In this paper we establish a Hasse principle concerning the linear dependence
over $\Z$ of nontorsion points in the Mordell-Weil group of an abelian variety
over a number field.Comment: First draft written on October 29, 2007. Submitted for publicatio

Banaszak, Grzegorz

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arXiv

Grzegorz Banaszak

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–714Number TheoryOn a Hasse principle for Mordell–Weil groupsGrzegorz BanaszakDepartment of Mathematics, Adam Mickiewicz University, 61614 Poznań, PolandReceived 7 April 2008; accepted after revision 17 March 2009Available online 7 May 2009Presented by Christophe SouléAbstractIn this Note we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the Mordell–Weilgroup of an abelian variety over a number field. To cite this article: G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.RésuméUn principe de Hasse pour les groupes de Mordell–Weil. Dans cette Note, on démontre un principe de Hasse concernant ladépendance linéaire sur Z des points d’ordre infini dans le groupe de Mordell–Weil d’une variété abélienne définie sur un corps denombres. Pour citer cet article : G. Banaszak, C. R. Acad. Sci. Paris, Ser. I 347 (2009).© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeSoit A une variété abélienne définie sur un corps de nombres F . Soient v un idéal premier de OF et kv := OF /v.Soit Av la réduction de A pour un idéal premier v de bonne réduction. Soit,rv :A(F) → Av(kv),le morphisme de réduction. On pose R := EndF (A). Soit Λ une sous-groupe de A(F) et soit P ∈ A(F). Une questionnaturelle est : La condition rv(P ) ∈ rv(Λ), pour presque tout idéal premier v de OF , implique-t-elle P ∈ Λ ? Cettequestion a eté posée par W. Gajda en 2002. Le résultat fondamental de cette Note est le théorème suivant :Théorème 0.1. Soient P1, . . . ,Pr des éléments de A(F) linéairement indépendants sur l’anneau R. Soit P un pointde A(F) tel que RP soit un R-module libre. Les conditions suivantes sont équivalentes :(1) P ∈ ∑ri=1 ZPi ;(2) rv(P ) ∈ ∑ri=1 Zrv(Pi) pour presque tout idéal premier v de OF .E-mail address: banaszak@amu.edu.pl.1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.03.014710 G. Banaszak / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–7141. IntroductionLet A be an abelian variety over a number field F . Let v be a prime of OF and let kv := OF /v. Let Av denote thereduction of A for a prime v of good reduction and let,rv :A(F) → Av(kv),be the reduction map. Put R := EndF (A). Let Λ be a subgroup of A(F) and let P ∈ A(F). A natural question iswhether the condition rv(P ) ∈ rv(Λ) for almost all primes v of OF implies that P ∈ Λ. This question was posed byW. Gajda in 2002. The main result of this Note is the following theorem:Theorem 1.1. Let P1, . . . ,Pr be elements of A(F) linearly independent over R. Let P be a point of A(F) such thatRP is a free R module. The following conditions are equivalent:(1) P ∈ ∑ri=1 ZPi ;(2) rv(P ) ∈ ∑ri=1 Zrv(Pi) for almost all primes v of OF .In the case of the multiplicative group F× the problem analogous to W. Gajda’s question has already been solvedby 1975. Namely, A. Schinzel, [16, Theorem 2, p. 398], proved that for any γ1, . . . , γr ∈ F× and β ∈ F× such thatβ = ∏ri=1 γ nv,ii mod v with nv,1, . . . , nv,r ∈ Z for almost all primes v of OF there are n1, . . . , nr ∈ Z such thatβ = ∏ri=1 γ nii . The theorem of A. Schinzel was proved again by Ch. Khare [11] using methods of C. Corralez-Rodrigáñez and R. Schoof [6]. Ch. Khare applied this theorem to prove that every family of one-dimensional strictlycompatible l-adic representations comes from a Hecke character.Theorem 1.1 strengthens the results of [2,8,19]. Namely T. Weston [19] obtained a result analogous to Theorem 1.1with coefficients in Z for R commutative. T. Weston did not assume that P1, . . . ,Pr are linearly independent over R,however there was some torsion ambiguity in the statement of his result. In [2], together with W. Gajda and P. Kra-soń, we proved Theorem 