On local-global divisibility by \(p^n\) in elliptic curves (Q2903280)

This article continues a research on the problem of the local-global divisibility of a point \(P\in{\mathcal A}(k)\) by a certain \(m\in{\mathbb N}\), where \({\mathcal A}\) is a commutative algebraic group defined over a number field \(k\). It is elementary to reduce this study to the case when \(m\) is any power of a fixed prime \(p\), so the paper considers just this case. We shall say that the \(p\)-local-global principle holds if it holds for any power of \(p\).NEWLINENEWLINEIn the case of elliptic curves, the reviewer and \textit{U. Zannier} [``On a local-global principle for the divisibility of a rational point by a positive integer'', Bull. Lond. Math. Soc. 39, No. 1, 27--34 (2007; Zbl 1115.14011)] gave a criterion for the validity of this local-global principle: if an elliptic curve \({\mathcal E}\) does not admit a \(k\)-isogeny of order \(p\), then the \(p\)-local-global principle holds. Theorems of Serre and Mazur allow to restrict the primes \(p\) for which the principle may not hold to a finite set, which is independent of \({\mathcal E}\) if \(k={\mathbb Q}\).NEWLINENEWLINESo the question arises whether this set can be made independent of \({\mathcal E}\) (the dependence on \(k\) being unavoidable) for a general number field \(k\). The main theorem of the paper states that if \(k\) does not contain the field \({\mathbb Q}(\zeta_p+\bar{\zeta_p})\) and the elliptic curve \({\mathcal E}\) does not admit a torsion point of order \(p\), then the \(p\)-local-global principle holds.NEWLINENEWLINECombining this with some results of Merel, the authors also give explicit constants \(C(d)\), where \(d=[k:{\mathbb Q}]\) and \(1\leq d\leq 5\), such that for all primes \(p>C(d)\) the \(p\)-local-global principle holds.