Local-global divisibility by 4 in elliptic curves defined over \(\mathbb Q\)
Let \({\mathcal A}\) be a commutative algebraic group defined over a number field \(k\). In a series of papers [Bull. Soc. Math. Fr. 129, No. 3, 317--338 (2001; Zbl 0987.14016), C. R., Math., Acad. Sci. Paris 338, No. 1, 47--50 (2004; Zbl 1035.14007), Bull. Lond. Math. Soc. 39, No. 1, 27--34 (2007; Zbl 1115.14011)], the reviewer and \textit{U. Zannier} have considered the following problem: Let \(r\) be a positive integer and let \(P\in {\mathcal A}(k)\). Suppose that for all but a finite number of primes \(v\) of \(k\), we have \(P=rD_v\) for some \(D_v\in {\mathcal A}(k_v)\). Can one conclude that there exists \(D\in {\mathcal A}(k)\) such that \(P=rD\)? One can clearly reduce the question to the case when \(r\) is a prime power. In the case when \({\mathcal A}={\mathcal E}\) is an elliptic curve, \(k={\mathbb Q}\) and \(r=4\), it is known that the Galois group of the field generated by the 4-torsion points of \({\mathcal E}\) over \({\mathbb Q}\) is of type \(({\mathbb Z}/2{\mathbb Z})^n\) with \(1\leq n\leq 4\). R. Dvornicich and U. Zannier have shown that the answer to the problem is affirmative for the cases when \(n=1,4\), but also that the answer can be negative, exhibiting a counterexample for the case \(n=2\). In the present paper the picture is completed by producing a counterexample also for the case \(n=3\).
scientific article; zbMATH DE number 5655731
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