Infinite family of elliptic curves of rank at least 4

Naskręcki, Bartosz

doi:10.2140/involve.2010.3.297

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Infinite family of elliptic curves of rank at least 4

Bartosz Naskręcki

Vol. 3 (2010), No. 3, 297–316

DOI: 10.2140/involve.2010.3.297

Abstract

We investigate $ℚ$ -ranks of the elliptic curve $E_{t}$ : $y^{2} + t x y = x^{3} + t x^{2} - x + 1$ , where $t$ is a rational parameter. We prove that for infinitely many values of $t$ the rank of $E_{t} (ℚ)$ is at least 4.

Keywords

elliptic curves, Mordell–Weil group, ranks in families

Mathematical Subject Classification 2000

Primary: 11D25, 11G05

Milestones

Received: 30 January 2010

Revised: 25 August 2010

Accepted: 4 September 2010

Published: 16 October 2010

Proposed: Bjorn Poonen

Communicated by Bjorn Poonen

Authors

Bartosz Naskręcki
Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań Poland

Vol. 3, No. 3, 2010