arXiv subjects

Mohammad Sadek

Publications and source records attributed to Mohammad Sadek.

At least 19 recordsLinked to original sources

arXiv2026

Rational Preperiodic Points of Quadratic Rational Maps over $\mathbb{Q}$ with Nonabelian Automorphism Groups

Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a quadratic rational map defined over the rational field $\mathbb{Q}$ with nonabelian automorphism group. We prove that no such map has a $\mathbb{Q}$-rational periodic point with exact period $N\ge 4$. We also give an explicit parametrization of such maps that have $\mathbb{Q}$-rational periodic points of period $1$, $2$, and $3$. In addition, we show that the number of $\mathbb{Q}$-rational preperiodic points of such a map $f$ cannot exceed $6$. As a result, we completely classify all portraits of $\mathbb{Q}$-rational preperiodic points for quadratic rational maps defined over $\mathbb{Q}$ with nonabelian automorphism showing that there are exactly $5$ such portraits.

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arXiv2025

Families of twists of tuples of hyperelliptic curves

Let $f \in \mathbb Q[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function $D\in\mathbb{Q}(v_1,v_2,v_3,v_4)$ such that the Jacobian of the quadratic twist of $y^2=f(x)$ and the Jacobian of the $m_i$-twist, respectively $2m_i$-twist, of $y^2=x^{m_i}+a_i^2$, $i=1,2,3$, by $D$ are all of positive Mordell-Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell-Weil rank.

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arXiv2024

On Torsion Subgroups of Elliptic Curves over Quartic, Quintic and Sextic Number Fields

The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree $d$, $d=4,5,6$, is not completely determined. However, the list of groups $\Phi^{\infty}(d)$, $d=4,5,6$, that can be realized as torsion subgroups for infinitely many non-isomorphic elliptic curves over these fields are known. We address the question of which torsion subgroups can arise over a given number field of degree $d$. In fact, given $G\in\Phi^{\infty}(d)$ and a number field $K$ of degree $d$, we give explicit criteria telling whether $G$ is realized finitely or infinitely often over $K$. We also give results on the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with torsion $G$. Finally, we give examples of number fields $K$ of degree $d$, $d=4,5,6$, over which the Mordell-Weil rank of elliptic curves with prescribed torsion is bounded from above.

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arXiv2024

Rational torsion on hyperelliptic jacobian varieties

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its Jacobian. Lepr\'{e}vost proved this conjecture with $\kappa=3$. In this work we prove that given an integer $N$ in the interval $[3g,4g+1]$, $g\ge 3$, satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order $N$. In particular, we establish the existence of such varieties for $N=4g+1$ when $g$ is odd and for $N=4g-1$ when $g$ is even. A few explicit applications of this result produce the first known infinite examples of torsion 13 when $g=3$, torsion 15 when $g=4$, and torsion $17,18,21$ when $g=5$. In fact, we show that infinitely many of the latter abelian varieties are absolutely simple.

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arXiv2024

Quadratic torsion orders on Jacobian varieties

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+ 2g -4$. For $N = 2g^{2} + 7g + 1$, we introduce a 1-parameter family of polynomials $f_{t}(x)$ of degree $2g+1$. For all but finitely many rational values of $t$, if the discriminant of $f_{t}(x)$ is nonzero, then the hyperelliptic curve defined by $y^{2} = f_{t}(x)$ has a rational point of order $N$ on its Jacobian.

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arXiv2024

Explicit construction of decomposable Jacobians

In this note we give explicit constructions of decomposable hyperelliptic Jacobian varieties over fields of characteristic $0$. These include hyperelliptic Jacobian varieties that are isogenous to a product of two absolutely simple hyperelliptic Jacobian varieties, a square of a hyperelliptic Jacobian variety, and a product of four hyperelliptic Jacobian varieties three of which are of the same dimension. As an application, we produce families of hyperelliptic curves with infinitely many quadratic twists having at least two rational non-Weierstrass points; and families of quadruples of hyperelliptic curves together with infinitely many square-free $d$ such that the quadratic twists of each of the curves by $d$ possess at least one rational non-Weierstrass point.