1.1 for elliptic curves without CM and more generally for a class of abelian varieties withEndF (A) = Z. We also got a general result for all abelian varieties [2, Theorem 2.9], in the direction of Theorem 1.1.However in Theorem 2.9 [2], the coefficients associated with points P1, . . . ,Pr are in R and the coefficient associatedwith the point P is in the set of positive integers N. W. Gajda and K. Górnisiewicz [8, Theorem 5.1], strengthenedTheorem 2.9 of [2] by implementing some techniques of M. Larsen and R. Schoof [12]. They proved that the coef-ficient associated with the point P is equal to 1. Nevertheless the coefficients associated with points P1, . . . ,Pr in[8, Theorem 5.1], are still in R. Recently A. Perucca [13, Corollary 5], has proven Theorem 5.1 of [8] using her l-adicsupport problem result. At the end of this paper we reprove Theorem 5.1 of [8] by arguments presented in the proofof Theorem 1.1.Although not explicitly presented in our proofs, this paper essentially applies results on Kummer Theory for abelianvarieties, originally developed by K. Ribet [15], and results of F. Bogomolov [5], G. Faltings [7], J.-P. Serre andJ. Tate [17], A. Weil [18], J. Zarhin [20] and other important results about abelian varieties. The application of theseresults comes by referring to [1,2,4,14] where Kummer Theory and the results of [5,7,17,18,20] are key ingredients.2. Proof of Theorem 1.1Let L/F be an extension of number fields. Let Sl be the following set of primes w in OL.Sl := {w: w|l} ∪ {w: w|v for a prime v of bad reduction for A/F }.Let Gl denote the l-torsion part of an abelian group G. The reduction map,rw :A(L)l → Aw(kw)l,is injective for every w /∈ Sl [10] pp. 501–502, [9] Theorem C.1.4 p. 263.The following lemma is a result of S. Barańczuk which is a refinement of Theorem 3.1 of [2] and Proposition 2.2of [3]. This is also a result of R. Pink [14, Corollary 4.3]. Recall [13, Proposition 2.2], that a nontorsion point Q ∈A(F) is independent over R if and only if the subgroup ZQ is Zariski dense in A.G. Banaszak / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–714 711Lemma 2.1. ([4], Th. 5.1, [14], Cor. 4.3.) Let l be a prime number. Let m1, . . . ,ms ∈ N ∪ {0}. Let L/F be a finiteextension and let Q1, . . . ,Qs ∈ A(L) be independent over R. There is a family of primes w of OL of positive densitysuch that rw(Qi) has order lmi in Aw(kw)l for all 1  i  s.The following corollary follows also from [14], Theorem 4.1:Corollary 2.2. Let m ∈ N. Let Q1, . . . ,Qs ∈ A(F) be independent over R and let T1, . . . , Ts ∈ A[lm]. Let L :=F(A[lm]). There is a family of primes w of OL of positive density such that for the prime v of OF below w:(1) rw(T1), . . . , rw(Ts) ∈ Av(kv) ⊂ Aw(kw),(2) rw(Ti) = rv(Qi) in Av(kv)l for all 1  i  s.Proof. Observe that the points Q1 − T1, . . . ,Qs − Ts are linearly independent over R in A(L). Hence it followsby Lemma 2.1 that there is a family of primes w of OL of positive density such that rw(Qi − Ti) = 0 in Aw(kw)l .Since Q1, . . . ,Qs ∈ A(F), it follows that rw(Qi − Ti) = rw(Qi) − rw(Ti) = rv(Qi) − rw(Ti) for the prime v of OFbelow w. Hence we get rw(Ti) = rv(Qi) ∈ Av(kv)l for all 1  i  s. Proof of Theorem 1.1. It is enough to prove that (2) implies (1). By Theorem 2.9 [2] there is an a ∈ N and elementsα1, . . . , αr ∈ R such thataP =r∑i=1αiPi. (1)Step 1. Assume that αi ∈ Z for all 1  i  r . We will show (cf. the proof of Theorem 3.12 of [2]) that P ∈ ∑ri=1 ZPi .Let lk be the largest power of l that divides a. Lemma 2.1 shows that for any 1  i  r there are infinitely many primesv such that rv(P1) = · · · = rv(Pi−1) = rv(Pi+1) = · · · = rv(Pr) = 0 and rv(Pi) has order equal to lk in Av(kv)l . By (1)we obtain arv(P ) = αirv(Pi). Moreover by assumption (2) of the theorem, rv(P ) = βirv(Pi) for some βi ∈ Z. Hence(αi−aβi)rv(Pi) = 0in Av(kv)l . This implies that lk divides αi for all 1  i  r . So by (1) we obtain:alkP =r∑i=1αilkPi + T , (2)for some T ∈ A(F)[lk]. Again, by Lemma 2.1 there are infinitely many primes v in OF such that