4 problems

Uniform $S$-unit orbit conjecture. There should exist a uniform bound, depending only on $K$, $S$, and $d$, for the number of $S$-units occurring in an orbit of any rational map of…

Let $B$ be a nonempty open subscheme of $\operatorname{Spec}(\mathbb Z)$, let $X\to B$ be a smooth, proper, geometrically connected genus-one curve, and let $Y=X-D$ where…

Let $a$ define the Mordell elliptic curve $E_a$, let $r$ be the rank of the free part of $E_a(\mathbb Q)$, and let $Y_a(\mathcal O)$ be the set of solutions of the corresponding Mo…

Mahler's conjecture. There are only finitely many points $P\in E(K)$ such that $x(P)\in\mathcal{O}_{KS}$.