Abelian varieties with isogenous reductions
aa r X i v : . [ m a t h . N T ] O c t ABELIAN VARIETIES WITH ISOGENOUS REDUCTIONS
CHANDRASHEKHAR B. KHARE AND MICHAEL LARSEN
Abstract.
Let A and A be abelian varieties over a number field K . We prove that ifthere exists a non-trivial morphism of abelian varieties between reductions of A and A ata sufficiently high percentage of primes, then there exists a non-trivial morphism A → A over ¯ K . Along the way, we give an upper bound for the number of components of a reductivesubgroup of GL n whose intersection with the union of Q -rational conjugacy classes of GL n is Zariski-dense. This can be regarded as a generalization of the Minkowski-Schur theoremon faithful representations of finite groups with rational characters. R´esum´e . Soient A et A deux vari´et´es ab´eliennes sur un corps de nombres K . Nous mon-trons que, s’il existe un morphisme non trivial de vari´et´es ab´eliennes entre r´eductions de A et A pour une proportion suffisamment grande d’id´eaux premiers, il existe un morphisme nontrivial A → A sur ¯ K . Nous donnons ´egalement une majoration du nombre du composantesd’un sous-groupe r´eductif de GL n dont l’intersection avec l’union des classes de conjugaison Q -rationnelles de GL n est dense pour la topologie de Zariski; c´est une g´en´eralisation d’unth´eor`eme de Minkowski-Schur sur les repr´esentations fid`eles des groupes finis a caract`ererationnel. In this note, we answer a recent question of Dipendra Prasad and Ravi Raghunathan[PR20, Remark 1]. We are grateful to Dipendra Prasad and Jean-Pierre Serre for helpfulcorrespondence. We would also like to thank the referee for several improvements andcorrections.Let K be a number field and A and A abelian varieties over K . If ℘ is a prime of K , wedenote by k ℘ the residue field of ℘ . If ℘ is a prime of good reduction for A i , we denote by A i℘ the reduction and by Frob ℘ the Frobenius element regarded as an automorphism, welldefined up to conjugacy, of the ℓ -adic Tate module of A i or, dually, of H ( ¯ A i , Z ℓ ). Theorem 1.
Let A and A be abelian varieties over a number field K . Suppose that fora density one set of primes ℘ of K , there exists a non-trivial morphism of abelian varietiesover ¯ k ℘ from A ℘ to A ℘ . Then there exists a non-trivial morphism of abelian varieties from A to A defined over ¯ K . Let G be a connected reductive algebraic group over an algebraically closed field F ofcharacteristic 0, and let V be a finite dimensional representation of G . Let T be a maximaltorus of G and W the Weyl group of G with respect to T . If V is irreducible, we say it is minuscule if W acts transitively on the weights of V with respect to T . The highest weightof V with respect to any choice of Weyl chamber has multiplicity 1, so every element of theWeyl orbit has multiplicity one.For general finite dimensional representations V , we say V is minuscule if each of itsirreducible factors is so. Regarding the character of a representation V as a function f V from W -orbits in X ∗ ( T ) to non-negative integers, when V is minuscule, for any dominant ML was partially supported by NSF grant DMS-2001349. weight λ , the multiplicity in V of the irreducible G -representation V λ with highest weight λ is the value of f V on the W -orbit containing λ . Proposition 2.
Let V and V be minuscule representations of G . If dim Hom T ( V , V ) > ,then dim Hom G ( V , V ) > .Proof. If dim Hom T ( V , V ) >
0, then V and V must have a common T -irreducible fac-tor, and that means they have a common weight χ with respect to T . If λ is the domi-nant weight in the orbit of χ , then V and V each contain V λ as a subrepresentation, sodim Hom G ( V , V ) > (cid:3) Now let A and A denote abelian varieties over a number field K with absolute Galoisgroup G K := Gal( ¯ K, K ). Let ℓ be a fixed rational prime, and let F = ¯ Q ℓ . Let V i = H ( ¯ A i , F ), regarded as G K -modules. Let V := V ⊕ V as G K -module and G the Zariskiclosure of G K in Aut F ( V ). By the semisimplicity of Galois representations defined byabelian varieties [Fa83], G is reductive. Let G denote the identity component G ◦ . Proposition 3.
There exists a positive density set of primes ℘ of K such that A × A hasgood reduction at ℘ , and Frob ℘ generates a Zariski dense subgroup of a maximal torus of G .Proof. The condition that Frob ℘ lies in the identity component G has density [ G : G ] − > X of G such that Frob ℘ ∈ G \ X implies that Frob ℘ generates a Zariski-densesubgroup of a maximal torus of G . However, by a second theorem of Serre [Se81, Th´eor`eme10], the set of ℘ such that Frob ℘ ∈ X has density 0. (cid:3) We can now prove the main theorem.
Proof.
