On the rank of an elliptic curve in elementary 2-extensions
The author studies the behaviour of the rank of an elliptic curve \(E\) over a number field \(k\) under transition from \(k\) to an elementary abelian 2-extension \(K\) of \(k\). By adjoining to \(k\) the square roots of a suitable set \(\Sigma\) of integers, the rank of \(E\) over a tower of elementary abelian 2-extensions can be shown to tend to infinity. As an application of the corresponding quantative result to the family of elliptic curves \(E_ n:\) \(y^ 2 = x^ 3 - n^ 2x\) \((n \in \mathbb{N}\) squarefree) arising from the congruent number problem, some prime numbers \(n \in \mathbb{N}\) and sets \(\Sigma\) are specified for which the rank of \(E_ n\) becomes arbitrarily large as one adjoins to \(k = \mathbb{Q}\) a sufficient number of square roots of numbers from \(\Sigma\). As another application it is shown that, for prime numbers \(p\), \(q\), and \(r \equiv 3 \pmod 8\), the curves \(E_ r\) of rank 0 over \(\mathbb{Q}\) have rank 0 also over \(K = \mathbb{Q} (\sqrt p, \sqrt q)\). \{Reviewer's remark: In her diploma thesis ``Rangsprünge elliptischer Kurven mit nichttrivialem 2-Teilungspunkt beim Übergang vom rationalen Zahlkörper \(\mathbb{Q}\) zu multiquadratischen Zahlkörpern'', Saarbrücken 1994, \textit{Margit Sens} considered similar questions. She constructed, among other things, an elliptic curve of rank \(\geq 9\) over \(\mathbb{Q}\), whose rank becomes \(\geq 28\) over the composite of 19 quadratic fields\}.
tower of elementary abelian 2-extensions
scientific article; zbMATH DE number 597335