Explicit points on the Legendre curve
نویسنده
چکیده
We study the elliptic curve E given by y = x(x+1)(x+ t) over the rational function field k(t) and its extensions Kd = k(μd, t). When k is finite of characteristic p and d = p + 1, we write down explicit points onE 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 X(E/Kd). When k has characteristic 0, we show that E has rank 0 over Kd for all d.
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