27 problems

Generalized Lehmer conjecture. There exists a positive constant $c=c(A,K,\mathcal L)$ such that

Let $K$ be a number field, and let $A/K$ be an abelian variety. The discreteness conjecture for abelian varieties. 1. $A(K^{\rm ab})$ has the discreteness property. 2. If $A$ does…

Let $d\geq 2$, let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a degree-$d$ dynamical system, let $h_{\mathcal{M}}$ be a height on the moduli space $\mathcal{M}_d$ of degree-$d$ dynamical s…

Let $E/\mathbb{Q}$ be an elliptic curve with $j$-invariant $j_E$ and minimal discriminant $\Delta_E$. Let $\hat h$ denote the canonical height and $h$ the relevant height on ration…

Let $\mathcal{M}_d^N$ be the moduli space of degree-$d$ endomorphisms of $\mathbb{P}^N$, let $h_{\mathcal{M}}$ be an ample moduli height, and let $\widehat h^{\mathrm{crit}}$ be th…

Let $K/\mathbb{Q}$ be a number field, let $\mathcal{M}_d^N$ be the moduli space of degree-$d$ endomorphisms of $\mathbb{P}^N$, and let $h_{\mathcal{M}}$ be an ample height on…

Let $X/\overline{\mathbb{Q}}$ be a smooth projective variety, let $f:X\to X$ be a dominant morphism with $\delta(f)>1$, and define … with the infimum of the empty set defined as…

Let $\mathbb{F}$ be a number field and let $(X_{\mathbb{F}},D_{\mathbb{F}})$ be a log pair such that $pm K_{(X_{\mathbb{F}},\Delta_{\mathbb{F}})}$ is ample. A polarized model…

Silverman's conjecture. For every $a\in\mathbb{P}^1(\overline{\mathbb{Q}})$, $\hat{h}_f(a)$ is either $0$ or transcendental.

Let $A\to T^0$ be a non-isotrivial family of abelian varieties over a number field $K$, with generic fiber $A_{\eta}$ simple of dimension at least two. A section $P:T^0\to A$ is no…

The sextic-twist height conjecture. For fixed $\epsilon>0$, the set of $d\in S_6(X)\setminus\{0\}$ such that

For integers $A,B$ with $4A^3+27B^2\neq 0$, consider the quadratic twists … for positive square-free integers $d$, and let $\eta_d(A,B)$ be defined by … with $\eta_d(A,B)=+\infty$…

Let $A$ be an abelian variety defined over a number field $K$, let $\mathcal{L}$ be a symmetric ample line bundle on $A/K$, and let $L/K$ be a finite extension. For the Néron–Tate…

Let $\mathsf{M}_d^N$ be the moduli space of endomorphisms of $\mathbb{P}^N$, let $\mathsf{P}_d^N\subseteq\mathsf{M}_d^N$ be the moduli space of regular polynomial endomorphisms, an…

Le Boudec's conjecture. For every $\alpha>0$, there exists a constant $c_{A,B}(\alpha)>0$ such that

Positive lower ample canonical height conjecture. If $O_f(x)$ is dense in the Zariski topology, then $\underline h_f(x)>0$. The source describes this as a weaker conjecture and as…

Let $X$ be a smooth projective variety over $\overline{\mathbb Q}$, let $f$ be an endomorphism of $X$ with $\delta_f>1$, and let $K$ be a number field. Write $Z_f(K)$ for the set o…

Zhang's conjecture. The asserted finiteness holds for every such family, line bundle, and non-torsion section.

Let $A$ be an abelian variety and let $\lambda:A\dashrightarrow\mathbb{P}^1$ be a rational function, both defined over a number field $K$. Let $h$ be the standard height on…

Average canonical height conjecture. For any $P\in E(K)$,

Let $K$ be the global function field in the paper's setting. For every finite extension $L/K$, every integer $r\geq 1$, every Drinfeld module $\phi/L$ of rank $r$, and every non-to…

Let $f:{\mathbf{A}}^2\to{\mathbf{A}}^2$ be a dominant polynomial mapping over $\overline{{\mathbf{Q}}}$ with first dynamical degree $\lambda_1>1$ and second dynamical degree…

Let $\varphi:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ be a dominant rational map defined over $\overline{\mathbb{Q}}$, with dynamical degree $\delta_\varphi>1$. Let $\ell_\varphi$…

Let $g\geq 1$, let $k$ be a number field, let $A/k$ be an abelian variety of dimension $g$, let $\mathcal{D}\in\operatorname{Div}(A)$ be an ample divisor, and let $P\in A(k)$ such…

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field, and let $Y$ be a subvariety of $X$ with canonical subvariety height $\hat h_{\mathscr L}(Y)$. A…