Low Degree Places on the Modular Curve X1(N)

2020

A positive integer N is called a θ-congruent number if there is a -triangle (a,b,c) with rational sides for which the angle between a and b is equal to θ and its area is N √(r^2-s^2), where θ∈ (0, π), cos(θ)=s/r, and 0 ≤ |s|<r are coprime integers. It is attributed to Fujiwara <cit.> that N is a -congruent number if and only if the elliptic curve E_N^: y^2=x (x+(r+s)N)(x-(r-s)N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N≠ 1,2,3,6 is a -congruent number if and only if rank of E_N^() is greater than zero. In this paper, we answer positively to a question concerning the existence of methods to create new rational θ-triangle for a θ-congruent number N from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E_N^θ(ℚ)...

2020

A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r − s, where θ ∈ (0, π), cos(θ) = s/r, and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara [4] that N is a θ-congruent number if and only if the elliptic curve E N : y 2 = x(x+ (r + s)N)(x− (r − s)N) has a point of order greater than 2 in its group of rational points. Moreover, a natural number N 6= 1, 2, 3, 6 is a θ-congruent number if and only if rank of E N (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ-triangle for a θ-congruent number N from given ones by generalizing the Fermat’s algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula...

We present here a result on congruent numbers elliptic curves. We construct an isomorphism class of elliptic curves associated to congruent numbers. We show that, two elliptic curves defined overQ and associated to congruent numbers which are the areas of two congruent right-angled triangles are Q-isomorphic. We prove a relation on the discriminants of congruent numbers elliptic curves, and we pose a conjecture on the conductors of congruent numbers elliptic curves.

Rocky Mountain J. Math., 2014

Consider the elliptic curves given by E n,θ : y 2 = x 3 + 2snx 2 − (r 2 − s 2 )n 2 x where 0 < θ < π, cos(θ) = s/r is rational with 0 ≤ |s| < r and gcd(r, s) = 1. These elliptic curves are related to the θ-congruent number problem as a generalization of the congruent number problem. For xed θ this family corresponds to the quadratic twist by n of the curve E θ : y 2 = x 3 + 2sx 2 − (r 2 − s 2 )x. We study two special cases θ = π/3 and θ = 2π/3. We have found a subfamily of n = n(w) having rank at least 3 over Q(w) and a subfamily with rank 4 parametrized by points of an elliptic curve with positive rank. We also found examples of n such that E n,θ has rank up to 7 over Q in both cases.

2018

In this paper, we give the elliptic curve (E) given by the equation: y 2 = x 3 + px + q (1) with p, q ∈ Z not null simultaneous. We study a part of the conditions verified by (p, q) so that ∃ (x, y) ∈ Z 2 the coordinates of a point of the elliptic curve (E) given by the equation (1).

2019

We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular numbers t = t(t + 1)/2. We prove that the rank of Et over the function field Q(t) is 1, while the rank is 0 over Q(t). We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.

In this paper, we give an elliptic curve (E) given by the equation: y² = φ(x) = x^3 + px + q, with p, q ∈ Z not null simultaneous. We study the conditions verified by (p, q) so that ∃ (x, y) ∈ ZxZ the coordinates of a point of the elliptic curve (E) given by the equation above.

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1996

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere: cujes rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Publ. Math. Debrecen, 2000