Elliptic Curves for Data Provenance

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 for Applications

Elliptic Nets and Elliptic Curves

Elliptic divisibility sequences are integer recurrence sequences, each of which is associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base fields. Suppose E is an elliptic curve over a field K, and P1, . . . , Pn are points on E defined over K. To this information we associate an ...

Elliptic Curves

This is a introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher genus. Readers desiring a more systematic development should consult one of the references for further ...

Elliptic Curves

1. Throughout P = C ∪ {∞} denotes the Riemann sphere, H denotes the upper half plane, C∗ denotes the multiplicative group of complex numbers, and P = (C \ {0})/C∗ denotes n dimensional complex projective space. For w ∈ C \{0} let [w] := wC∗ denote the corresponding point of P. For A ∈ GLn+1(C) let MA denote the corresponding automorphism of projective space so that MA([w]) = [Aw]. Identify P an...