Quadratic twists of elliptic curves with 3-Selmer rank 1. (English) Zbl 1297.14037
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) and, for any nonzero squarefree integer \(d\), let \(E^{(d)}\) be the associated quadratic twist. A (weak) form of a conjecture of D. Goldfeld [in: Number theory, Proc. Conf., Carbondale 1979, Lect. Notes Math. 751, 108–118 (1979; Zbl 0417.14031)] states that for a fixed \(E\) a positive proportion of the \(E^{(d)}\) have rank 1 over \(\mathbb{Q}\). The paper under review provides conditions on \(E\) and \(d\) which, assuming the Birch and Swinnerton-Dyer conjecture, yield \(\text{rank}(E^{(d)})=1\) and gives examples of 4 infinite families of elliptic curves which verify the conditions.
The proof relies on the computation of the 3-Selmer group \(\mathrm{Sel}_3(E^{(d)}/\mathbb{Q})\). The hypotheses on \(E\) (semistability, the type of bad reduction and \(E\simeq E'/\langle P\rangle\) for some \(P\in E'(\mathbb{Q})[3]-\{O\}\,\)) and on \(d\) (3 does not divide the class number of \(\mathbb{Q}(\sqrt{d})\) and a series of congruences modulo the primes of bad reduction) are used to reduce the computation of \(\mathrm{Sel}_3(E^{(d)}/\mathbb{Q})\) to the one of certain cohomology classes unramified outside 3 (or unramified everywhere). If furthermore the root number \(\omega(E^{(d)})\) is \(-1\) and \(E^{(d)}(\mathbb{Q})\) has trivial torsion, then the author is able to prove that \(\# \mathrm{Sel}_3(E^{(d)}/\mathbb{Q})=3\) and, assuming the Birch and Swinnerton-Dyer conjecture, that \(\text{rank}(E^{(d)})=1\). The fact that the conditions hold for a positive proportion of \(d\) is obvious for the congruences and is guaranteed by the Nakagawa-Horie estimates (see J. Nakagawa and K. Horie [Proc. Am. Math. Soc. 104, No. 1, 20–24 (1988; Zbl 0663.14023)]) for quadratic fields with class number prime to 3 for the remaining condition.
The final section provides 4 families of type \[ E_{m,n}\;:\;y^2+xy=x^3+H(m,n)x+J(m,n) \] for \(m=1\), 7, 13 and 19 (formulas for \(n\), \(H(m,n)\) and \(J(m,n)\) are given explicitly in Section 3), which verify all conditions for infinitely many \(n\). Actually the author starts with 14 values for \(m\) and restricts to the 4 above only to ensure the final condition \(\omega(E^{(d)})=-1\).
The proof relies on the computation of the 3-Selmer group \(\mathrm{Sel}_3(E^{(d)}/\mathbb{Q})\). The hypotheses on \(E\) (semistability, the type of bad reduction and \(E\simeq E'/\langle P\rangle\) for some \(P\in E'(\mathbb{Q})[3]-\{O\}\,\)) and on \(d\) (3 does not divide the class number of \(\mathbb{Q}(\sqrt{d})\) and a series of congruences modulo the primes of bad reduction) are used to reduce the computation of \(\mathrm{Sel}_3(E^{(d)}/\mathbb{Q})\) to the one of certain cohomology classes unramified outside 3 (or unramified everywhere). If furthermore the root number \(\omega(E^{(d)})\) is \(-1\) and \(E^{(d)}(\mathbb{Q})\) has trivial torsion, then the author is able to prove that \(\# \mathrm{Sel}_3(E^{(d)}/\mathbb{Q})=3\) and, assuming the Birch and Swinnerton-Dyer conjecture, that \(\text{rank}(E^{(d)})=1\). The fact that the conditions hold for a positive proportion of \(d\) is obvious for the congruences and is guaranteed by the Nakagawa-Horie estimates (see J. Nakagawa and K. Horie [Proc. Am. Math. Soc. 104, No. 1, 20–24 (1988; Zbl 0663.14023)]) for quadratic fields with class number prime to 3 for the remaining condition.
The final section provides 4 families of type \[ E_{m,n}\;:\;y^2+xy=x^3+H(m,n)x+J(m,n) \] for \(m=1\), 7, 13 and 19 (formulas for \(n\), \(H(m,n)\) and \(J(m,n)\) are given explicitly in Section 3), which verify all conditions for infinitely many \(n\). Actually the author starts with 14 values for \(m\) and restricts to the 4 above only to ensure the final condition \(\omega(E^{(d)})=-1\).
Reviewer: Andrea Bandini (Parma)
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