ ( T ) isindependent of λ .Let v / ∈ S ′ be the prime with least norm such that F rob v does not act asscalars on M λ . (Note that F rob v is unique up to conjugation.) Since ρ λ is a semi-simple representation, this is equivalent to P v ( T ) = ( T − θ v ) d forany θ v ∈ C . Thus by compatibility, F rob v does not act as a scalar on M λ for any λ .Denote the eigenvalues of F rob v by { α i,v } for i = 1 to d . Let us choose ℓ unramified in E so that the distinct eigenvalues of F rob v remain distinctmodulo ℓ . This can be done by choosing an ℓ that does not divide thediscriminant of E and the norm of the product of the differences of any twodistinct eigenvalues of F rob v . We have (cid:12)(cid:12)(cid:12)Y ( α i,v − α j,v ) (cid:12)(cid:12)(cid:12) ≤ (2( N v ) w ) d where the product is over pairs of distinct eigenvalues of F rob v . Thus, wecan find such an ℓ satisfying ℓ ≪ log { d E (2( N v ) w ) d } ≪ wd log N v + log d E , (17)where the implied constant is absoulte.Here, we are using the fact that for an integer m >
1, there exists a primewhich is O (log m ) that does not divide m . Indeed, by the prime numbertheorem, the product of all the primes less than 3 log m (say) would belarger than m . Hence, at least one of these primes does not divide m .It then follows that for at least one of the places λ of E that lie over ℓ ,the distinct eigenvalues of F rob v remain distinct modulo λ . Thus, F rob v does not act as a scalar on M λ = M λ /λM λ .Let ρ λ be the natural map from G K to P GL ( M λ ), the projective generallinear group associated to M λ . By above, the image of F rob v to P GL ( M λ ) isnot the identity. By applying Theorem 4.1 to this representation, it followsthat there exists a prime v ′ (say) for which the image of F rob v ′ in P GL ( M λ )is not the identity. In particular, v ′ does not split completely in the fixedfield L (say) of the kernel of ρ λ . We know that L is unramified outside ℓN and as log n L n K ≤ [ E : Q ] d log ℓ t follows by (14) of section 4,log d L ≪ n L [ E : Q ] d log N ℓ + n L n K log d K . (18)Then by (6) of section 4 and n = n L /n K ≥
2, we can choose v ′ satisfyinglog N v ′ ≪ f ( K ) + n K n L log d L ≪ f ( K ) + n K [ E : Q ] d log N ℓ. (19)By (17), we havelog
N v ′ ≪ f ( K ) + n K [ E : Q ] d log( N wd log N v + N log d E ) . (20)Since F rob v ′ does not act as a scalar on M λ , it does not act as a scalar on M λ , and hence on M λ either, and as v is a prime of least norm with thisproperty, we have an absolute and effective constant c > N v ≤ N v ′ ≤ (cid:16) e f ( K ) { N ( wd log N v + log d E } n K [ E : Q ] d (cid:17) c . (21)Hence, we have proved that N v is bounded by ≪ ( e f ( K ) ( N wd log d E ) n K n E d ) c ( f ( K )+ n K n E d log( N wd log d E )) cn K n E d +1 . (22)Here the implied constant is absolute and effective. Thus, N v ≤ c B ( N wd log d E , K, n E , d ) c . This contradicts our assumption and proves the lemma. (cid:3) Proof of Theorem (1.1)
By assumption, ω ∈ H mℓ ( A ). Let M ℓ be the Q ℓ [ G K ] sub-module gen-erated by ω inside H mℓ ( A ). Let λ be a place of E lying over ℓ . Let M λ := M ℓ ⊗ Q ℓ E λ . Then, by Theorem (3.2), M λ is isomorphic (as E λ [ G K ] module) to a sum of H J,λ for certain subsets J of Ψ of size 2 m .Let us denote this set of subsets J of Ψ by J . Thus, M λ = M ℓ ⊗ Q ℓ E λ = ⊕ J ∈J H J,λ . (23)The right hand side of (23) can be realised as the λ component of a familyof λ -adic representations, say { M λ } , as follows: For any finite place λ of E ,let M λ := ⊕ J ∈J H J,λ . By the Corollary (3.1), { M λ } is a strictly compatible family of semi-simple λ -adic representations of G K . It is easy to see from the definition that theconductor of this family is bounded by the conductor of the family { H mλ } .Moreover, the conductor of H mλ can be bounded in terms of m and theconductor N of A . In particular, we can get a bound depending on N andthe dimension d of A that majorizes the conductor of all the { H mλ } .We have n E ≤ (2 d )! and the discriminant satisfieslog d E ≤ (2 d )! log d F . otice that w = m ≤ d, and the dimension of the M λ is equal to d m ! ≤ d . Thus, if we set C ( N, d, F, K ) = c B (2 d (2 d + 1)! N log d F , K, (2 d )! , d ) c (24)where c, c and B are as in Lemma 5.2, then applying Lemma 5.2 to { M λ } ,we get that G K acts as scalars on M λ = Q ℓ [ G K ]( ω ) ⊗ Q ℓ E λ . This impliesthat M λ = Q ℓ · ω ⊗ Q ℓ E λ . Thus, 1 = dim E λ M λ = dim Q ℓ M ℓ , provingTheorem (1.1). (cid:3) Proof of Theorem 1.2
