Related papers: On the elliptic curve endomorphism generator

We study the distribution of algebraic points on curves in abelian varieties over finite fields.

Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\, a^5-a\not\equiv 0$ mod $p$, or,…

A random geometric digraph $G_n$ is constructed by taking $\{X_1,X_2,... X_n\}$ in $\mathbb{R}^2$ independently at random with a common bounded density function. Each vertex $X_i$ is assigned at random a sector $S_i$ of central angle…

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that…

Let $E_\lambda$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-\lambda)$. We recently proved that, given $n$ linearly independent points $P_1(\lambda), \dots,P_n(\lambda)$ on $E_\lambda$ with coordinates in…

Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class…

Let E be an elliptic curve over Q (or, more generally, a number field). Then on the one hand, we have the finitely generated abelian group E(Q), on the other hand, there is the Shafarevich-Tate group Sha(Q,E). Descent is a general method of…

Let $p$ be a prime and let $K$ be a finite extension of $\mathbb{Q}_p$. Let $E/K$ be an elliptic curve with additive reduction. In this paper, we study the topological group structure of the set of points of good reduction of $E(K)$. In…

In this paper, we study the problem of sampling random supersingular elliptic curves with unknown endomorphism rings. This problem has recently gained considerable attention as many isogeny-based cryptographic protocols require such…

We describe an explicit method for constructing pseudo-automorphisms of a space $X$ which is obtained by blowing up points of $P^k$ (or a product $P^k \times \cdots \times P^k$). The centers of blowup are chosen to lie on an elliptic normal…

We give a generalization of a theorem of Silverman and Stephens regarding the signs in an elliptic divisibility sequence to the case of an elliptic net. We also describe applications of this theorem in the study of the distribution of the…

Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…

It is well-known that if $E$ is an elliptic curve over the finite field $\mathbb{F}_p$, then $E(\mathbb{F}_p)\simeq\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/mk\mathbb{Z}$ for some positive integers $m, k$. Let $S(M,K)$ denote the set of pairs…

We describe the automorphism groups of elliptic Poisson algebras on polynomial algebras in three variables and give an explicit set of generators and defining relations for this group.

A $\textit{polygonal curve}$ is a collection of $m$ connected line segments specified as the linear interpolation of a list of points $\{p_0, p_1, \ldots, p_m\}$. These curves may be obtained by sampling points from an oriented curve in…

Let $\mathrm{PG}(1,E)$ be the projective line over the endomorphism ring $E=End_q({\mathbb F}_{q^t})$ of the $\mathbb F_q$-vector space ${\mathbb F}_{q^t}$. As is well known there is a bijection $\Psi:\mathrm{PG}(1,E)\rightarrow{\cal…

Let $C$ be an elliptic curve defined over $\mathbb Q$ by the equation $y^2=x^3+Ax+B$ where $A,B\in\mathbb Q$. A sequence of rational points $(x_i,y_i)\in C(\mathbb Q),\,i=1,2,\ldots,$ is said to form a sequence of consecutive squares on $C$…

In literature, there are two different definitions of elliptic divisibility sequences. The first one says that a sequence of integers $\{h_n\}_{n\geq 0}$ is an elliptic divisibility sequence if it verifies the recurrence relation…

We study isomorphism invariant point processes of $\mathbb{R}^d$ whose groups of symmetries are almost surely trivial. We define a 1-ended, locally finite tree factor on the points of the process, that is, a mapping of the point…

In this paper, we study partial automorphisms and, more generally, injective partial endomorphisms of a finite undirected path from Semigroup Theory perspective. Our main objective is to give formulas for the ranks of the monoids…