相关论文: Formal groups, elliptic curves, and some theorems …

This paper gives additional background in algebraic geometry as an accompaniment to the article, ``Formal Groups, Elliptic Curves, and some Theorems of Couveignes'' [arXiv:math.NT/9708215]. Section 1 discusses the addition law on elliptic…

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal…

This article gives an elementary computational proof of the group law for Edwards elliptic curves. The associative law is expressed as a polynomial identity over the integers that is directly checked by polynomial division. Unlike other…

This article is a short introduction to the theory of the groups of points of elliptic curves over finite fields. It is concerned with the elementary theory and practice of elliptic curves cryptography, the new generation of public key…

In analogy with the classical group law on a plane cubic curve, we define a group law on a smooth plane tropical cubic curve. We show that the resulting group is isomorphic to $S^1$.

This article gives an elementary computational proof of the group law for Edwards elliptic curves following Bernstein, Lange, et al., Edwards, and Friedl. The associative law is expressed as a polynomial identity over the integers that is…

We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the…

We give an elementary proof of the group law for elliptic curves using explicit formulas.

As it is well known, one can define an abelian group on the points of an elliptic curve, using the so called chord-tangent law \cite{dale}, and a chosen point. However, that very chord-tangent law allows us to define a rather more obscure…

We provide a generalization of an algebraic linear combination for the trace of certain elliptic modular forms, and through specializing the expression at a suitable pair consisting of an elliptic curve over algebraic number fields and its…

I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number…

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational…

We revisit the group structure on elliptic curves and give a simple and elementary proof of the associativity of the addition. We do this by providing an explicit formula for the sum of three points, only using the explicit definition of…

The study of alternative models for elliptic curves has found recent interest from cryptographic applications, once it was recognized that such models provide more efficiently computable algorithms for the group law than the standard…

This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n=3 for…

In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve…

In this article we present a characterization of elliptic curves defined over a finite field Fq which possess a rational subgroup of order three. There are two posible cases depending on the rationality of the points in these groups. We…

We explicitly construct and investigate a number of examples of $\mathbb{Z}/p^r$-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and $p$-divisible groups, as well as some other related…