16 problems

Mazur's uniform Mordell–Lang conjecture. Let $g\ge 2$ be an integer. Then there exists a constant $c(g)\ge 1$ such that, for every smooth curve $C$ of genus $g$ defined over $F$, e…

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

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 $\mathcal{A}=(\mathbb{G}_{a}^{m},\Phi)$ 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 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…

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…