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

Dvornicich, Roberto; Zannier, Umberto

Bull. Lond. Math. Soc. 39, No. 1, 27-34 (2007).

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 \(k([A[q])\).

Reviewer: Boris Kunyavskii (Ramat Gan)

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

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