arXiv subjects

Sajad Salami

Publications and source records attributed to Sajad Salami.

23 records · Page 2Linked to original sources

arXiv2019

A correction to the paper "On Curves with Split Jacobian"

In [5], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively in [1] to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the formula given in [5] works only under certain conditions.

math.AG↗

arXiv2019

On special matrices related to Cauchy and Toeplitz matrices

In this paper, we are going to calculate the determinant of a certain type of square matrices, which are related to the well-known Cauchy and Toeplitz matrices. Then, we will use the results to determine the rank of special non-square matrices.

math.CO↗

arXiv2018

The rational points on certain Abelian varieties over function fields

In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.

math.NT↗

arXiv2014

On the high rank $π/3$ and $2π/3$-congruent number elliptic curves

Consider the elliptic curves given by $ E_{n,θ}:\quad y^2=x^3+2s n x^2-(r^2-s^2) n^2 x $ where $0 < θ< π$, $\cos(θ)=s/r$ is rational with $0\leq |s| <r$ 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 fixed $θ$ this family corresponds to the quadratic twist by $n$ of the curve $E_θ: \,\, y^2=x^3+2s x^2-(r^2-s^2) x.$ 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 ${\mathbb Q}(w)$ 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 $\mathbb Q$ in both cases.

math.NT↗

arXiv2014

On $θ$-congruent numbers on real quadratic number fields

Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < θ< π$ has rational cosine, say $\cos (θ)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A positive integer $n$ is called a $(\mathbb K,θ)$-congruent number if there is a triangle, called the $(\mathbb K,θ, n)$-triangles, with sides in $\mathbb K$ having $θ$ as an angle and $nα_θ$ as area, where ${α_θ}=\sqrt{r^2-s^2}$. Consider the $(\mathbb K,θ)$-congruent number elliptic curve $E_{n,θ}: y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over $\mathbb K$. Denote the squarefree part of positive integer $t$ by ${\rm sqf}(t)$. In this work, it is proved that if $m\neq {\rm sqf}(2r(r-s))$ and $mn\neq 2, 3, 6$, then $n$ is a $(\mathbb K,θ)$-congruent number if and only if the Mordell-Weil group $E_{n,θ}(\mathbb K)$ has positive rank, and all of the $(\mathbb K,θ, n)$-triangles are classified in four types.

math.NT↗