arXiv subjects

Mohammad Sadek

Publications and source records attributed to Mohammad Sadek.

At least 37 records · Page 2Linked to original sources

arXiv2020

Rational Points in Geometric Progression on the Unit Circle

A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we prove the existence of infinitely many rational numbers $r$ such that for each $r$ there exist infinitely many $r$-geometric progression sequences on the unit circle $x^2 + y^2 = 1$ of length at least $3$.

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arXiv2020

Counting elliptic curves with bad reduction over a prescribed set of primes

Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of elliptic curves over $\mathbb Q$, when ordered by height, with Kodaira type $T$ at a prime $p\ge5$ inside the set of all elliptic curves. This proportion is a rational function in $p$. For instance, we show that $\displaystyle\frac{p^8(p-1)}{p^9-1}$ of all elliptic curves with bad reduction at $p$ are of multiplicative reduction. Furthermore, we prove that the prime-to-$6$ part of the conductors of a majority ($=ζ(10)/ζ(2)\approx 0.6$) of elliptic curves are squarefree, where $ζ$ is the Riemann-zeta function.

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arXiv2020

Strong rational Diophantine D(q)-triples

We show that for infinitely many square-free integers q there exist infinitely many triples of rational numbers {a, b, c} such that a^2 + q, b^2 + q, c^2 + q, ab + q, ac + q and bc + q are squares of rational numbers.

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arXiv2019

Rational sequences on different models of elliptic curves

Given a set $S$ of elements in a number field $k$, we discuss the existence of planar algebraic curves over $k$ which possess rational points whose $x$-coordinates are exactly the elements of $S$. If the size $|S|$ of $S$ is either $4,5$, or $6$, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of $S$ are realized as the $x$-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves.

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arXiv2019

On quadratic progression sequences on smooth plane curves

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of $\overline{k}$-points with field of definition of degree at most $n$, $n\ge 3$.

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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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arXiv2018

On rational periodic points of $x^d+c$

We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we show that the density of such polynomials with a rational periodic point of any period among all polynomials $f(x)=x^d+c$, $c\in\Q$, is zero. Furthermore, we establish the connection between polynomials $f$ with periodic points and two arithmetic sequences. This yields necessary conditions that must be satisfied by $c$ and $d$ in order for the polynomial $f$ to possess a rational periodic point of exact period $N$, and a lower bound on the number of primitive prime divisors in the critical orbit of $f$ when such a rational periodic point exists. The note also introduces new results on the irreducibility of iterates of $f$.

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arXiv2018

Sequences of consecutive squares on quartic elliptic curves

Let $C: y^2=ax^4+bx^2+c$, be an elliptic curve defined over $\mathbb Q$. A set of rational points $(x_i,y_i) \in C(\mathbb Q)$, $i=1,2,\cdots,$ is said to be a sequence of consecutive squares if $x_i= (u + i)^2$, $i=1,2,\cdots$, for some $u\in \mathbb Q$. Using ideas of Mestre, we construct infinitely many elliptic curves $C$ with sequences of consecutive squares of length at least $6$. It turns out that these 6 rational points are independent. We then strengthen this result by proving that for a fixed $6$-term sequence of consecutive squares, there are infinitely many elliptic curves $C$ with the latter sequence forming the $x$-coordinates of six rational points in $C(\mathbb Q)$.

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arXiv2017

Minimal regular models of quadratic twists of genus two curves

Let $K$ be a complete discrete valuation field with ring of integers $R$ and residue field $k$ of characteristic $p>2$. We assume moreover that $k$ is algebraically closed. Let $C$ be a smooth projective geometrically connected curve of genus $2$. If $K(\sqrt{D})/K$ is a quadratic field extension of $K$ with associated character $χ$, then $C^χ$ will denote the quadratic twist of $C$ by $χ$. Given the minimal regular model $\mathcal X$ of $C$ over $R$, we determine the minimal regular model of the quadratic twist $C^χ$. This is accomplished by obtaining the stable model $\mathcal{C}^χ$ of $C^χ$ from the stable model $\mathcal{C}$ of $C$ via analyzing the Igusa and affine invariants of the curves $C$ and $C^χ$, and calculating the degrees of singularity of the singular points of $\mathcal{C}^χ$.