Denote by P ( A v , T ) the characteristic polynomial of F rob v . Writing P ( A v , T ) = Y (1 − α i T ) ∈ Z [ T ]we see that the α i are algebraic of degree ≤ deg P ( A v , T ) = 2 dim A v =2 dim A .Moreover, for any n ≥
1, if k v,n denotes the extension of k v of degree n , P ( A v /k v,n , T ) = Y (1 − α ni T ) . Now, dim T kℓ ( A v , k v,n ) = { I : α nI = q kn } , here α I = Q α i ∈ I α i where I runs over subsets of { α , · · · , α d } of cardinality2 k . The right hand side is equal to { I : α I = q k ζ n for some ζ n ∈ µ n } But α I is an algebraic number of degree ≤ b k ( A ) = dim H kℓ ( A ) ≤ (cid:0) d k (cid:1) .Hence if α I = q k ζ n then n is bounded. In other words, all Tate classes aredefined over an extension of k v of degree bounded independently of v . (cid:3) Proof of Theorem 1.3
By definition T kℓ ( A ) consists of cohomology classes x ∈ H kℓ ( A ) on which G K acts by the character χ kℓ , where χ ℓ is the cyclotomic character of G K acting on the Tate module Z ℓ (1). Note that we are assuming that K issufficiently large so that all the Tate classes appear over K . By Theorems(3.1) and (3.2), we see that G K acts on the 1-dimensional subspaces H I bythe Hecke character ψ I . Here, ψ I = k Y j =1 ψ i j , here I is the subset { i , . . . , i k } of { , · · · , d } . Thus,dim T kℓ ( A ) = { I | ψ I = χ kℓ } . (25)Similiarly, for a finite place v of K of good reduction for A ,dim T kl ( A v ) = { I : ψ I ( F rob v ) = ζ n ( N v ) k } (26)where ζ n is some n -th root of unity where n ≤ (cid:0) d k (cid:1) is bounded indepedentlyof v as proved in Theorem (1.2). Indeed, by Theorems (3.1) and (3.2), wesee that F rob v acts on the 1-dimensional vector subspaces H I by multipli-cation by ψ I ( F rob v ). On the other hand, by above and by Theorem (1.2), T kℓ ( A v ) consists precisely of those cohomology classes on which F rob v actsby χ kℓ ( F rob v ) = ζ n ( N v ) k , and so (26) follows.Thus, for all finite places v of good reduction, (25) and (26) implies: S := { I : ψ I = χ kℓ } ⊂ S v := { I : ψ I ( F rob v ) = ( N v ) k } Let T := { v | S ( S v } . For any v ∈ T , there exists some J / ∈ S ( J depends on v ) of cardinality 2 k such that ψ J ( F rob v ) = ζ n ( N v ) k , where ζ n is some n -th root of unity. Wecan write T = ∪ J / ∈ S T J , T J := { v ∈ T | ψ J ( F rob v ) = ζ n ( N v ) k } We want to prove that T is of density 0. Suppose not. By above, sincethere are only finitely many index sets J , there exists a J / ∈ S such that δ ( T J ) >
0. Hence, for all v ∈ T J , ψ nJ ( F rob v ) = ( N v ) kn = χ knℓ ( F rob v ).Applying Theorem 2, p. 163 of [9] to the characters ψ nJ and χ knℓ , we deducethat ψ nJ = χ knℓ χ where χ is a character of finite order. Hence ψ nJ = χ knℓ when restrictedto G L := Gal( K/L ), where L is the fixed field cut out by Ker ( χ ). Thus, ψ := ψ nJ · χ − kℓ is a character of G L of order n . Thus ψ J = χ kℓ whenrestricted Gal ( K/L ψ ), where L ψ is the number field cut out by Ker ( ψ ).This implies that J contributes to a new Tate cycle on A . However, thisis a contradiction, since by assumption all the Tate cycles are defined over K and that J / ∈ S . This proves that T has density 0 and that proves theresult. (cid:3) References [1] N. M. Katz and W. Messing, Some consequences of the Riemann hypothesis forvarieties over finite fields, Inventiones Math., (1974), 73–77.[2] J. Lagarias and A. Odlyzko, Effective versions of the Chebotarev Density Theorem,in: Algebraic Number Fields , pp. 409-464, ed. A. Fröhlich, Academic Press, NewYork, 1977.[3] S. Lang, Complex Multiplication, Springer-Verlag, 1983.[4] X. Li, The smallest prime that does not split completely in a number field,
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