rv(Pi) = 0 in Av(kv)lfor all 1  i  r . In addition rv(P ) ∈ ∑ri=1 Zrv(Pi) for almost all v. So (2) implies that rv(T ) = 0, for infinitely manyprimes v. This contradicts the injectivity of rv , unless T = 0. Hence,alkP =r∑i=1αilkPi . (3)Repeating the above argument for primes dividing alkshows that condition (1) holds.Step 2. Fix an embedding of F into C. Assume αi /∈ Z for some i. Observe that αi is an endomorphism of the Riemannlattice L, such that A(C) ∼= Cg/L. To make the notation simple, we will denote again by αi the endomorphism αi ⊗ 1acting on Tl(A) ∼= L ⊗ Zl . Let P(t) := det(t IdL −αi) ∈ Z[t], be the characteristic polynomial of αi acting on L. LetK be the splitting field of P(t) over Q. We take l such that it splits in K and l does not divide primes of bad reduction.Since P(t) has all roots in OK and is also the characteristic polynomial of αi on Tl(A), we see that P(t) has allroots in Zl by the assumption on l. If P(t) has at least two different roots in OK , we easily find a vector u ∈ Tl(A)which is not an eigenvector of αi on Tl(A). If P(t) has a single root λ ∈ OK then P(t) = (t − λ)2g and we musthave λ ∈ Z because we are in characteristic 0. Hence P(t) = (t − λ)2g is the characteristic polynomial of αi as anendomorphism of L. Since αi /∈ Z we find easily u ∈ L such that u is not an eigenvector of αi acting on Tl(A). Inany case there is u ∈ Tl(A) which is not an eigenvector of αi acting on Tl(A). Rescaling if necessary, we can assumethat u is not divisible by l in Tl(A). Hence for m ∈ N and m big enough we can see that the coset u + lmTl(A) is not712 G. Banaszak / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–714an eigenvector of αi acting on Tl(A)/lmTl(A). Indeed, if αiu ≡ cmu mod lmTl(A) for cm ∈ Z/lm for each m ∈ N,then cm+1u ≡ cmu mod lmTl(A). Because u is not divisible by l in Tl(A), this implies that cm+1 ≡ cm mod lm foreach m ∈ N. But this contradicts the fact that u is not an eigenvector of αi acting on Tl(A). Consider the naturalisomorphism of Galois and R modules Tl(A)/lmTl(A) ∼= A[lm]. We put T ∈ A[lm] to be the image of the cosetu+ lmTl(A) via this isomorphism. Put L := F(A[lm]). By Corollary 2.2 we choose a prime v below a prime w of OLsuch that(i) rw(T ) ∈ Av(kv)l ,(ii) rv(Pj ) = 0 for all j 	= i and rv(Pi) = rw(T ) in Av(kv)l .From (1) and (ii) we get arv(P ) = αirv(Pi) = αirw(T ) in Av(kv)l . Hence for the prime w in OL over v we get inAw(kw)l the following equality:arw(P ) = αirw(Pi) = αirw(T ). (4)By assumption (2) and (ii) there is d ∈ Z, such that arv(P ) = adrv(Pi) = adrw(T ) in Av(kv)l . Hence, for the primew in OL over v, we get in Aw(kw)l the following equality:arw(P ) = adrw(Pi) = adrw(T ). (5)Since rw is injective, the equalities (4) and (5) give:αiT = adT in A[lm].But this contradicts the fact that T is not an eigenvector of αi acting on A[lm]. It proves that αi ∈ Z for all 1  i  r ,but this case has already been taken care of in step 1 of our proof. Corollary 2.3. Let A be a simple abelian variety. Let P1, . . . ,Pr be elements of A(F) linearly independent over R.Let P be a nontorsion point of A(F). The following conditions are equivalent:(1) P ∈ ∑ri=1 ZPi ,(2) rv(P ) ∈ ∑ri=1 Zrv(Pi) for almost all primes v of OF .Proof. This is an immediate consequence of Theorem 1.1. Indeed, for a nontorsion point P the R-module RP is afree R-module since D = R ⊗Z Q is a division algebra because A is simple. Corollary 2.4. Let A be a simple abelian variety. Let P and Q be nontorsion elements of A(F). The followingconditions are equivalent:(1) P = mQ for some m ∈ Z,(2) rv(P ) = mvrv(Q) for some mv ∈ Z for almost all primes v of OF .Proof. This is an immediate consequence of Corollary 2.6 because RQ is a free R-module since A is simple. The following proposition is Theorem 5.1 of [8]. We give a new proof of this theorem using arguments presentedin the proof of Theorem 1.1.Proposition 2.5. Let A be an abelian variety over F . Let P1, . . . ,Pr be elements of A(F) linearly independent over R.Let P be a point of A(F) such that RP is a