Families of Elliptic Curves and Geometric Properties of Torsion Points

Abstract

In this chapter we consider families of elliptic curves by studying cubics in normal form with coefficients depending on a parameter. The most important example is the Legendre family E: y ² = x(x-1)(x-λ) over k(λ), where k is a field of characteristic unequal to 2. The points ·(0, 0), (1, 0), (λ, 0)× are the three 2-torsion points on E _λ for each value of λ ∈ k - ·0, 1×, and they are specified with a given ordering.