Let A be an abelian variety defined over a number field k. Let P be a point
in A(k) and let X be a subgroup of A(k). Gajda in 2002 asked whether it is true
that the point P belongs to X if and only if the point (P mod p) belongs to (X
mod p) for all but finitely many primes p of k. We answer negatively to Gajda's
question.Comment: 3 page

Jossen, Peter

Perucca, Antonella

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 9–10Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comNumber TheoryA counterexample to the local–global principle of linear dependence forAbelian varietiesUn contre-exemple au principe de la dépendance linéaire des variétés abéliennesPeter Jossen a, Antonella Perucca ba NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germanyb Section des mathématiques, École polytechnique fédérale de Lausanne, EPFL station 8, Ch-1015 Lausanne, Switzerlanda r t i c l e i n f o a b s t r a c tArticle history:Received 4 July 2009Accepted after revision 23 November 2009Available online 23 December 2009Presented by Jean-Pierre SerreLet A be an Abelian variety defined over a number field k. Let P be a point in A(k) andlet X be a subgroup of A(k). Gajda and Kowalski asked in 2002 whether it is true that thepoint P belongs to X if and only if the point (P mod p) belongs to (X mod p) for all butfinitely many primes p of k. We provide a counterexample.© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoient k un corps de nombres, A une variété abélienne sur k, P un point de A(k) et X unsous-groupe de A(k). En 2002 Gajda et Kowalski ont demandé s’il est vrai que le point Pappartient à X si et seulement si le point (P mod p) appartient à (X mod p) pour presquetoute place finie p de k. Nous donnons une réponse négative à cette question.© 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Let A be an Abelian variety defined over a number field k. Let P be a point in A(k) and let X be a subgroup of A(k).Suppose that for all but finitely many primes p of k the point (P mod p) belongs to (X mod p). Is it true that P belongsto X? This question, which was formulated by Gajda and by Kowalski in 2002, was named the problem of detecting lineardependence. The problem was addressed in several papers [1–4,6,9–13] but the question was still open. In a recent note,[7], the first author stated that the answer to this problem is always affirmative, but this is wrong. In this note we presenta counterexample.A counterexample to the analogous statement for tori was given by Schinzel in [12]. We have recently been informedthat Banaszak and Krasoń found different counterexamples, which will appear in a new version of [3]. In his Ph.D. thesis,[8], the first author shows that for simple Abelian varieties the answer is positive.Let k be a number field and let E be an elliptic curve over k such that there are points P1, P2, P3 in E(k) which areZ-linearly independent. Define A := E3, and let P ∈ A(k) and X ⊆ A(k) be the following:P :=( P1P2P3)X := {M P ∈ A(k) ∣∣ M ∈ Mat(3,Z), tr M = 0}E-mail addresses: peter.jossen@gmail.com (P. Jossen), antonella.perucca@epfl.ch (A. Perucca).1631-073X/$ – see front matter © 2009 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2009.11.01910 P. Jossen, A. Perucca / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 9–10So the group X consists of the images of the point P via the subgroup of the endomorphisms of A consisting of the matriceswith integer coefficients and trace zero. Since the points Pi are Z-independent, the point P does not belong to X . Noticethat no non-zero multiple of P belongs to X .Claim. Let p be a prime of k where E has good reduction. The image of P under the reduction map modulo p belongs to the image of X .For the rest of this note, we fix a prime p of good reduction for E . We write κ for the residue field of k at p. To easenotation, we now let E denote the reduction of the given elliptic curve modulo p and write P1, P2, P3, P for the image ofthe given points under the reduction map modulo p.Our aim is to find an integer matrix M of trace zero such