Maximal ranks and integer points on a family of elliptic curves. II.
type: Item
instance of: scholarly article
title: Maximal ranks and integer points on a family of elliptic curves. II.
zbMATH Open document ID: 1234.11074
DOI: 10.1216/RMJ-2011-41-1-311
published in: Rocky Mountain Journal of Mathematics
publication date: 2011-03-01T00:00:00Z
review text: Let \(p\) be an odd prime number. Assume that there are rational points \((x_1,y_1), (x_2,y_2)\) on the elliptic curve \(E_{-2p}:y^2=x^3-2px\) and a rational point \((x_3,y_3)\) on the elliptic curve \(E_{8p}:y^2=x^3+8px\) with \(y_i>0\), \(x_i=d_iu_i^2\), \(d_i\) squarefree, \(u_i\in{\mathbb Q}\), \(1\leq i\leq 3\), \((d_1,d_2)\in\{(-1,2),(-1,-2),(-1,p),(2,-2),(2,p)\}\), \(d_3\in\{2,p\}\). The author proves that the Mordell-Weil rank of \(E_{-2p}\) is \(3\). There are at most \(99\) integer points \((x,y)\) on \(E_{-2p}\) with \(|x|>128\cdot 10^{12}p^{\frac{33}{2}}\) and \(y>0\). For Part I see Glas. Mat., III. Ser. 44, No. 1, 83--87 (2009; Zbl 1213.11125).
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 5858893
zbMATH Keywords: elliptic curve
zbMATH Keywords: rational points
zbMATH Keywords: Mordell-Weil rank
zbMATH Keywords: integer points
author: P. G. Walsh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5858893