Related papers: Dynamical modular curves for quadratic polynomial …

Given a number field $K$ and a polynomial $f(z) \in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $\alpha \to \beta$ if and only…

We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic…

Given a number field $K$ and a polynomial $f(z) \in K[z]$, one can naturally construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points of $f$, with an edge $\alpha \to \beta$ if and only if $f(\alpha)…

Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over…

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\bold Q$, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we…

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action…

Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic…

To each quadratic number field $K$ and each quadratic polynomial $f$ with $K$-coefficients, one can associate a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, and whose edges reflect the…

The dynatomic modular curves parametrize polynomial maps together with a point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form $f_c(z)…

We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$-th Gleason polynomial $G_n\in\mathbb{Q}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an…

Consider the moduli space, $\mathcal{M}_{3},$ of cubic polynomials over $\mathbb{C}$, with a marked critical point. Let $\mathscr{S}_{k,n}$ be the set of all points in $\mathcal{M}_{3}$ for which the marked critical point is strictly…

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a…

Bounding the number of preperiodic points of quadratic polynomials with rational coefficients is one case of the Uniform Boundedness Conjecture in arithmetic dynamics. Here, we provide a general framework that may reduce finding periodic…

Fix $d \ge 2$ and a field $k$ such that $\mathrm{char}~k \nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^d+c$…

Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $\phi(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely…

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…

Let $f \in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an…

For every nonconstant polynomial $f\in\mathbb Q[x]$, let $\Phi_{4,f}$ denote the fourth dynatomic polynomial of $f$. We determine here the structure of the Galois group and the degrees of the irreducible factors of $\Phi_{4,f}$ for every…

We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this…

We prove here the smoothness and the irreducibility of the periodic dynatomic curves $ (c,z)\in \C^2$ such that $z$ is $n$-periodic for $z^d+c$, where $d\geq2$. We use the method provided by Xavier Buff and Tan Lei in \cite{BT} where they…