that P = M P in A(κ).For i = 1,2,3 call J i the subgroup of the integers defined as follows: n belongs to J i if and only if nPi is in the subgroupof E(κ) generated by the other two points. Call αi the positive generator of J i . There are integers mij such thatα1 P1 + m12 P2 + m13 P3 = Om21 P1 + α2 P2 + m23 P3 = Om31 P1 + m32 P2 + α3 P3 = OAssume that the greatest common divisor of α1, α2 and α3 is 1 (we prove this assumption later). We can thus find integersa1,a2,a3 such that3 = α1a1 + α2a2 + α3a3Write mii := 1 − αiai , so that in particular m11 + m22 + m33 = 0. Then we have( m11 −a1m12 −a1m13−a2m21 m22 −a2m23−a3m31 −a3m32 m33)( P1P2P3)=( P1P2P3)Notice that the above matrix has integer entries and trace zero. Hence we are left to prove that the greatest common divisorof α1, α2 and α3 is indeed 1, or in other words that the ideals J1, J2 and J3 generate Z.Fix a prime number  and let us show that  does not divide gcd(α1,α2,α3). Suppose on the contrary that  dividesgcd(α1,α2,α3). By definition of the ideals J i , this is equivalent to saying that  divides all the coefficients appearing in anylinear relation between P1, P2 and P3. In particular, this implies that  divides the order of P1, P2 and P3 in E(κ).Let Z denote the subgroup of E(κ) generated by P1, P2 and P3. It is well known that the group E(κ)[] is either trivial,isomorphic to Z/Z or isomorphic to (Z/Z)2. In any case, the intersection Z ∩ E(κ)[] is generated by two elements orless. Without loss of generality, let us suppose that the subgroup of Z generated by P2 and P3 contains Z ∩ E(κ)[].We are supposing that  divides all the coefficients appearing in any linear relation of the points Pi . Let α1 = x1 andwrite x1P1 + x2P2 + x3P3 = O for some integers x2 and x3. It follows thatx1 P1 + x2 P2 + x3 P3 = Tfor some point T in Z ∩ E(κ)[]. The point T is a linear combination of P2 and P3 hence x1 ∈ J1. Since α1 generates J1,we have a contradiction.In our counterexample, the only requirement for the elliptic curve E is that E(k) has rank at least 3. According to JohnCremona’s database [5], the elliptic curve given by the equationE: y2 + y = x3 − 7x + 6has rank 3 over Q. The three points P1 := (−2,3), P2 := (−1,3) and P3 := (0,2) on E are Z-linearly independent.References[1] S. Barańczuk, On a generalization of the support problem of Erdös and its analogues for abelian varieties and K -theory, Journal Pure Appl. Algebra 214(2010) 380–384.[2] G. Banaszak, On a Hasse principle for Mordell–Weil groups, C. R. Acad. Sci. Paris, Ser. I 347 (2009) 709–714.[3] G. Banaszak, P. Krasoń, On arithmetic in Mordell–Weil groups, arXiv:math/0904.2848, 2009.[4] G. Banaszak, W. Gajda, P. Krasoń, Detecting linear dependence by reduction maps, J. Number Theory 115 (2) (2005) 322–342.[5] J. Cremona, Elliptic curve data, http://www.warwick.ac.uk/staff/J.E.Cremona/, 2009.[6] W. Gajda, K. Górnisiewicz, Linear dependence in Mordell–Weil groups, J. Reine Angew. Math. 630 (2009) 219–233.[7] P. Jossen, Detecting linear dependence on a semiabelian variety, arXiv:math/0903.5271, 2009.[8] P. Jossen, On the arithmetic of 1-motives, Ph.D. thesis, Central European University Budapest, July 2009.[9] C. Khare, Compatible systems of mod p Galois representations and Hecke characters, Math. Res. Lett. 10 (2003) 71–83.[10] E. Kowalski, Some local–global applications of Kummer theory, Manuscripta Math. 111 (1) (2003) 105–139.[11] A. Perucca, On the problem of detecting linear dependence for products of abelian varieties and tori, Acta Arith., in press.[12] A. Schinzel, On power residues and exponential congruences, Acta Arith. 27 (1975) 397–420.[13] T. Weston, Kummer theory of abelian varieties and reduction of Mordell–Weil groups, Acta Arith. 110 (2003) 77–88.

A counterexample to the local–global principle of linear dependence for Abelian varieties

Peter Jossen

Antonella Perucca

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2009.11.019.pdf

A counterexample to the local-global principle of linear dependence for abelian varieties

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