12 problems

Uniform boundedness conjecture. For any complex function field $L$ and any integer $g\geq 2$ there exists a number $B_g(L)$ such that for every $C\in{f{\mathcal C}_g(L)}$ one has

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 $\operatorname{Per}(\varphi,\mathbb{P}^1(K))$ denote the set of periodic points of a morphism $\varphi:\mathbb{P}^1\to\mathbb{P}^1$ defined over a number field $K$. Periodic-po…

Let $\varphi:\mathbb{P}^1(\mathbb{Q})\to\mathbb{P}^1(\mathbb{Q})$ be a morphism of degree $d$, and let $\operatorname{Per}(\varphi,\mathbb{P}^1(\mathbb{Q}))$ denote its set of peri…

Let $n\geq 1$ and $d\geq 2$ be integers. Let $K/\mathbb{Q}$ be a number field of degree at most $n$, and let $\varphi:\mathbb{P}^1(K)\to\mathbb{P}^1(K)$ be a morphism of degree $d$…

Fix $q$, $r\geq 1$, and $d\geq 1$. Let $L$ range over extensions of $F$ of degree at most $d$, and let $\phi$ range over rank-$r$ Drinfeld $A$-modules over $L$. Poonen's Drinfeld-m…

Let $k$ be a field, let $K$ be the function field of an integral curve over $k$, and let $\operatorname{Rat}_d(\overline K)$ denote the degree-$d$ rational functions over an algebr…

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 $K$ be a number field, let $d$ be an integer, and let $S=\{\phi_1,\phi_2,\dots,\phi_s\}$ be a set of rational maps defined over $K$ such that $2\leq\deg(\phi)\leq d$ for all…

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…

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…