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arXiv2024

Arithmetic progressions in polynomial orbits

Let $f$ be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit $\operatorname{Orb}_f(t)=\{t,f(t),f(f(t)),\cdots\}$, where $t$ is an integer, using arithmetic progressions each of which contains $t$. Fixing an integer $k\ge 2$, we prove that it is impossible to cover $\operatorname{Orb}_f(t)$ using $k$ such arithmetic progressions unless $\operatorname{Orb}_f(t)$ is contained in one of these progressions. In fact, we show that the relative density of terms covered by $k$ such arithmetic progressions in $\operatorname{Orb}_f(t)$ is uniformly bounded from above by a bound that depends solely on $k$. In addition, the latter relative density can be made as close as desired to $1$ by an appropriate choice of $k$ arithmetic progressions containing $t$ if $k$ is allowed to be large enough.

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arXiv2023

A Dynamical Analogue of a Question of Fermat

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter quadratic polynomials with orbits containing three consecutive squares. In addition, we show that there exists at least one polynomial of the form $x^2+c$ with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.

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arXiv2023

Genus two curves with everywhere good reduction over quadratic fields

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic number field and $C$ is a genus $2$ curve with everywhere good reduction over $K$. We provide the first infinite sequence of pairs $(K,C)$ where $K$ is a real (complex) quadratic field and $C$ has everywhere good reduction over $K$. Moreover, we show that the Jacobian of $C$ is an absolutely simple abelian variety.

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arXiv2023

Genus 2 curves with bad reduction at one odd prime

In this article we consider smooth projective curves $C$ of genus two described by integral equations of the form $y^2=xh(x)$, where $h(x)\in\mathbb{Z}[x]$ is monic of degree $4$. It turns out that if $h(x)$ is reducible, then the absolute discriminant of $C$ can never be an odd prime, except when $h(x)=(x-b)g(x)$ and $g(x)$ is irreducible. In this case we obtain a complete description of such genus $2$ curves. In fact, we prove that there are two one-parameter families $C_t^i$, $i=1,2$, of such curves such that if $C$ is a genus two curve with an odd prime absolute discriminant, then $C$ is $C_t^i$, for some $i$, and $t\in\mathbb{Z}$. Moreover, we show that $C_t^i$ has an odd prime absolute discriminant, $p$, if and only if a certain degree-$4$ irreducible polynomial $f^i(t)\in\mathbb{Z}[t]$ takes the value $p$ at $t$. Hence there are conjecturally infinitely many such curves. When $h(x)$ is irreducible, we give explicit examples of one-parameter families of genus $2$ curves $C_t$ such that $C_t$ has an odd prime absolute discriminant for conjecturally infinitely many integer values $t$.

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arXiv2023

Eventual Stability of pure polynomials over the rational field

A polynomial with rational coefficients is said to be pure with respect to a rational prime $p$ if its Newton polygon has one slope. In this article, we prove that the number of irreducible factors of the $n$-th iterate of a pure polynomial over the rational field $\mathbb Q$ is bounded independent of $n$. In other words, we show that pure polynomials are {\em eventually stable}. Consequently, several eventual stability results available in literature follow; including the eventual stability of the polynomial $x^d+c\in\mathbb{Q}[x]$, where $c\ne 0,1$, is not a reciprocal of an integer. In addition, we establish the dynamical irreducibility, i.e., the irreducibility of all iterates, of a subfamily of pure polynomials, namely Dumas polynomials with respect to a rational prime $p$ under a mild condition on the degree. This provides iterative techniques to produce irreducible polynomials in $\mathbb{Q}[x]$ by composing pure polynomials of different degrees. During the course of this work, we characterize all polynomials whose degrees are large enough that are not pure, yet they possess pure iterates. This implies the existence of polynomials whose shifts are all dynamically irreducible in $\mathbb{Z}[x]$.

