Rational Points on an Elliptic Curve

2007 ◽  
Keyword(s):  
Elliptic Curve ◽  
Rational Points

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Number of rational points of elliptic curves

2021 ◽  
pp. 2250017
Author(s):  
Viliam Ďuriš ◽  
Timotej Šumný
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Current Knowledge ◽  
American Mathematical Society ◽  
Rational Points ◽  
Modern Theory ◽  
Cambridge University ◽  
Weil Theorem ◽  
A Finite Group

In the modern theory of elliptic curves, one of the important problems is the determination of the number of rational points on an elliptic curve. The Mordel–Weil theorem [T. Shioda, On the Mordell–Weil lattices, Comment. Math. University St. Paul. 39(2) (1990) 211–240] points out that the elliptic curve defined above the rational points is generated by a finite group. Despite the knowledge that an elliptic curve has a final number of rational points, it is still difficult to determine their number and the way how to determine them. The greatest progress was achieved by Birch and Swinnerton–Dyer conjecture, which was included in the Millennium Prize Problems [A. Wiles, The Birch and Swinnerton–Dyer conjecture, The Millennium Prize Problems (American Mathematical Society, 2006), pp. 31–44]. This conjecture uses methods of the analytical theory of numbers, while the current knowledge corresponds to the assumptions of the conjecture but has not been proven to date. In this paper, we focus on using a tangent line and the osculating circle for characterizing the rational points of the elliptical curve, which is the greatest benefit of the contribution. We use a different view of elliptic curves by using Minkowki’s theory of number geometry [H. F. Blichfeldt, A new principle in the geometry of numbers, with some applications, Trans. Amer. Math. Soc. 15(3) (1914) 227–235; V. S. Miller, Use of elliptic curves in cryptography, in Proc. Advances in Cryptology — CRYPTO ’85, Lecture Notes in Computer Science, Vol. 218 (Springer, Berlin, Heidelberg, 1985), pp. 417–426; E. Bombieri and W. Gubler, Heights in Diophantine Geometry, Vol. 670, 1st edn. (Cambridge University Press, 2007)].

scholarly journals Trinomials defining quintic number fields

2017 ◽  
Vol 13 (07) ◽  
pp. 1881-1894 ◽  
Author(s):  
Jesse Patsolic ◽  
Jeremy Rouse
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Number Fields ◽  
Rational Points

Given a quintic number field K/ℚ, we study the set of irreducible trinomials, polynomials of the form x5 + ax + b, that have a root in K. We show that there is a genus 4 curve CK whose rational points are in bijection with such trinomials. This curve CK maps to an elliptic curve defined over a number field, and using this map, we are able (in some cases) to determine all the rational points on CK using elliptic curve Chabauty.

scholarly journals Complete addition laws on abelian varieties

2012 ◽  
Vol 15 ◽  
pp. 308-316 ◽  
Author(s):  
Christophe Arene ◽  
David Kohel ◽  
Christophe Ritzenthaler
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Galois Group ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Rational Points ◽  
Absolute Galois Group ◽  
Projective Embedding ◽  
Complete Set

AbstractWe prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

Rational Points in Arithmetic Progressions on y2 = xn + k

2012 ◽  
Vol 55 (1) ◽  
pp. 193-207 ◽  
Author(s):  
Maciej Ulas
Keyword(s):  
Elliptic Curve ◽  
Arithmetic Progression ◽  
Hyperelliptic Curve ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Multiple Roots

AbstractLet C be a hyperelliptic curve given by the equation y2 = f(x) for f ∈ ℤ[x] without multiple roots. We say that points Pi = (xi, yi) ∈ C(ℚ) for i = 1, 2, … , m are in arithmetic progression if the numbers xi for i = 1, 2, … , m are in arithmetic progression.In this paper we show that there exists a polynomial k ∈ ℤ[t] with the property that on the elliptic curve ε′ : y2 = x3+k(t) (defined over the field ℚ(t)) we can find four points in arithmetic progression that are independent in the group of all ℚ(t)-rational points on the curve Ε′. In particular this result generalizes earlier results of Lee and Vélez. We also show that if n ∈ ℕ is odd, then there are infinitely many k's with the property that on curves y2 = xn + k there are four rational points in arithmetic progressions. In the case when n is even we can find infinitely many k's such that on curves y2 = xn +k there are six rational points in arithmetic progression.

scholarly journals Minimal models for -coverings of elliptic curves

2014 ◽  
Vol 17 (A) ◽  
pp. 112-127
Author(s):  
Tom Fisher
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Minimal Model ◽  
Rational Points ◽  
Minimal Models ◽  
Integer Coefficients ◽  
Field Extensions ◽  
Cubic Forms ◽  
New Formula

AbstractIn this paper we give a new formula for adding $2$-coverings and $3$-coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$-coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.

scholarly journals Kummer theory on the product of an elliptic curve by the multiplicative group

1981 ◽  
Vol 22 (1) ◽  
pp. 83-88 ◽  
Author(s):  
D. Bertrand
Keyword(s):  
Elliptic Curve ◽  
Multiplicative Group ◽  
Algebraic Groups ◽  
Rational Points ◽  
Galois Groups ◽  
Kummer Theory ◽  
Fixed Set

This note extends classical results on certain Galois groups attached to onedimensional algebraic groups. We prove that the fields arising from the division of a fixed set of rational points on the product of an elliptic curve by the multiplicative group are as “large” as possible.

scholarly journals The real field with the rational points of an elliptic curve

2011 ◽  
Vol 211 (1) ◽  
pp. 15-40 ◽  
Author(s):  
Ayhan Günaydın ◽  
Philipp Hieronymi
Keyword(s):  
Elliptic Curve ◽  
Rational Points ◽  
Real Field ◽  
The Real
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