22 problems

Let $X$ be a quasi-projective variety defined over $\mathbb{C}$, let $\Phi$ be an endomorphism of $X$, let $V\subset X$ be a closed subvariety, and let $x\in X(\mathbb{C})$. Dynami…

Mordell–Lang conjecture. The Zariski closure of $S_A(K)$ and $S_B(K)$ consists of a finite union of elliptic curves and points.

Let $A$) be an abelian variety over $\bar{Q}$ equipped with a Néron–Tate height $chat h$, let $Gamma$ be a finitely generated subgroup of $A(\bar{Q})$, and define its division grou…

Let $g$ be a positive integer, and let $\phi_1:A\to K\{\tau\},\dots,\phi_g:A\to K\{\tau\}$ be Drinfeld modules. An algebraic $(\phi_1,\dots,\phi_g)$-submodule of $\mathbb{G}_a^g$ i…

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

Let $k$ be a number field, let $A/k$ be an abelian variety, and let $X$ be a non-torsion subvariety of $A$, meaning that $X$ is not the translate of an abelian subvariety by a tors…

Pink–Mordell–Lang conjecture. If $Z$ is not contained in any translation of a proper algebraic subgroup of $G$, then $Z\cap(\Gamma+G^{[>d]})$ is not Zariski dense in $Z$.

Let $S$ be a Shimura variety, let $obreakboldsymbol{Sigma} obreakto S$ be a structure of finite rank, and call a weakly special subvariety $obreakboldsymbol{Sigma} obreak$-specia…

Let $S$ be a Shimura variety. For a subset $obreakboldsymbol{Xi} obreakto S$, let $obreakboldsymbol{bar{Xi}} obreak$ be the union in $S$ of the generalised Hecke orbits of the po…

Let $p$ be a prime, let $k$ be a finite field of characteristic $p$, let $K/k$ be a regular extension with algebraic closure $K\subseteq\overline K$, and let $A$ be a $K$-abelian v…

Let $K/K_0$ be an extension of algebraically closed fields, let $A/K$ be an abelian variety, let $X\subseteq A$ be a subvariety, and let $\Lambda\subseteq A(K)$ be a subgroup of fi…

Let $G$ be a finite-rank subgroup of a semiabelian complex variety, equipped with the first-order structure induced from the complex field. Mordell–Lang conjecture. This induced fi…

Uniform dynamical Mordell--Lang conjecture. The set of pairs $(n,m)$ such that

Let $\mathcal{A}=(\mathbb{G}_{a}^{m},\Phi)$ be a uniformizable $T$-module such that there exists $i\in\mathbb{N}\setminus\{0\}$ for which the leading coefficient matrix of…

Generic Denis–Mordell–Lang conjecture.

Denis–Mordell–Lang conjecture. There exists an $A$-module $G ot i H$ such that

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…

Let $d\ge 1$, $g\ge 2$, and $r\ge 0$. Let $K$ be a number field of degree $d$, and let $C$ be a curve over $K$ of genus $g$ with Jacobian $J$. Uniform Mordell–Lang conjecture for r…

Let $g\ge 2$ and $r\ge 0$. For a curve $C$ over $\mathbb{C}$ of genus $g$, let $J$ be its Jacobian and let $\iota\colon C\to J$ be an embedding. Let $\Gamma\subset J(\mathbb{C})$ b…

Let $K$ be the function field and $\mathbf A_K$ its adele ring. Let $X\subseteq {\mathbb G}_a^g$ be a subvariety and let $\Gamma\subseteq K^g$ be a finitely generated $\Phi(R)$-mod…

Let $K$ be a field extension of ${\mathbb F}_p(t)$, let $\Phi$ be a Drinfeld module of generic characteristic over $K$, let $X\subseteq {\mathbb G}_a^g$ be a subvariety, and let…