## 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 $$k([A[q])$$.

### MSC:

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

### Keywords:

commutative algebraic group; rational point; divisibility

### Citations:

Zbl 0987.14016; Zbl 1035.14007
Full Text:

### References:

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