Explicit points on the Legendre curve III

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...

Congruences concerning Legendre polynomials III

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,

Sziklai's conjecture on the number of points of a plane curve over a finite field III

In the paper [11], Sziklai posed a conjecture on the number of points of a plane curve over a finite field. Let C be a plane curve of degree d over Fq without an Fq-linear component. Then he conjectured that the number of Fq-points Nq(C) of C would be at most (d− 1)q+1. But he had overlooked the known example of a curve of degree 4 over F4 with 14 points ([10], [1]). So we must modify this conj...

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.