𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves

✍ Scribed by Gwynneth Coogan; Jorge Jimenéz-Urroz

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
146 KB
Volume
96
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

Motivated by a conjecture of Mazur, Kuwata and Wang proved that for elliptic curves E 1 and E 2 whose j-invariants are not simultaneously 0 or 1728, there exist infinitely many square-free integers d for which the rank of the Mordell-Weil group of the d-quadratic twists of E 1 and E 2 satisfy: rkðE 1 d ; QÞ > 0 and rkðE 2 d ; QÞ > 0: Here we present results for the related questions: Are there infinitely many square-free integers d for which: rkðE 1 d ; QÞ ¼ 0 and rkðE 2 d ; QÞ ¼ 0? And, are there infinitely many square-free integers d for which: rkðE 1 d ; QÞ ¼ 0 and rkðE 2 d ; QÞ > 0? # 2002 Elsevier Science (USA)

📜 SIMILAR VOLUMES

On the Existence of Elliptic Curves of M
On the Existence of Elliptic Curves of Mordell–Weil Rank 6 with Four Parameters
✍ Hizuru Yamagishi 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 92 KB

We construct an elliptic curve defined over Q with Mordell᎐Weil rank G 6 as a generic twist by a certain quadratic extension. Moreover, since they have four independent parameters, they give us rather a large supply of elliptic curves defined over Q with rank G 6. As an application, we find infinite

The Average Analytic Rank of a Family of
The Average Analytic Rank of a Family of Elliptic Curves
✍ L. Mai 📂 Article 📅 1993 🏛 Elsevier Science 🌐 English ⚖ 333 KB

This paper studies the family of elliptic curves \(E_{m}: X^{3}+Y^{3}=m\) where \(m\) is a cubefree integer. Assuming the Generalized Rieman Hypothesis, the average rank of \(E_{m}\) 's with even analytic rank is proved to be \(\leqslant 5 / 2\), asymptotically. We also obtain some results for the c