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arXiv2023

Divisibility of orders of reductions of elliptic curves

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set of primes $p$ of density $1$ is determined by the torsion subgroups of elliptic curves that are $\mathbb Q$-isogenous to $E$. In this work, we give explicit families of elliptic curves $E$ over $\mathbb Q$ together with integers $m_E$ such that the congruence class of $|\widetilde{E}_p(\mathbb{F}_p)|$ modulo $m_E$ can be computed explicitly. In addition, we can estimate the density of primes $p$ for which each congruence class occurs. These include elliptic curves over $\mathbb Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $\mathbb Q$-isogeny class of $E$. We also exhibit elliptic curves over $\mathbb Q(t)$ for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than $1$ are divisible by given small integers.

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arXiv2022

Construction of Polynomials with prescribed divisibility conditions on the critical orbit

We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rational field $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primes that can appear as primitive prime divisors of $f_{d,c}^n(0)$ for some $c\in\Q$ is positive. In fact, under certain assumptions, we explicitly calculate the latter density when $d=2$. Furthermore, fixing $d,n\ge 2$, we show that for a given integer $N>0$, there is $c\in \Q$ such that $\f^n(0)$ has at least $N$ primitive prime divisors each of which is appearing up to any predetermined power. This shows that there is no uniform upper bound on the number of primitive prime divisors in the critical orbit of $\f(x)$ that does not depend on $c$. The developed results provide a method to construct polynomials of the form $\f(x)$ for which the splitting field of the $m$-th iteration, $m\ge1$, has Galois group of maximal possible order. During the course of this work, we give explicit new results on post-critically finite polynomials $\f(x)$ over local fields.

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arXiv2022

Rational points on quadratic elliptic surfaces

We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we prove that the Mordell-Weil rank of each such elliptic surface is at most $6$ over $\mathbb Q$. In fact, we show that the Mordell-Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over $\mathbb Q$.

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arXiv2022

Simultaneous Rational Periodic Points of Degree-2 Rational Maps

Let $S$ be the collection of quadratic polynomial maps, and degree $2$-rational maps whose automorphism groups are isomorphic to $C_2$ defined over the rational field. Assuming standard conjectures of Poonen and Manes on the period length of a periodic point under the action of a map in $S$, we give a complete description of triples $(f_1,f_2,p)$ such that $p$ is a rational periodic point for both $f_i\in S$, $i=1,2$. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two nonzero rational numbers $a, b $ are periodic points of the map $ϕ_{t_1,t_2}(z)=t_1 z + t_2/z$ for infinitely many nonzero rational pairs $(t_1, t_2)$ if and only if $a^2 = b^2$.

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arXiv2022

Divisibility by 2 on quartic models of elliptic curves and rational Diophantine $D(q)$-quintuples

Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This provides an analogue to the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that although the rational $ D(16t+9) $-quadruple $\{t, 16t+8,2 25t+14, 36t+20 \}$ can not be extended to a polynomial $ D(16t+9) $-quintuple using a linear polynomial, there are infinitely many rational values of $t$ for which the aforementioned rational $ D(16t+9) $-quadruple can be extended to a rational $ D(16t+9) $-quintuple. Moreover, these infinitely many values of $t$ are parametrized by the rational points on a certain elliptic curve of positive Mordell-Weil rank.

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arXiv2022

Moments of Gaussian hypergeometric functions over finite fields

We prove explicit formulas for certain first and second moment sums of families of Gaussian hypergeometric functions $_{n+1}F_n$, $n\ge1$, over finite fields with $q$ elements where $q$ is an odd prime. This enables us to find an estimate for the value $_6F_5(1)$. In addition, we evaluate certain second moments of traces of the family of Clausen elliptic curves in terms of the value $_3F_2(-1)$. These formulas also allow us to express the product of certain $_2F_1$ and $_{n+1}F_n$ functions in terms of finite field Appell series which generalizes current formulas for products of $_2F_1$ functions. We finally give closed form expressions for sums of Gaussian hypergeometric functions defined using different multiplicative characters.

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arXiv2020

Families of polynomials of every degree with no rational preperiodic points

Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb Q$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no rational preperiodic points for any $t\in K$.

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