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Arman Shamsi Zargar

Publications and source records attributed to Arman Shamsi Zargar.

At least 19 recordsLinked to original sources
arXiv2025

The splitting fields and Generators of Shioda's elliptic surfaces $y^2=x^3 +t^{m} +1$ (I)

The splitting field of an elliptic surface $\mathcal E$ defined over ${\mathbb Q}(t)$ is the smallest subfield $\mathcal K$ of $\mathbb C$ such that ${\mathcal E}({\mathbb C}(t))\cong {\mathcal E}({\mathcal K}(t))$. In this paper, we determine the splitting field ${\mathcal K}_m$ and a set of linearly independent generators for the Mordell--Weil lattice of Shioda's elliptic surface with generic fiber given by ${\mathcal E}_m: y^2=x^3 +t^{m} +1$ over ${\mathbb Q}(t)$ for positive integers $1\leq m\leq 12$.

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arXiv2025

On Properly $θ$-Congruent Numbers Over Real Number Fields

The notion of $θ$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $θ$-congruent if it is the area of a rational triangle with an angle $θ$ whose cosine is rational. Das and Saikia [2] established criteria for numbers to be $θ$-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $θ$-congruent and properly $θ$-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to $6$, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree~$6$, and examine the exceptional cases $n=1, 2, 3$ and $6$.

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arXiv2025

Integer triangles with a rational ratio of circumcircle radius to excircle radius

We consider the problem of finding integer triangles with $R/r$ a positive rational, where $R$ and $r$ are the radii of the circumcircle and an excircle, respectively. We show that for general triangles $R/r>1/4$ applies. The equation $R/r=N$ turns out to be related to the elliptic curve $\mathcal{E}_N$ given by $v^2=u^3+2(2N^2+2N-1)u^2-(4N-1)u$. If $N>1/4$ is rational, then the torsion group of $\mathcal{E}_N$ is $\mathbb Z/2\mathbb Z\times\mathbb Z/6\mathbb Z$ if $N(N+2)$ is a square and $\mathbb Z/6\mathbb Z$ otherwise. We show that a rational triangle with rational ratio $R/r=N$ exists if and only if $N>1/4$ and there exists a rational non-torsion point on the curve $\mathcal{E}_N$ which satisfies a certain condition. Furthermore, we show that the rank of $\mathcal{E}_N$ is positive when $N = m^2 \pm 1>1/4$ for a rational $m$. We also show that on every curve $\mathcal{E}_N$ whose rank is positive, there are infinitely many rational points which lead to infinitely many non-similar integer triangles with $R/r=N$.

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arXiv2025

Generators and splitting fields of certain elliptic K3 surfaces

Let $k \subset {\mathbb C}$ be a number field and ${\mathcal E}$ be an elliptic curve defined over $k(t)$, the rational function field of the projective line ${\mathbb P}^1_k$, is isomorphic to the generic fiber of an elliptic surface $π:= \Sc_\Ee \rightarrow {\mathbb P}^1_k$. For any subfield ${\mathcal K}\subseteq {\mathbb C}$ of $k$, the set ${\mathcal E}({\mathcal K}(t))$ of ${\mathcal K}(t)$-rational points of ${\mathcal E}$ is known to be a finitely generated abelian group. The splitting field of ${\mathcal E}$ defined over $k(t)$ is the smallest finite extension ${\mathcal K} \subset {\mathbb C}$ of $k$ such that ${\mathcal E} ({\mathbb C} (t)) \iso {\mathcal E} ({\mathcal K}(t))$. In this paper, we consider the elliptic $K3$ surfaces defined over $k={\mathbb Q}$ with the generic fiber given by the Weierstrass equation ${\mathcal E}_n: \displaystyle y^2=x^3 + t^n + 1/t^n$, $1\leq n\leq 6$, and determine the splitting field ${\mathcal K}_n$, and find an explicit set of independent generators for ${\mathcal E}_n ({\mathcal K_n}(t))$ for $1\leq n \leq 6$.