free R module. The following conditions are equivalent:(1) P ∈ ∑ri=1 RPi ;(2) rv(P ) ∈ ∑ri=1 Rrv(Pi) for almost all primes v of OF .Proof. Again we need to prove that (2) implies (1). Let us assume (2). By [2], Theorem 2.9 there is an a ∈ N andelements α1, . . . , αr ∈ R such that equality (1) holds. Let l be a prime number such that lk||a for some k > 0. PutG. Banaszak / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–714 713L := F(A[lk]) and take arbitrary T ∈ A[lk]. By Corollary 2.2 we can choose a prime v below a prime w of OL suchthat(i) rw(T ) ∈ Av(kv)l ,(ii) rv(Pj ) = 0 for all j 	= i and rv(Pi) = rw(T ) in Av(kv)l .From (1) and (ii) we get arv(P ) = αirv(Pi) = αirw(T ) in Av(kv)l . Hence we have the following equality in Aw(kw)l :arw(P ) = αirw(Pi) = αirw(T ). (6)By assumption (2) and (ii) there is δ ∈ R, such that arv(P ) = aδrv(Pi) = aδrw(T ) = 0 in Av(kv)l . Hence we havethe following equality in Aw(kw)l :arw(P ) = aδrw(Pi) = aδrw(T ) = 0. (7)By injectivity of rw , the equalities (6) and (7) imply:αiT = 0 in A[lk].This shows that αi maps to zero in EndGF (A[lk]). It is easy to observe that the natural map,R/lk R → EndGF(A[lk]),is an embedding for every prime number l and every k ∈ N. Recall [20, Corollary 5.4.5], that this map is an iso-morphism for l  0 and all k ∈ N cf. the proof of Lemma 2.2 of [2]. It follows that αi ∈ lk R for all 1  i  r .SoalkP =r∑i=1βiPi + T ′, (8)where T ′ ∈ A(F)[lk] and βi ∈ R for all 1  i  r . By Lemma 2.1 there are infinitely many primes v in OF suchthat rv(Pi) = 0 in Av(kv)l for all 1  i  r . In addition rv(P ) ∈ ∑ri=1 Rrv(Pi) for almost all v. So (8) implies thatrv(T′) = 0, for infinitely many primes v. Hence T ′ = 0 by the injectivity of rv [10] pp. 501–502, [9] Theorem C.1.4p. 263. HencealkP =r∑i=1βiPi. (9)Repeating the above argument for primes dividing alkfinishes the proof of the proposition. 3. Remark on Mordell–Weil R systemsLet R be a ring with identity. In the paper [1] the Mordell–Weil R systems have been defined. In [2] we inves-tigated Mordell–Weil R systems satisfying certain natural axioms A1 − A3 and B1 − B4. We also assumed that Rwas a free Z-module. Let us consider Mordell–Weil R systems which are associated to families of l-adic representa-tions ρl :GF → GL(Tl) such that ρl(GF ) contains an open subgroup of homotheties. Since Theorem 2.9 of [2] andTheorem 5.1 of [4] were proven for Mordell–Weil R systems, then Proposition 2.8 and its proof generalize for theMordell–Weil R systems. This shows that Theorem 2.9 of [2], which is stated for Mordell–Weil R systems, holdswith a = 1. Let us also assume that there is a free Z-module L such that R ⊂ EndZ(L) and for each l there is anisomorphism L ⊗ Zl ∼= Tl such that the action of R on Tl comes from its action on L. Abelian varieties are principalexamples of Mordell–Weil R systems satisfying all the requirements stated above with R = EndF (A). Then Theo-rem 1.1 generalizes also for Mordell–Weil R systems satisfying the above assumptions because we can apply againTheorem 2.9 of [2] and Theorem 5.1 of [4].AcknowledgementsThe author would like to thank the referees for valuable comments and suggestions. The research was partiallyfinanced by the research grant N N201 1739 33 of the Polish Ministry of Science and Education and the grant MRTN-CT-2003-504917 of the Marie Curie Research Training Network “Arithmetic Algebraic Geometry”.714 G. Banaszak / C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–714References[1] G. Banaszak, W. Gajda, P. Krasoń, Support problem for the intermediate Jacobians of l-adic representations, J. Number Theory 100 (1) (2003)133–168.[2] G. Banaszak, W. Gajda, P. Krasoń, Detecting linear dependence by reduction maps, J. Number Theory 115 (2) (2005) 322–342.[3] G. Banaszak, W. Gajda, P. 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On a Hasse principle for Mordell–Weil groups

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On a Hasse principle for Mordell-Weil groups

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