19 problems

Pink's conjecture. The intersection

Let $K$ be a field complete with respect to a non-archimedean absolute value, let $A/K$ be a semiabelian variety, and let $X \subset A$ be a closed subvariety. Tate–Voloch conjectu…

Let $K$ be a number field, and let $G$ be either $\mathbb{G}_m^t$ for some $t\in\mathbb{Z}_{\geq 0}$ or an abelian variety over $K$. Let $\Gamma\subset G(\overline{K})$ be a subgro…

Chai's additivity conjecture. Assume that $T$ is a torus and that $A$ is an Abelian variety. Then

Poonen's Mordell–Lang–Bogomolov conjecture. For $i \geq 1$, suppose

Geometric divisibility sequence conjecture. The geometric divisibility sequence $(D_{n\mathcal{P}})_{n\geq 1}$ corresponding to $\mathcal{P}$ satisfies

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 $G$ be a semiabelian variety over $K$, and let $\Gamma\subset G(\overline{K})$ be a subgroup of finite rank. For a subvariety $V\subset G$, write…

Relative compactification conjecture. For some $n_0\in{\mathbf N}$, $P_{n_0}$ is a relative compactification of ${\mathcal G}$ and

Let $G$ be a semiabelian variety defined over $\overline{\mathbb{Q}}$ and let $X\subseteq G$ be an algebraic subvariety. For an algebraic subgroup $H\subseteq G$, write…

Let $G$ be a semiabelian variety of dimension $g+t$, let $u$ be a place of the number field $K$, and let $(xi)in G^mathbb{N}$ be a sequence of closed points. The sequence is **gen…

Let $X/{\mathbb F}_q$ be a semiabelian variety and let $Y\subset X$ be a closed subvariety. Let $Z$ be the union of all translates of positive-dimensional semiabelian varieties…

Bounded height conjecture. The height $h_L$ is bounded from above on

Let $K$ be an algebraically closed field extending $k_0$, and let $B=\mathbb{B}(K)$ be a semiabelian variety defined over a field $k_0$ of characteristic zero. For a subvariety…

Let $L_0$ be a field finitely generated over ${\mathbb F}_p$, let $L$ be an algebraic closure of $L_0$, and let $L_0^\text{per}$ be the perfect closure of $L_0$ in $L$. Let $A$ be…

Let $L$ be an algebraically closed field of positive characteristic. Let $A$ be a semiabelian variety over $L$, let $X\subset A$ be an irreducible subvariety, and let…

Pink–Zilber conjecture. The intersection $V\cap\mathcal{S}^{[2]}$ is finite.

Let $U$ be a smooth complex variety, and define the space of Beilinson-Hodge cycles by … Beilinson-Hodge conjecture. The regulator map induces a surjection … This is the restricted…