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arXiv2017

On large $F$-Diophantine sets

Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $ S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $ S$ is an $(F,2)$-Diophantine set. In addition, we show that the degree of $F$ can be as small as $\displaystyle 4\lfloor n/3\rfloor$.

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arXiv2016

High rank quadratic twists of pairs of elliptic curves

Given a pair of elliptic curves $E_1$ and $E_2$ over the rational field $\mathbb Q$ whose $j$-invariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals $d$ such that the $d$-quadratic twists of $E_1$ and $E_2$ are both of positive rank. We construct infinite families of pairs of elliptic curves $E_1$ and $E_2$ over $\mathbb Q$ such that for each pair there exist infinitely many square-free rationals $d$ for which the $d$-quadratic twists of $E_1$ and $E_2$ are both of rank at least 2.

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arXiv2016

On rational triangles via algebraic curves

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between each family and the set of rational points of an algebraic curve. These algebraic curves are: a curve of genus 0, an elliptic curve, and a genus 3 curve. We study the set of rational points on each of these curves and describe some of its rational points explicitly.

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arXiv2016

On dependence of rational points on elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb Q$. Let $Γ$ be a subgroup of $E(\mathbb Q)$ and $P\in E(\mathbb Q)$. In [1], it was proved that if $E$ has no nontrivial rational torsion points, then $P\inΓ$ if and only if $P\in Γ$ mod $p$ for finitely many primes $p$. In this note, assuming the General Riemann Hypothesis, we provide an explicit upper bound on these primes when $E$ does not have complex multiplication and either $E$ is a semistable curve or $E$ has no exceptional prime.

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arXiv2016

On geometric progressions on hyperelliptic curves

Let $C$ be a hyperelliptic curve over $\mathbb Q$ described by $y^2=a_0x^n+a_1x^{n-1}+\ldots+a_n$, $a_i\in\mathbb Q$. The points $P_{i}=(x_{i},y_{i})\in C(\mathbb{Q})$, $i=1,2,...,k,$ are said to be in a geometric progression of length $k$ if the rational numbers $x_{i}$, $i=1,2,...,k,$ form a geometric progression sequence in $\mathbb Q$, i.e., $x_i=pt^{i}$ for some $p,t\in\mathbb Q$. In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight.

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arXiv2016

On sequences of consecutive squares on elliptic curves

Let $C$ be an elliptic curve defined over $\mathbb Q$ by the equation $y^2=x^3+Ax+B$ where $A,B\in\mathbb Q$. A sequence of rational points $(x_i,y_i)\in C(\mathbb Q),\,i=1,2,\ldots,$ is said to form a sequence of consecutive squares on $C$ if the sequence of $x$-coordinates, $x_i,i=1,2,\ldots$, consists of consecutive squares. We produce an infinite family of elliptic curves $C$ with a $5$-term sequence of consecutive squares. Furthermore, this sequence consists of five independent rational points in $C(\mathbb Q)$. In particular, the rank $r$ of $C(\mathbb Q)$ satisfies $r\ge 5$.

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arXiv2015

Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves

We study the character sums \[ϕ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } ψ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left((x^{m}+a)(x^{n}+b)\right)\] where $ϕ$ is the quadratic character defined over $\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties.

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arXiv2015

Formal Groups and Invariant Differentials of Elliptic Curves

In this paper, we find a power series expansion of the invariant differential $ω_E$ of an elliptic curve $E$ defined over $\mathbb{Q}$, where $E$ is described by certain families of Weierstrass equations. In addition, we introduce several congruence relations satisfied by the trace of the Frobenius endomorphism of $E$.

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arXiv2015

Edwards Curves and Gaussian Hypergeometric Series

Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\, a^5-a\not\equiv 0$ mod $p$, or, $ax^2+y^2=1+dx^2y^2,\,ad(a-d)\not\equiv0$ mod $p$, respectively. We express the number of rational points of $E$ over $\mathbb{F}_p$ using the Gaussian hypergeometric series $\displaystyle {_2F_1}\left(\begin{matrix} ϕ&ϕ {} & ε \end{matrix}\Big| x\right)$ where $ε$ and $ϕ$ are the trivial and quadratic characters over $\mathbb{F}_p$ respectively. This enables us to evaluate $|E(\mathbb{F}_p)|$ for some elliptic curves $E$, and prove the existence of isogenies between $E$ and Legendre elliptic curves over $\mathbb{F}_p$.

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