7 problems

Lang–Silverman conjecture. There exists a positive quantity $c_{23}=c_{23}(K,g)$ such that

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $V_X$ be the smallest translate of a subtorus containing $X$. Write $\widehat{\mu}^{\mathrm{ess}}(X)$ for…

Let $X\subset \mathbf{G}_m^N(\overline{\mathbf{Q}})$ be a variety, and let $U_X^{\mathbf{Q}}$ be the smallest union of torsion varieties containing $X$ that is defined over…

Let $\mathcal{A}\rightarrow C$ be an abelian scheme on a curve over a number field $K$ whose generic fiber is geometrically simple of dimension at least $2$. Let…

Strong Lang–Silverman conjecture. There exist positive quantities $c_{33}=c_{33}(K,g)$ and $c_{34}=c_{34}(K,g)$ such that either there is an abelian subvariety $B\subset A$, with…

Let $K$ be a number field, let $E/K$ be an elliptic curve, and let $P\in E(K)$. Let $L=3(O)$, where $O$ is the origin of $E$, and write $\widehat{h}(P)=\widehat{h}_{E,L}(P)$ for th…

Morton–Silverman canonical-height conjecture. There is a positive constant $M'=M'(d)>0$ such that, for every polynomial $\phi\in\mathbb{Q}[z]$ of degree $d$ and every point…