33 problems

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 $f$ be a quadratic polynomial defined over ${\mathbb Q}$, and let a rational periodic point have exact period $N$. Poonen's conjecture. Such a polynomial cannot have a rational…

Let $f,g$ be polynomials of the same degree $d\geq 2$, and let $\operatorname{Prep}(f)$ denote the set of preperiodic points of $f$. DeMarco–Krieger–Ye's conjecture. There should e…

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…

Baker–DeMarco's conjecture. The parameters $c\in\mathbb{C}$ for which both $0$ and $1$ are preperiodic for $f_c$ are exactly

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 $f$ be a quadratic rational map of degree $2$ defined over ${\mathbb Q}$ whose automorphism group is cyclic of order $2$, denoted $\mathfrak{C}_2$. The conjecture. Such a map c…

Quadratic portrait classification conjecture. Fix an integer $d\geq 1$. There exists a minimal finite set $\overline{\Gamma}=\overline{\Gamma}(d)$ of portraits such that for every…

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 $d\ge 2$ and let $\operatorname{Rat}_d(\mathbb{Q})$ be the set of rational maps of degree $d$ defined over $\mathbb{Q}$. For each $\phi\in\operatorname{Rat}_d(\mathbb{Q})$, let…

Let $\phi_c(z)=z^2+c$ be a quadratic polynomial map defined over $\mathbb{Q}$. Quadratic unicritical uniform boundedness conjecture. There exists a constant $C$ such that, for ever…

Let $d\geq 2$. For a rational function $f:\mathbb{P}^1\to\mathbb{P}^1$, let $\operatorname{Prep}(f)$ be the set of points $x\in\mathbb{P}^1(\overline{\mathbb{C}})$ such that…

For each $d\ge 2$, let … where $\vartheta_2=1/2$ and $\vartheta_d=2(d+1)/(5d+1)$ for $d\ge 3$. Le Boudec–Mavraki conjecture. The error term can be improved to … for some constant…

Let $D,N\ge 1$ and $d\ge 2$. For an algebraic variety $V$, write … Let $H$ be an absolute multiplicative height on $\operatorname{End}^N_d$, and let $\operatorname{Num}(A,X)$ count…

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…

Average preperiodic-point conjecture. There exists $\gamma_d>0$ such that

For $k\in\mathbb Q$, define the degree-four rational map … A rational periodic point has minimal period $n$ if $n$ is the least positive integer such that the $n$-fold iterate retu…

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…

Ingram–Silverman conjecture. If $F$ is a number field and $\alpha\in F$ is a wandering point for $\varphi$, then $\alpha$ realizes all but finitely many portraits for $\varphi$.

Consider a rational map of degree $2$ on $\mathbb{P}^1$ defined over a quadratic number field, and call a point preperiodic if its forward orbit is finite. Quadratic-field preperio…

Finiteness conjecture. The morphism $\varphi$ has only finitely many $\mathbb{F}_q((T))$-valued preperiodic points.

Benedetto et al.'s 14-point conjecture. One has

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…