39 problems

Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a degree-$d$ map over $\overline{\mathbb{Q}}$, and let $c$ range over its critical points, counted with multiplicity. Define the critical hei…

Let $f:X\to X$ be a morphism of a smooth projective variety over $\overline{\mathbb{Q}}$, let $D$ be an ample eigendivisor satisfying $f^*D\sim\delta D$ for some $\delta>1$, and le…

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 $E$ be an elliptic curve defined over a number field $K$. For a point $P\in E(\overline{K})$, let $\widehat{h}(P)$ denote the canonical height, and suppose that $P$ is non-tors…

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 $K$ be a global function field, let $\boldsymbol{\phi}$ be a Drinfeld module of rank $d$ over $K$, let $M_K$ be the set of places of $K$, and let $\boldsymbol{\phi}_Q$ denote t…

Minimum-height conjecture. These bounds are the correct values:

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 $\mathcal{M}_d^N$ 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 $\mathbb{F}$ 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…

Let $X$ be a projective variety, let $f:X\to X$ be a polarized endomorphism, and let $\overline{L}_f$ be the canonical adelic line bundle associated with a polarization. For a subv…

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

Let $\mathbf{K}$ be a function field of characteristic zero. Let $(X,f,L)$ be a non-isotrivial polarized endomorphism defined over $\mathbf{K}$, where $X$ is normal. Let…

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…

Under the hypotheses of Section 2, let $X$ be the underlying variety, let $\mathcal{F}$ be the given system of rational self-maps, let $X_{\mathcal{F}}(\overline{K})$ denote the po…

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