Formulae for null curves deriving from elliptic curves

Generalization of Vélu’s Formulae for Isogenies between Elliptic Curves

Abstract Given an elliptic curve E and a finite subgroup G, Vélu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P +G as the difference between the abscissa of IG(P ) and the first elementary symmetric polynomial on the abscissas of th...

On Silverman's conjecture for a family of elliptic curves

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...

Elliptic Curves from Sextics

Let N be the moduli space of sextics with 3 (3,4)-cusps. The quotient moduli space N/G is one-dimensional and consists of two components, Ntorus/G and Ngen/G. By quadratic transformations, they are transformed into one-parameter families Cs and Ds of cubic curves respectively. First we study the geometry of Nε/G, ε = torus, gen and their structure of elliptic fibration. Then we study the Mordel...

Pseudorandom sequences from elliptic curves

In this article we will generalize some known constructions to produce pseudorandom sequences with the aid of elliptic curves. We will make use of both additive and multiplicative characters on elliptic curves.

Elliptic Curves for Applications