Related papers: Elliptic Loops

Let $K$ be a global field of finite characteristic $p\geq2$, and let $E/K$ be a non-isotrivial elliptic curve. We give an asympotoic formula of the number of places $\nu$ for which the reduction of $E$ at $\nu$ is a cyclic group. Moreover…

A pointwise-elliptic subset of a topological group is one whose elements all generate relatively-compact subgroups. A connected locally compact group has a dense pointwise-elliptic subgroup if and only if it is an extension by a compact…

For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use…

An \emph{automorphic loop} (or \emph{A-loop}) is a loop whose inner mappings are automorphisms. Every element of a commutative A-loop generates a group, and $(xy)^{-1} = x^{-1}y^{-1}$ holds. Let $Q$ be a finite commutative A-loop and $p$ a…

Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic…

In this paper, we will talk about the titled elliptic curve defined over imaginary quadratic fields such as $\mathbb{Q}(\sqrt{-q})$, where $q$ is congruent to 3 modulo 8 and $(p,q)=1$.

A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End(R_Z) of the additive group R_Z, taking any r in R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this…

In this paper we consider the plane elliptic motion which occurs if the moving centrode is a circle of radius $r$ and the fixed centrode a circle of radius $2r$. Every point of the moving plane generates an ellipse in the fixed plane. Let a…

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set…

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in one variable over a finite field is finiteley generated.

This master thesis describes how Selmer groups can be used to determine the Mordell-Weil group of elliptic curves over a number field K. The Mordell-Weil Theorem states that $E(K) = E(K)_{tors} \times Z^r$, where $r$ is the rank of $E$, and…

Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various…

We give an infinite family of congruent number elliptic curves, each with rank at least two, which are related to integral solutions of $m^2=n^2+nl+l^2$.

Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, \rho_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of \rho_E for a…

Consider elliptic curves $ E=E_\sigma: y^2 = x (x+\sigma p) (x+\sigma q), $ where$ \sigma =\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(\phi)}(E/{\mathbf{Q})}$, and $\…

Let K be a number field and let $\mathcal{E}$ be an elliptic curve defined over $K$. Let $m$ be a positive integer. We denote by $K(\mathcal{E}[m])$ the number fields obtained by adding to $K$ the coordinates of the $m$-torsion points of…

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb…

A new definition of the elliptic algebra U_{q,p}(g^) associated with an untwisted affine Lie algebra g^ is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of…

Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of…

By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.