101 problems

Let $A$ be an abelian variety over an algebraically closed globally valued field $K$. If $K$ is non-archimedean, let $k$ be its constant field and let $(A^{K/k},\operatorname{tr})$…

Geometric Bogomolov conjecture. $X$ contains Zariski dense sets of small points if and only if $X$ is special.

Bogomolov conjecture. If $X_n$ is Zariski dense in $X$ for every $n\geq 1$, then $X$ is a torsion subvariety of $\mathbb{G}_a^g$. This is presented as a consequence of the proposed…

Let $K$ be a finitely generated field, let $A$ be the coefficient ring of a Drinfeld module, and let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module. For a point…

Let $A$ be an abelian variety defined over a number field $k$, endowed with an ample symmetric line bundle $\mathcal{L}$. Let $V$ be a proper $k$-irreducible subvariety of $A$ that…

Let $\Gamma$ be a finitely generated subgroup of $\overline{\mathbb{Q}}^*$ and let … be its division group. Rémond's generalized Lehmer conjecture. There is a positive constant $c$…

Let $K$) be a number field of degree $d=[K:{\mathbb Q}]\geq 1$, with discriminant $\Delta_K$, and let $\theta\in K$ be primitive when $K={\mathbb Q}(\theta)$. Let $h$ denote the ab…

Let $A$ be an abelian variety defined over a number field $K$, let $L$ be a symmetric ample line bundle on $A$, and let $h_L$ be the associated height on subvarieties of $A$, coinc…

For an elliptic curve $E$ over $\mathbb{Q}$, let $h(E)$ denote its Faltings height and let $N_E$ be its conductor. Height conjecture. There is a constant $\kappa$ such that for all…

Bogomolov conjecture. There should exist an $\epsilon>0$ such that $C(\epsilon)$ is finite.

Let $K$ be a number field and let $c>0$. Fix a type $\Phi=(w,(h^r)_{r\in\mathbb{Z}})$ of motives, with $h^r=\dim \operatorname{gr}_{dR}^r$, and choose $a,b$ such that $h^r=0$ unles…

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field, and let $Y$ be a subvariety of $X$. Let the maximal preperiodic subvarieties contained in $Y$ be…

Let $X$ be a smooth proper variety over a number field $k$, let $D$ be a normal crossings divisor, and let $A$ be an ample line bundle on $X$. Let $r$ be a positive integer and let…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic, let $g\geq 1$, and let $\{x_k\}\subset\mathbb{G}_a^g(K^{\operatorname{sep}})$ be a strict sequence…

Let $n\geq 1$ be an integer, and let $\boldsymbol{\alpha}\in{\mathbb G}_m^n$ have multiplicatively independent coordinates. The obstruction index…

Zhang's conjecture. If $Y$ is an integral closed subset of $V$ and $\hat{\rm h}_L(Y)=0$, then $Y$ is preperiodic.

Denominator conjecture. There are $k_\ell\in\mathbb Z$ for $1\leq \ell\leq 2c(G)-2$ such that

Let $K=\mathbb C(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety over $K$ containing no rational curve. Let $h_L$ be a Weil height attached to an a…

Let $K=k(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety of general type over $K$. A sequence of $K$-rational points in $X$ is generic if every inf…

Let $K=k(B)$, where $B$ is a smooth projective curve, let $X$ be a projective variety of general type over $K$, and choose a projective model and a relatively ample line bundle def…

Multi-indexed generalized Fermat's conjecture. If there is $I_1\geqslant I_0$ such that $Y_I(K)$ is finite for every $I\geqslant I_1$, then there exists $I_2\geqslant I_0$ such tha…

Generalized Fermat's conjecture. If there is $N_1\geqslant N_0$ such that $Y_N(K)$ is finite for every integer $N\geqslant N_1$, then there exists $N_2\geqslant N_0$ such that…

Let $G$ be a semiabelian variety over a globally valued field $K$, with torus part identified with $\mathbb{G}_m^t$, abelian quotient $A$, ample symmetric line bundle $M$ on $A$, a…

Let $G$ be a semiabelian variety over an algebraically closed globally valued field $K$, with abelian quotient $\pi:G\to A$. If $K$ is non-archimedean, let $k$ be its constant fiel…

Let $K$ be a number field, let $\mu$ be an adelic measure defined over $K$, let $h_\mu$ be its associated height on $\mathbb{P}^1(\overline{K})$, and let $\mathcal{L}(\mu)$ denote…