9 problems

David's conjecture. For every $\varepsilon>0$, there exists a constant $c(A/K,\mathcal{L})>0$ such that

Amoroso–Zannier's conjecture. There exists a constant $c(K)>0$ such that

Let $K$ be a number field and let $E$ be an elliptic curve over $K$. Write $E_{\text{tor}}$ for the torsion subgroup of $E(\overline{\mathbb{Q}})$, and let $K(E_{\text{tor}})$ be t…

Amoroso–David's obstruction-index conjecture. There exists a real number $c(n)>0$ such that

Product conjecture. If $E$ and $F$ have the Bogomolov property, then the group product $E^*F^*$ also has the Bogomolov property.

Let $B$ be a smooth projective curve, let $A/K$ be an abelian variety with a model $f\colon\mathcal{A}\to B$, where $\mathcal{A}_K=A$. Let $D\subset A$ be a reduced effective ample…

Rémond's generalized Lehmer conjecture. The following assertions hold:

For each integer $d\ge 3$, let $f\in\mathbb{Z}[x]$ have degree $d$ and nonzero discriminant, and let $h(f)$ be the maximum of the logarithmic heights of the coefficients of $f$. Vo…

Let $E/\mathbb{Q}$ be an elliptic curve in minimal Weierstrass form, let $P \in E(\mathbb{Q})$ be an integral point of infinite order, let $\widehat{h}(P)$ denote its canonical hei…