24 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

Fix integers $N\geq1$, $d\geq2$, and $D\geq1$. Dynamical uniform boundedness conjecture. There is a positive integer $C(N,d,D)$ such that for every degree-$d$ morphism…

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…

Strong uniform boundedness conjecture. For fixed $A$, $r \ge 1$, and $d \ge 1$, there is a uniform bound on $\#({}^{\phi}L)_{\operatorname{tors}}$ as $L$ ranges over extensions of…

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

Geometric Uniform Boundedness Conjecture. Given integers $D\geq 1$, $N\geq 1$, and $d\geq 2$, there exists $C=C(d,N,D)>0$ such that if $f$ is a degree $d$ non-isotrivial endomorphi…

Uniform Boundedness for iterated preimages. For each $D\geq 1$, there exist $C=C(f,a,D)>0$ and a proper subvariety $V_{f,a}\subset\Lambda\times{\mathbb P}^N$ over…

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

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{C}$, with double covers … such that the origin of each $E_j$ is a ramification point of $\pi_j$. Define … where…

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 $K/\mathbb Q$, let $\mathcal M_d^N{\mathcal P}$ denote the $K$-rational points of the moduli space associated to an unweighted…

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…

Conjecture Br(AV). There is a constant $C=C(d,g)$ such that, if $A$ is an abelian variety of dimension $g$ defined over a number field of degree $d$, then

Conjecture Br(K3). There is a constant $C=C(d)$ such that, if $X$ is a K3 surface defined over a number field of degree $d$, then

Coleman's discriminant conjecture. The quantity $\operatorname{discr}(A)$ is uniformly bounded for abelian varieties $A$ of bounded dimension defined over number fields of bounded…

Let $D\geq 1$, $N\geq 1$, and $d\geq 2$. For a number field $K$ and a preperiodic portrait ${\mathcal P}$, let $\mathcal{M}_d^N[{\mathcal P}]$ denote the dynamical moduli space cla…

Let $K$ be a number field, let $D=[K:\mathbb{Q}]$, and let $\Phi\colon\mathbb{P}^N\to\mathbb{P}^N$ be a morphism of degree $d\geq 2$ defined over $K$. A point of $\mathbb{P}^N(K)$…

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 $k$ be a number field of degree $d$, let $g\geq 1$, and let $A/k$ be an abelian variety of dimension $g$. Strong torsion conjecture. There is a positive constant $c(k,g)$, depe…

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…