2010/02/17 by Douglas Ulmer, Ulmer, Douglas
Mathematics · #11G05 #11G40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G05 #msc:11G40
paper · pdf · doi:10.48550/arxiv.1002.3313
v2: major revision with many more details. 24 pages. v3: minor changes
arxiv created 2013/09/19 · arxiv updated 2013/09/23
We study the elliptic curve E given by y2=x(x+1)(x+t) over the rational function field k(t) and its extensions Kd=k(μd,t1/d). When k is finite of characteristic p and d=pf+1, we write down explicit points on E and show by elementary arguments that they generate a subgroup Vd of rank d-2 and of finite index in E(Kd). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over Kd, and we relate the index of Vd in E(Kd) to the order of the Tate-Shafarevich group \sha(E/Kd). When k has characteristic 0, we show that E has rank 0 over Kd for all d.