Related papers: An Algorithm to Find Rational Points on Elliptic C…
The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well known. We show how this correspondence can be extended to the…
We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.
This article discusses two versions of elliptic equations obtained from a system of equations describing a rational cuboid. Analysis of elliptic equations shows that they are equivalent, and that there are rational points on the elliptic…
Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…
The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a…
In this article, we are interested in finding rational points on certain superelliptic curves.
L-function and rational points on an elliptic curve via the classical number theory.
We present a simple and efficient algorithm to compute the sum of the algebraic conjugates of a point on an elliptic curve.
A formalism is given to count integer and rational solutions to polynomial equations with rational coefficients. These polynomials $P(x)$ are parameterized by three integers, labeling an elliptic curve. The counting of the rational…
In this paper, the proof of the existence of a rational point on an elliptic curve is transformed into the proof of the existence of an integer solution for a Diophantine equation. By a new formula for calculating the number of elements in…
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the…
In this paper, we analyze the theta series associated to the quadratic form $Q(\mathbf{x}) := x_1^2 + x_2^2 + x_3^2 + x_4^2$ with congruence conditions on $x_i$ modulo $2, 3, 4$, and $6$. By employing special operators on modular,…
In this paper, we propose a numerical method to solve isotropic elliptic equations on point cloud by generalizing the point integral method. The idea of the point integral method is to approximate the differential operators by integral…
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular…
We look at the elliptic curve E(q), where q is a fixed rational number. A point (p,r) on E(q) is called a rational point if both p and r are rational numbers. We introduce the concept of conjugate points and show that not both can be…
In this paper, we describe a new Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The algorithm depends on a property of the group of rational points of an elliptic curve and is thus not a generic algorithm. The…
We present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are: (i) application of classical addition theorems for…
An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.
This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm.
We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the…