Maximal ranks and integer points on a family of elliptic curves. II.
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).
scientific article; zbMATH DE number 5858893