16 problems

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, and let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree at least…

The $4d-1$ bound conjecture. One can take

Morton–Silverman conjecture. There exists a constant $C=C(d,N,[k:{\mathbb Q}])>0$ such that every degree $d$ endomorphism $f$ of ${\mathbb P}^N$ over $k$ satisfies

Let $K/\mathbb{Q}$ be a quadratic field, let $f\in K[z]$ be a quadratic polynomial, and let $\operatorname{PrePer}(f,K)$ denote its set of $K$-rational preperiodic points. Doyle's…

Let $K$ be a number field, and let $S$ be a finite set of places of $K$ containing all archimedean places. Let $\varphi:\mathbb{P}_K^{1}\to\mathbb{P}_K^{1}$ be a nonconstant ration…

Let $K$ be a number field, let $S$ be a finite set of places containing all the archimedean ones, let $f\colon \mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree at leas…

Uniform boundedness conjecture. For each degree $d\geq 2$, there exists a constant $B=B(d)$ such that either

Let $D,N\ge 1$ and $d\ge 2$ be integers. For a number field $K$ with $[K:\mathbb{Q}]=D$ and a morphism $\phi:\mathbb{P}^N\to\mathbb{P}^N$ of degree $d$ defined over $K$, a point of…

Let $D\ge1$, $N\ge1$, and $d\ge2$. For a number field $K/\mathbb Q$, write $[K:\mathbb Q]$ for its degree, and let $\operatorname{End}_d^N(K)$ be the degree-$d$ endomorphisms of…

Let $K/\mathbb{Q}$ be a quadratic field and let $f\in K[z]$ be a quadratic polynomial. Let $\operatorname{PrePer}(f,K)$ be the set of $K$-rational preperiodic points of $f$, and le…

Let $V$ be an irreducible quasiprojective complex algebraic variety, and let $f:V\times\mathbb{P}^1\to\mathbb{P}^1$ be an algebraic family of rational maps of degree $d\geq 2$, wit…

Let $g(z)$ be any rational map, and for an integer $d$ define … For a number field $K$, let $\operatorname{Preper}(f_{d,g},K)$ denote the set of $K$-rational preperiodic points of…

For an integer $d\geq 2$ and $c\in\mathbb{Q}$, let $f_{d,c}(z)=z^d+c$. Generalized Poonen conjecture. For $n>3$, there is no $f_{d,c}$ defined over $\mathbb{Q}$ with a $\mathbb{Q}$…

For $c\in\mathbb{C}$, let $f_c(z)=z^2+c$, and let $S_{0,1}$ denote the set of parameters $c$ for which both $0$ and $1$ are preperiodic under iteration of $f_c$. Simultaneous prepe…

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field $K$, with weight $d$, dimension $\dim X$, and polarization degree $c_1(\mathscr L)^{\dim X}$. Uni…

Let $(X,f,\mathscr L)$ be a polarized dynamical system over an algebraically closed field of characteristic zero. Let $Y$ be an irreducible subvariety of $X$, and let $Y_0$ be the…