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arXiv2025

Partitions into Triples with Equal Products and Families of Elliptic Curves

Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers having the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between a subset of $S_l(M,N)$ with (integral) parametric elements and a family of elliptic curves. When $\ell=2$ and $3$, we use certain known subsets of $S_l(M,N)$ with parametric elements and respectively find families of elliptic curves of generic rank $\geq 5$ and $\geq 6$, while for $\ell=4$ we first obtain a subset of $S_l(M,N)$ with parametric elements, then construct a family of elliptic curves of generic rank $\geq 8$. Finally, we perform a computer search within these families to find specific curves with rank $\geq 11$ and in particular we found two curves of rank $14$.

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arXiv2024

Pairing Powers of Pythagorean Pairs

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2$ is a square. A pythagorean pair $(a, b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$, such that $(a^hk, b^hl)$ is a pythagorean pair. To each pythagorean pair $(a, b)$ we assign an elliptic curve $Γ_{a^h ,b^h}$ for $h\ge 3$ with torsion group isomorphic to $\mathbb Z/2\mathbb Z \times \mathbb Z/4\mathbb Z$ such that $Γ_{a^h,b^h}$ has positive rank over $\mathbb Q$ if and only if $(a,b)$ is a pythapotent pair of degree $h$. As a side result, we get that if $(a, b)$ is a pythapotent pair of degree $h$, then there exist infinitely many pythagorean pairs $(k,l)$, not multiples of each other, such that $(a^hk,b^hl)$ is a pythagorean pair. In particular, we show that any pythagorean pair is always a pythapotent pair of degree 3. In a previous work, pythapotent pairs of degrees 1 and 2 have been studied.

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arXiv2024

The diophantine equation $x^4+y^4=z^4+w^4$

Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees~$74$, $88$ and $132$ respectively.

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arXiv2022

An octic diophantine equation and related families of elliptic curves

We obtain two parametric solutions of the diophantine equation $ϕ(x_1, x_2, x_3)=ϕ(y_1, y_2, y_3)$ where $ϕ(x_1, x_2, x_3)$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. These parametric solutions yield infinitely many examples of two equiareal triangles whose sides are perfect squares of integers. Further, each of the two parametric solutions leads to a family of elliptic curves of rank~$5$ over $\mathbb{Q}(t)$. We study one of the two families in some detail and determine a set of five free generators for the family.

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arXiv2021

A geometric approach to elliptic curves with torsion groups $\mathbb{Z}/10\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, $\mathbb{Z}/14\mathbb{Z}$, and $\mathbb{Z}/16\mathbb{Z}$

We give new parametrisations of elliptic curves in Weierstrass normal form $y^2=x^3+ax^2+bx$ with torsion groups $\mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$ over $\mathbb{Q}$, and with $\mathbb{Z}/14\mathbb{Z}$ and $\mathbb{Z}/16\mathbb{Z}$ over quadratic fields. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three infinite families of elliptic curves with torsion group $\mathbb{Z}/12\mathbb{Z}$ and positive rank. Furthermore, we found elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ and rank $3$, which is a new record for such curves, as well as some new elliptic curves with torsion group $\mathbb{Z}/16\mathbb{Z}$ and rank $3$.

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arXiv2021

Rational $θ$-parallelogram envelopes via $θ$-congruent elliptic curves

We introduce a new generalization of $θ$-congruent numbers by defining the notion of rational $θ$-parallelogram envelope for a positive integer $n$, where $θ\in (0, π)$ is an angle with rational cosine. Then, we study more closely some problems related to the rational $θ$-parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of T.~Ochiai, where only the case $θ=π/2$ was considered. Moreover, we answer the open questions in his paper and their generalizations for any Pythagorean angle.

