On the elliptic curves of the form $ y^2=x^3-3px $ (original) (raw)

On the Elliptic Curves of the Form y2=x3−pqxy^2 = x^3 − pqxy2=x3−pqx

Iranian Journal of Mathematical Sciences and Informatics, 2015

By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. This paper studies the rank of the family Epq : y 2 = x 3 − pqx of elliptic curves, where p and q are distinct primes. We give infinite families of elliptic curves of the form y 2 = x 3 − pqx with rank two, three and four, assuming a conjecture of Schinzel and Sierpinski is true.

On The Rank Of Congruent Elliptic Curves

2017

In this paper, ppp and qqq are two different odd primes. First, We construct the congruent elliptic curves corresponding to ppp, 2p2p2p, pqpqpq, and 2pq,2pq,2pq, then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.

Complete characterization of the Mordell-Weil group of some families of elliptic curves

Bulletin of The Iranian Mathematical Society, 2016

The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime ppp the rank of elliptic curve y2=x3−3pxy^2=x^3-3pxy2=x3−3px is at most two‎. ‎In this‎ ‎paper we go further‎, ‎and using height function‎, ‎we will determine the Mordell-Weil group of a‎ ‎family of elliptic curves of the form y2=x3−3nxy^2=x^3-3nxy2=x3−3nx‎, ‎and give‎ ‎a set of its generators under certain conditions‎. ‎We will‎ ‎introduce an infinite family of elliptic curves with rank at least‎ ‎two‎. ‎The full Mordell-Weil group and the generators of a‎ ‎family (which is expected to be infinite under the assumption of a standard conjecture) of elliptic curves with exact rank two will be described‎.

Ranks of elliptic curves with prescribed torsion over number fields

Int. Math. Res. Not. IMRN, 2014

We study the structure of the Mordell-Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group T and a quartic field K such that among the elliptic curves over K with torsion subgroup T , there are curves of positive rank, but none of rank 0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call false complex multiplication, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

Elliptic Curves of Type y2=x3−3pqx Having Ranks Zero and One

Malaysian Journal of Mathematical Sciences

The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr×G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E:y2=x3−3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p≡q≡5(mod24), then E has rank zero. Furthermore, if p≡5(mod12) is considered instead of a modulus of 24, then E has rank zero or one. Lastly, for primes of the form p=24k+17 and q=24ℓ+5, where 9k+3ℓ+7 is a perfect square, we show that E has rank one.

Mordell-Weil theorem and the rank of elliptical curves

2007

An elliptic curve over a field is a nonsingular cubic curve in two variables with a rational point over the field. The set of these rational points forms an abelian group by the suitable definition of the group operation. If the field is an algebraic number field, Mordell-Weil theorem states that the group of rational points is finitely generated. The rank of an elliptic curve is the size of the smallest torsion-free generating set. The rank is very important in the study of elliptic curves. The rank is involved with many significant open questions on elliptic curves these days including the Birch and Swinnerton-Dyer Conjecture, which is one of the seven Millennium Prize problems established by the Clay Mathematics Institute. By using the proof of Mordell-Weil theorem, a formula for the rank of the elliptic curves in certain cases over algebraic number fields can be obtained and computable. This formula was first observed by J. Tate. The objective of this thesis is to give a detaile...

On the family of elliptic curves \varvec{y^2=x^3-m^2x+p^2}$$y2=x3-m2x+p2

Proceedings - Mathematical Sciences, 2018

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of E m, p : y 2 = x 3 − m 2 x + p 2 , where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of E m, p (Q) is trivial for both the cases {m ≥ 1, m ≡ 0 (mod 3)} and {m ≥ 1, m ≡ 0 (mod 3), with gcd(m, p) = 1}. We also show that given any odd prime p and for any positive integer m with m ≡ 0 (mod 3) and m ≡ 2 (mod 32), the lower bound for the rank of E m, p (Q) is 2. Finally, we find curves of rank 9 in this family.

On the group orders of elliptic curves over finite fields (Compositio Mathematica 85 (1993) 229-247)

Given a prime power q, for every pair of positive integers m and n with m|gcd(n, q − 1) we construct a modular curve over Fq that parametrizes elliptic curves over Fq along with Fq-defined points P and Q of order m and n, respectively, with P and n mQ having a given Weil pairing. Using these curves, we estimate the number of elliptic curves over Fq that have a given integer N dividing the number of their Fq-defined points.

A family of elliptic curves of rank ≥ 4

Involve, a Journal of Mathematics

In this paper we consider a family of elliptic curves of the form y 2 = x 3 −c 2 x +a 2 , where (a, b, c) is a primitive Pythagorean triple. First we show that the rank is positive. Then we construct a subfamily with rank ≥ 4.