A well-known theorem of Tate [Ta66] asserts that the existence of a non-trivial F q -morphism between abelian varieties over F q is equivalent to the existence of a Frob q -stablemorphism of their ℓ -adic Tate modules. By the easy direction of this result, the existence ofa non-trivial morphism defined over ¯ F q implies the existence of a Frob mq -stable morphism oftheir Tate modules for some positive integer m .By Proposition 3, the hypothesis of the theorem therefore implies thatdim Hom( V , V ) Frob m℘ > ℘ for which Frob ℘ generates a Zariski-dense subgroup of a maximal torus T of G and some positive integer m . As T is connected, Frob m℘ likewise generates a Zariski-dense subgroup of T . Thus dim Hom T ( V , V ) >
0. By a theorem of Pink [Pi98, Corol-lary 5.11], the G -representations V and V are minuscule. Thus Proposition 2 impliesthat dim Hom G ( V , V ) >
0. Finally, Faltings’ proof of Tate’s Conjecture [Fa83] impliesHom ¯ K ( A , A ) is non-zero. (cid:3) Remark 4.
One might ask whether there exists a non-trivial homomorphism A → A defined over K itself if for a density one set of ℘ there exists a non-trivial k ℘ -homomorphism A ℘ → A ℘ . D. Prasad pointed out the following counterexample to us. Let E be an ellipticcurve over Q which does not have complex multiplication. Let E n denote the quadratic twistof E by n ∈ Q × . Let A = E , A = E × E × E . For every rational prime p >
3, either2, 3, or 6 lies in F × p , so if E has good reduction at p , the same is true for both A and A ,and there exists an F p -isomorphism from ( A ) p to least one of ( E ) p , ( E ) p , and ( E ) p , and BELIAN VARIETIES WITH ISOGENOUS REDUCTIONS 3 therefore a non-trivial F p -homomorphism to ( A ) p . On the other hand, there is no Q -isogenyfrom A to any one of E , E , or E , and therefore no non-trivial Q -homomorphism to A .We can prove a stronger version of Theorem 1 in analogy with the theorem of C. S. Rajan[Ra98]. Theorem 5.
Let n be a positive integer. If A and A are abelian varieties of dimension ≤ n over a number field K and the set of primes ℘ of K for which there exists a non-trivial ¯ k ℘ -morphism of abelian varieties from A ℘ to A ℘ has upper density > − e − n n ! n , then thereexists a non-trivial ¯ K -morphism of abelian varieties from A to A . The only additional ingredient necessary to prove Theorem 5 is an upper bound, dependingonly on n , on the number of components of G . This is an immediate consequence of thefollowing theorem. Theorem 6.
Let n be a positive integer, F a field of characteristic , and G ⊂ GL n areductive F -subgroup. If the set of ¯ F -points of G consisting of matrices whose characteristicpolynomials lie in Q [ x ] is Zariski-dense, then | G/G ◦ | < e n n ! n . We remark that without the rationality assumption, this statement fails even for n = 1,where G could be an arbitrarily large cyclic group. Proof.
The locus of ¯ F -points of G whose characteristic polynomials lie in Q [ x ] is G F -stable,so the Zariski-closure does not change when the base field is changed from F to ¯ F . Thisjustifies assuming that F is algebraically closed.We can write G ◦ = DZ ◦ , where D and Z := Z ( G ◦ ) are the derived group and thecenter of G ◦ respectively. By [Sp79, Corollary 2.14], the outer automorphism group of D iscontained in the automorphism group of the Dynkin diagram ∆ of D . Every automorphismof ∆ preserves the set of isomorphic components. We claim that | Aut ∆ | ≤ n !. It sufficesto prove this when ∆ consists of m mutually isomorphic connected diagrams ∆ of rank r = n/m . The claim obviously holds when r = 1. It is easily verified for n ≤
4. For n ≥ | Aut(∆ ) | /r ≤ √ < n/
2, so if r ≥ | Aut(∆) | = | Aut(∆ ) | n/r ( n/r )! < ( n/ n/ ⌊ n/ ⌋ ! < n ! . Any automorphism of G ◦ is determined by its restrictions to the characteristic subgroups D and Z ◦ . An automorphism which is inner on D and trivial on Z ◦ is inner. Thus, thehomomorphism Aut( G ◦ ) → Aut( D ) × Aut( Z ◦ ) gives an injective homomorphismOut( G ◦ ) → Out( D ) × Aut( Z ◦ ) = Out( D ) × GL k ( Z ) , where k = dim Z ◦ ≤ n . By Minkowski’s theorem [Se16, Theorem 9.1], every finite subgroupof GL k ( Z ) has order at most M ( k ) := Y p p P i ≥ (cid:4) k ( p − pi (cid:5) . We have log M ( k ) ≤ k +1 X p =2 kp log p ( p − = k k X i =1 ( i + 1) log( i + 1) i ≤ k , since ( i + 1) log( i + 1) ≤ i for all i ≥
1. Thus, any finite subgroup of Out( G ◦ ) has order ≤ n ! e n . C. B. KHARE AND MICHAEL LARSEN
