Singular Plane Curves and Mordell-Weil Groups of Jacobians

This note explores the method of A. Néron [5] for constructing elliptic curves of (fairly) high rank over Q . Néron’s basic idea is very simple: although the moduli space of elliptic curves is only 1-dimensional, the vector space of homogeneous cubic polynomials in three variables is 10-dimensional. Therefore, one can construct elliptic curves which pass through any given 9 rational points. With luck, these points will be linearly independent in the Mordell-Weil group. There are other vector spaces of homogeneous polynomials, besides the obvious one, which generically define elliptic curves. For example, given two fixed points P and Q in CP, consider the set of homogeneous polynomials