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arXiv2020

The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

A positive integer $N$ is called a $θ$-congruent number if there is a $\ta$-triangle $(a,b,c)$ with rational sides for which the angle between $a$ and $b$ is equal to $θ$ and its area is $N \sqrt{r^2-s^2}$, where $θ\in (0, π)$, $\cos(θ)=s/r$, and $0 \leq |s|<r$ are coprime integers. It is attributed to Fujiwara \cite{fujw1} that $N$ is a $\ta$-congruent number if and only if the elliptic curve $E_N^\ta: y^2=x (x+(r+s)N)(x-(r-s)N)$ has a point of order greater than $2$ in its group of rational points. Moreover, a natural number $N\neq 1,2,3,6$ is a $\ta$-congruent number if and only if rank of $E_N^\ta(\Q)$ 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 $E_N^θ({\mathbb Q})$. Then, based on the addition of two distinct points in $E_N^θ({\mathbb Q})$, we provide a way to find new rational $\ta$-triangles for the $θ$-congruent number $N$ using given two distinct ones. Finally, we give an alternative proof for Fujiwara's theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $E_N^θ({\mathbb Q})$ with corresponding rational $θ$-triangles

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arXiv2019

A sextic diophantine chain and a related Mordell curve

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain $ ϕ(x_1,\,y_1,\,z_1)=ϕ(x_2,\,y_2,\,z_2)=ϕ(x_3,\,y_3,\,z_3)=k$ where $ϕ(x,\,y,\,z)$ is a sextic form defined by $ϕ(x,\,y,\,z)$ $=x^6+y^6+z^6-2x^3y^3-2x^3z^3-2y^3z^3$ and $k$ is an integer. Each numerical solution of such a sextic chain yields, in general, nine rational points on the Mordell curve $y^2=x^3+k/4$. While all of these nine points are not independent in the group of rational points of the Mordell curve, we have constructed a parameterized family of Mordell curves of generic rank $\geq 6$ using the aforementioned parametric solution of the sextic diophantine chain. Similarly, the numerical solutions of the sextic chain yield additional examples of Mordell curves whose rank is $\geq 6$.

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arXiv2018

Evaluation of Gaussian hypergeometric series using Huff's models of elliptic curves

A Huff curve over a field $K$ is an elliptic curve defined by the equation $ax(y^2-1)=by(x^2-1)$ where $a,b\in K$ are such that $a^2\ne b^2$. In a similar fashion, a general Huff curve over $K$ is described by the equation $x(ay^2-1)=y(bx^2-1)$ where $a,b\in K$ are such that $ab(a-b)\ne 0$. In this note we express the number of rational points on these curves over a finite field $\mathbb{F}_q$ of odd characteristic in terms of Gaussian hypergeometric series $\displaystyle {_2F_1}(λ):={_2F_1}\left(\begin{matrix} ϕ&ϕ& ε\end{matrix}\Big| λ\right)$ where $ϕ$ and $ε$ are the quadratic and trivial characters over $\mathbb{F}_q$, respectively. Consequently, we exhibit the number of rational points on the elliptic curves $y^2=x(x+a)(x+b)$ over $\mathbb{F}_q$ in terms of ${_2F_1}(λ)$. This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of ${_2F_1}$. Finally, we present the exact value of $_2F_1(λ)$ for different $λ$'s over a prime field $\mathbb{F}_p$ extending previous results of Greene and Ono.

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arXiv2016

A parametrised family of Mordell curves

An elliptic curve defined by an equation of the type $y^2=x^3+d$ is called a Mordell curve. We obtain a parametrised family of Mordell curves whose rank, in general, is at least three, and whose torsion group is $\mathbb{Z}/3\mathbb{Z}$.

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arXiv2016

On Parametric Spaces of Bicentric Quadrilaterals

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as tangents. Consider a bicentric quadrilateral with rational sides. We discuss the problem of finding such quadrilaterals where the ratio of the radii of the circumcircle and incircle is rational. We show that this problem can be formulated in terms of a family of elliptic curves given by $E_a:y^2=x^3+(a^4-4a^3-2a^2-4a+1)x^2+16a^4x$ which have, in general, \(\mathbb Z/8\mathbb Z\), and in rare cases \(\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z\) as torsion subgroups. We show the existence of infinitely many elliptic curves $E_a$ of rank at least two with torsion subgroup $\mathbb Z/8\mathbb Z$, parameterized by the points of an elliptic curve of rank at least one, and give five particular examples of rank $5$. We, also, show the existence of a subfamily of $E_a$ whose torsion subgroup is $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$.

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