13 problems

Let $K\subset\overline{\mathbb{Q}}\subset\mathbb{C}$ be a number field, let $S$ be an irreducible algebraic variety over $K$, and let $\pi:A\rightarrow S$ be a family of abelian va…

Relative Lehmer conjecture for elliptic curves. There exists a constant $c(E/K)>0$ such that

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…

Adelic obstruction-index conjecture. There exists a constant $c(n)>0$ such that

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.

Néron–Tate height conjecture for families. One may conjecture that (i) $\widehat{h}_{\iota}(X)<\infty$ for every subvariety $X\subseteq A$; (ii) the limsup in the definition of…

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…

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…

Lang–Silverman conjecture, version 2. There exist positive constants $c_1=c_1(d,g)$ and $c_2=c_2(d,g)$ such that either there is a proper subabelian variety $B\subset A$ with…