Integer Points and Independent Points on the Elliptic Curve $y^2=x^3-p^kx$

Abstract

Let $E_k$ be the elliptic curve given by $y^2=x^3-p^k x$, where $p$ is a prime number and $k \in \{1,2,3\}$. In this paper, we first give a necessary and sufficient condition for the rank of $E_k(\mathbf{Q})$ to equal one or two, respectively, and in the rank two case, explicitly describe independent points of free part of the Mordell-Weil group $E_k(\mathbf{Q})$. Secondly, we show several subfamilies of $E_k$ whose integer points and ranks can be completely determined.