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The 2-primary torsion on elliptic curves in the \(\mathbb Z_p\)-extensions of \(\mathbb Q\). (English) Zbl 1132.11029

Let \(E\) be an elliptic curve defined over \(\mathbb Q\). In 1981, K. Ribet has proved that if \(K\) is an abelian extension of \(\mathbb Q\) of infinite degree, the torsion subgroup \(E(K)_{tors}\) of the group \(E(K)\) of \(K\)-rational points on \(E\) is finite. Some results concerning the type of \(E(K)_{tors}\) have then been obtained if \(K\) is the maximal elementary abelian \(2\)-extension of \(\mathbb Q\). For instance, the author has recently proved that, in this case, there are exactly twenty possibilities for the type of \(E(K)_{tors}\).
Let \(p\) be a prime number and \(\mathbb Q_{p,\infty}\) be the \(\mathbb Z_p\)-extension of \(\mathbb Q\). In this paper, the author examines the \(2\)-primary part \(E(\mathbb Q_{p,\infty})_{(2)}\) of the group \(E(\mathbb Q_{p,\infty})\). He proves that if \(p\neq 3\), then \(E(\mathbb Q_{p,\infty})_{(2)}\) is isomorphic to a subgroup of \(\mathbb Z/2\mathbb Z\times \mathbb Z/8\mathbb Z\). As the author mentions, in case \(p\neq 3\) and \(E(\mathbb Q)\) does not have any point of order two, the group \(E(\mathbb Q_{p,\infty})_{(2)}\) is trivial; this is an immediate consequence of the fact that \(\mathbb Q_{p,\infty}\) contains no cubic fields. He also obtains a complete classification of the group \(E(\mathbb Q_{2,\infty})_{(2)}\) in terms of \(E(\mathbb Q)_{(2)}\). Furthermore, in case \(p\geq 5\), or if \(p=3\) and \(E(\mathbb Q)_{(2)}\neq \big\{ O\big\}\), he shows that \(E(\mathbb Q_{p,\infty})_{(2)} =E(\mathbb Q)_{(2)}\). Concerning this latter result, he in fact proves the more general following statement: let \(L\) be an extension of \(\mathbb Q\) which contains no quadratic fields. Let us suppose that either \(L\) contains no cubic fields, or that \(E(\mathbb Q)_{(2)}\neq \big\{ O\big\}\). Then, one has \(E(L)_{(2)}=E(\mathbb Q)_{(2)}\).

MSC:

11G05 Elliptic curves over global fields
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References:

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