Explicit points on the Legendre curve II

Explicit points on the Legendre curve III

Explicit points on the Legendre curve

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 Swinner...

The Ordering of Points on a Curve. Part II

(1) Let P, Q be subsets of E2 T, p1, p2, q1 be points of E2 T, f be a map from I into (E2 T) P, and s1 be a real number. Suppose that P is an arc from p1 to p2 and q1 ∈ P and q1 ∈ Q and f (s1) = q1 and f is a homeomorphism and f (0) = p1 and f (1) = p2 and 0 ≤ s1 and s1 ≤ 1 and for every real number t such that 0 ≤ t and t < s1 holds f (t) / ∈ Q. Let g be a map from I into (E2 T) P and s2 be a ...

Explicit Uniform Estimation of Rational Points Ii. Hypersurface Coverings

Explicit 4-descents on an Elliptic Curve

It is shown that the obvious method of descending from an element of the 2-Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the Weil-Chatelet group of E. Explicit algorithms for such a method are given.