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On positive proportion of rank-zero twists of elliptic curves over \(\mathbb{Q}\). (English) Zbl 1323.11038
For \(E/\mathbb{Q}\) an elliptic curve and \(r\) an integer let \(E^{(r)}\) be the \(r\)-twist of \(E\). In the paper under review the author gives a new proof of the following result of A. Dabrowski and M. Wieczorek (first proved in [J. Number Theory 124, No. 2, 364–379 (2007; Zbl 1117.11032)] under the assumption of a variant of the twin primes conjecture, and then reproved unconditionally by A. Dabrowski [C. R., Math., Acad. Sci. Paris 346, No. 9–10, 483–486 (2008; Zbl 1210.11067)]): for every \(k \geq 1\) there is a family \(E_1, \ldots, E_k\) of pairwise non-isogenous elliptic curves over \(\mathbb{Q}\) such that \(\text{rank} E_1^{(p)}(\mathbb{Q}) = \cdots = \text{rank} E_k^{(p)}(\mathbb{Q})=0\) for a positive proportion of primes \(p\).
The author constructs an explicit infinite family of elliptic curves \(E_{A,B}\) with integral coefficients, and uses complete 2-descent over \(\mathbb{Q}\) to prove that the equality \(\text{rank} E_{A,B}^{(p)}(\mathbb{Q})=0\) holds as long as \(p\) satisfies certain congruence conditions (depending on \(A\) and \(B\) in a simple way); by Dirichlet’s theorem on primes in arithmetic progressions, this is enough to establish the desired result.
The proof is very close in spirit to that of [Zbl. 1210.11067], but while it is immediate to show that the family of elliptic curves \(E_{A,B}\) is infinite, the corresponding statement for the families considered by Dabrowski depends on deep results of H. Iwaniec [Invent. Math. 47, 171–188 (1978; Zbl 0389.10031)] and J.-R. Chen [Sci. Sin. 16. 157–176 (1973; Zbl 0319.10056)].
MSC:
11G05 Elliptic curves over global fields
14H52 Elliptic curves
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