The conjugation action on G ◦ defines a homomorphism G/G ◦ → Out( G ◦ ). Let Γ denotethe kernel of this homomorphism and G the inverse image of Γ in G . Thus, the index of Γ in the component group G/G ◦ is ≤ n ! e n ≤ e n . Arguing by contradiction, we may assumethe order of Γ is at least e − n | G/G ◦ | ≥ e n n ! n . Let Γ := Z G ( G ◦ ) /Z ◦ , so Γ ∼ = Z G ( G ◦ ) /Z is a quotient group of Γ. Consider the shortexact sequence 0 → Z ◦ → Z G ( G ◦ ) → Γ → . The extension class α ∈ H (Γ , Z ◦ ) is annihilated by N := | Γ | . As Z ◦ ∼ = ( F × ) k is a divisiblegroup, it follows that the extension class α lies in the image of H (Γ , Z ◦ [ N ]), where Z ◦ [ N ]denotes the kernel of the N th power map on Z ◦ . We can therefore represent α by a 2-cocyclewith values in Z ◦ [ N ]. This means that there exists a set-theoretic section i : Γ → Z G ( G ◦ )such that the associated 2-cocycle takes values in Z ◦ [ N ], and it follows that ˜Γ := Z ◦ [ N ] i (Γ)is a finite subgroup of Z G ( G ◦ ) ⊂ G which maps onto Γ and therefore onto Γ .By Jordan’s theorem, ˜Γ contains an abelian normal subgroup ˜ A of index ≤ J ( n ), aconstant depending only on n . The optimal Jordan constant has been computed by MichaelCollins [Co07], and for all n , we have J ( n ) ≤ e n . Indeed, for n ≥
71, the bound, ( n + 1)!,is given by Theorem A, and( n + 1)! < ( n + 1) n < ( n ) n < (( e n ) ) n = e n . For 20 ≤ n ≤
70 and n ≤
19, the bounds are given by Theorems B and D respectively, andthey can be checked by machine to be less than e n in every case.Let T be a maximal torus of G ◦ , so ˜ A T is a commutative subgroup of G . As˜ A ∩ T ⊂ ˜ A ∩ G ◦ = ker ˜ A → Γ , we have | ˜ A T /T | = | ˜ A / ( ˜ A ∩ T ) | ≥ | Im ˜ A → Γ | ≥ | Γ | e n ≥ e n n ! n . Therefore, if M := e n n ! , then ˜ A T has at least M n components. Since ˜ A T /T is a quotientgroup of ˜ A ⊂ GL n ( F ), it contains no elementary p -group of rank > n , so it must have anelement of order ≥ M . Let g ∈ ˜ A map to such an element.By hypothesis, there exists t ∈ G ◦ × { g } such that the characteristic polynomial of gt has coefficients in Q . We can further assume that t is semisimple, so we can choose ourmaximal torus T to contain t . Let T ′ = h g i T . Every element of T ′ is the product of twocommuting elements, one which is of finite order, and one which belongs to T , so both aresemisimple, from which it follows that their product is semisimple. Thus T ′ is diagonalizable,so it is a closed subgroup of a maximal torus of GL n [Bo91, Proposition 8.4]. Without lossof generality, we may assume this maximal torus is the group GL n of invertible diagonalmatrices.The contravariant functor taking an algebraic group to its character group gives an equiv-alence of categories between diagonalizable groups and finitely generated abelian groups[Bo91, Proposition 8.12]. In particular, there is a bijective correspondence between sub-groups Λ ⊂ Z n and closed subgroups D Λ of the group GL n of diagonal matrices in GL n ,where D Λ = { ( x , . . . , x n ) ∈ GL n | λ ( x , . . . , x n ) = 1 ∀ λ ∈ Λ } . BELIAN VARIETIES WITH ISOGENOUS REDUCTIONS 5
Let Λ be the subgroup of Z n such that D Λ = T and Λ ′ the subgroup such that D Λ ′ = T ′ .The inclusion T ֒ → T ′ corresponds to the surjection Z n / Λ ′ → Z n / Λ and thus to the inclusionΛ ′ ⊂ Λ. As T ′ /T is cyclic, Λ / Λ ′ is cyclic of the same order k . Let λ ∈ Λ map to a generatorof Λ / Λ ′ . Then the smallest integer m such that λ (( gt ) m ) = 1 is the smallest such that λ ( g m ) = 1, which is k .Writing gt = ( x , . . . , x n ) ∈ GL ( F ) n ⊂ GL n ( F ), the x i are the eigenvalues of gt , so theyall lie in some Galois extension of Q of degree ≤ n !. Therefore λ ( gt ) lies in this extension.Since it is a primitive k th root of unity, this implies φ ( k ) ≤ n !. Now φ ( q ) ≥ √ q for all primepowers q except 2, and it follows from the multiplicativity of φ that φ ( k ) ≥ p k/ k ≥
1, so M ≤ k ≤ n ! , which is a contradiction. (cid:3) References [Bo91] Borel, Armand: Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126.Springer-Verlag, New York, 1991.[Co07] Collins, Michael J.: On Jordan’s theorem for complex linear groups.
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Invent. Math. (1966), 134–144. UCLA Department of Mathematics, Box 951555, Los Angeles, CA 90095, USA
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