Explicit points on the Legendre curve

We study the elliptic curve E   given by y2=x(x+1)(x+t)y2=x(x+1)(x+t) over the rational function field k(t)k(t) and its extensions Kd=k(μd,t1/d)Kd=k(μd,t1/d). When k is finite of characteristic p   and d=pf+1d=pf+1, we write down explicit points on E   and show by elementary arguments that they generate a subgroup VdVd of rank d−2d−2 and of finite index in E(Kd)E(Kd). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E   over KdKd, and we relate the index of VdVd in E(Kd)E(Kd) to the order of the Tate–Shafarevich group Ш(E/Kd)Ш(E/Kd). When k has characteristic 0, we show that E   has rank 0 over KdKd for all d.