On a local-global principle for the divisibility of a rational point by a positive integer

DOI10.1112/blms/bdl002zbMath1115.14011OpenAlexW2052567850MaRDI QIDQ3429604

Umberto Zannier, Roberto Dvornicich

Publication date: 2 April 2007

Published in: Bulletin of the London Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1112/blms/bdl002

zbMATH Keywords

rational pointdivisibilitycommutative algebraic group

Mathematics Subject Classification ID

Rational points (14G05) Elliptic curves over global fields (11G05) Arithmetic ground fields for curves (14H25) Galois cohomology (11R34)

Related Items (17)

On the local-global divisibility over abelian varieties ⋮ Counterexamples to the local-global divisibility over elliptic curves ⋮ On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility ⋮ Elliptic curves with \(\mathbb Q({\mathcal E}[3)=\mathbb Q(\zeta_3)\) and counterexamples to local-global divisibility by 9] ⋮ On the minimal set for counterexamples to the local-global principle ⋮ The local-global principle for divisibility in CM elliptic curves ⋮ Local-global principles for Weil–Châtelet divisibility in positive characteristic ⋮ On 7-division fields of CM elliptic curves ⋮ Local–global divisibility on algebraic tori ⋮ On local-global divisibility over ${\rm GL}_2$-type varieties ⋮ On the local-global divisibility of torsion points on elliptic curves and \(\operatorname{GL}_{2}\)-type varieties ⋮ Number fields generated by the 3-torsion points of an elliptic curve ⋮ Selmer groups are intersection of two direct summands of the adelic cohomology ⋮ Cohomology of integer matrices and local-global divisibility on the torus ⋮ Divisibility questions in commutative algebraic groups ⋮ Local-global divisibility by 4 in elliptic curves defined over \(\mathbb Q\) ⋮ Local-global questions for divisibility in commutative algebraic groups

Cites Work

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• Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
• Some local-global applications of Kummer theory
• The support problem for abelian varieties.
• An analogue for elliptic curves of the Grunwald--Wang example.
• Local-global problem for Drinfeld modules.
• Galois properties of points of finite order of elliptic curves
• Local-global divisibility of rational points in some commutative algebraic groups
• Principal Homogeneous Spaces Over Abelian Varieties

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