On a local-global principle for the divisibility of a rational point by a positive integer. (English) Zbl 1115.14011

The main object of the paper under review is a commutative algebraic group \(A\) defined over a number field \(k\). The authors are interested in the following local-global principle: if \(q\) is a given positive integer and \(P\in A(k)\) is such that, for almost all places \(v\) of \(k\) one has \(P\in qA(k_v)\), can one conclude that \(P\in qA(k)\)? This question was addressed in their earlier papers [Bull. Soc. Math. France 129, 317–338 (2001; Zbl 0987.14016), C. R. Acad. Sci. Paris, Sér. I 338, 47–50 (2004; Zbl 1035.14007)], where a cohomological obstruction to this principle was discovered. It is the kernel of the restriction map \(H^1(G, A[q])\to \prod H^1(C, A[q])\), where \(G= \text{Gal}(k([A[q])/k)\) and the product is taken over all cyclic subgroups \(C\) of \(G\).
In the paper under review, the authors provide more affirmative cases of the local-global principle following from vanishing of the above mentioned obstruction. Namely, Theorem 1 states that this is the case if \(A\) is an elliptic curve over \(\mathbb Q\) and \(q=p^n\) with \(p\) not belonging to \(S=\{2,3,5,7,11,13,17,19,37,43,67,163\}\). Theorem 2 states that the same is true for any elliptic curve \(E\) defined over any number field \(k\) provided \(p>p_0(E,k)\). It is shown that \(p_0\) must depend on \(k\); the question whether \(p_0\) must depend on \(E\) remains open. Theorem 3 says that, in a sense, vanishing of the cohomological obstruction is a necessary condition for the local-global principle to hold. More precisely, if it does not vanish, one can produce a counter-example for the group \(A\times _kL\) where \(L\) is some finite extension of \(k\) linearly disjoint from \(k([A[q])\).


14G05 Rational points
14H25 Arithmetic ground fields for curves
11G05 Elliptic curves over global fields
11R34 Galois cohomology
Full Text: DOI


[1] Cassels, Prolegomena to a middlebrow arithmetic of curves of genus 2 230 (1996) · Zbl 0857.14018
[2] Dvornicich, Local-global divisibility of rational points in some commutative algebraic groups, Bull. Soc. Math. France 129 pp 317– (2001) · Zbl 0987.14016
[3] Dvornicich, An analogue for elliptic curves of the Grunwald-Wang example, C. R. Acad. Sci. Paris, Ser. I 338 pp 47– (2004) · Zbl 1035.14007
[4] Fried, Field arithmetic 11 ((3)) (1986)
[5] van der Heiden, Local-global problem for Drinfeld modules, J. Number Theory 104 pp 193– (2004) · Zbl 1093.11040
[6] Kowalski, Some local-global applications of Kummer theory, Manuscripta Math. 111 pp 105– (2003) · Zbl 1089.11031
[7] Lang, Elliptic curves: diophantine analysis 231 (1978)
[8] Lang, Number theory III 60 (1991)
[9] Lang, Algebraic number theory 110 (1994)
[10] Lang, Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80 pp 659– (1958) · Zbl 0097.36203
[11] Larsen, The support problem for abelian varieties, J. Number Theory 101 pp 398– (2003) · Zbl 1039.11040
[12] Mazur, ’Rational isogenies of prime degree’ (with an appendix by D. Goldfeld), Invent. Math. 44 pp 129– (1978) · Zbl 0386.14009
[13] Merel, Bornes sur la torsion des courbes elliptiques sur les corps de nombres, in: Elliptic curves, modular forms, and Fermat’s last theorem pp 110– (1995)
[14] Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 pp 259– (1972) · Zbl 0235.14012
[15] Serre, Graduate Texts in Mathematics 67, in: Local fields (1976)
[16] Serre, Topics in Galois theory (1992)
[17] Silverman, The arithmetic of elliptic curves 106 (1986) · Zbl 0585.14026
[18] Wong, Power residues on Abelian varieties 102, in: Manuscripta Math. pp 129– (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.