arXiv subjects

Ali S. Janfada

Publications and source records attributed to Ali S. Janfada.

3 recordsLinked to original sources

arXiv2017

Dyadic Steenrod algebra and its applications

First, by inspiration of the results of Wood \cite{differential,problems}, but with the methods of non-commutative geometry and different approach, we extend the coefficients of the Steenrod squaring operations from the filed $\mathbb{F}_2$ to the dyadic integers $\mathbb{Z}_2$ and call the resulted operations the dyadic Steenrod squares, denoted by $Jq^k$. The derivation-like operations $Jq^k$ generate a graded algebra, called the dyadic Steenrod algebra, denoted by $\mathcal{J}_2$ acting on the polynomials $\mathbb{Z}_2[ξ_1, \dots, ξ_n]$. Being $\mathcal{J}_2$ an Ore domain, enable us to localize $\mathcal{J}_2$ which leads to the appearance of the integration-like operations $Jq^{-k}$ satisfying the $Jq^{-k}Jq^k=1=Jq^kJq^{-k}$. These operations are enough to exhibit a kind of differential equation, the dyadic Steenrod ordinary differential equation. Then we prove that the completion of $\mathbb{Z}_2[ξ_1, \dots, ξ_n]$ in the linear transformation norm coincides with a certain Tate algebra. Therefore, the rigid analytic geometry is closely related to the dyadic Steenrod algebra. Finally, we define the Adem norm $\| \ \|_A$ in which the completion of $\mathbb{Z}_2[ξ_1, \dots, ξ_n]$ is $\mathbb{Z}_2\llbracketξ_1,\dots,ξ_n\rrbracket$, the $n$-variable formal power series. We surprisingly prove that an element $f \in \mathbb{Z}_2\llbracket ξ_1,\dots,ξ_n\rrbracket$ is hit if and only if $\|f\|_A<1$. This suggests new techniques for the traditional Peterson hit problem in finding the bases for the cohit modules.

math.AT↗

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↗