14 problems

Growth-rate conjecture. The correct order of magnitude in $r$ for $\widetilde A(n,r)$ should be of the shape

Unlikely intersection bound. For every $\varepsilon>0$, there exists $c=c(n,\mathcal{V},\varepsilon,K)\in\mathbb{R}$ such that, for every flat subgroup scheme…

Let $\mathbb{K}$ be a number field, let $\Gamma\leq\mathbb{K}^{\times}$ be a non-torsion finitely generated subgroup, and define its divisible hull by … Put…

Schinzel's conjecture. There exists a nonzero vector ${\bf b}\in\mathbb{Z}^n$ orthogonal to ${\bf a}$ such that

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

Let $F$ be a field, and write $F^\times$ for its multiplicative group. A group is indecomposable if it cannot be expressed as a nontrivial direct product of groups. Nonexistence co…

There exist constants $\varepsilon_0>0$ and $A$ such that, for every prime $p$, every multiplicative subgroup $\mathcal G\subseteq\overline{\mathbb F}_p$ with…

Let $\bar{\mathbb{R}}$ be the real field, let $G$ be a subgroup of $\mathbb{C}^{\times}$, and let $S_\alpha$ denote the structure used in $(\bar{\mathbb{R}},S_\alpha)$. Write…

Let $g_1,\dots,g_n\in\overline{\mathbb{Q}}^\times$ and let $\Gamma=\langle g_1,\dots,g_n\rangle^{\mathrm{div}}$ be a finite-rank divisible subgroup of…

Let ${\mathbb G}_{\rm m}^2$ be the two-dimensional multiplicative algebraic group over ${\mathbb C}$, and call a point torsion if both coordinates are roots of unity. A torsion cos…

Let $X=\{a_1,\ldots,a_m\}$ be a finite set of algebraic numbers, and let $K$ be a number field containing all $a_i$. An $X$-base Fibonacci-Wieferich prime is a prime ideal…

Bogomolov conjecture. If, for each $\epsilon>0$, the set $V(\overline{\mathbb Q})\cap S_{\epsilon}$ is Zariski dense in $V$, then $V$ is a torsion translate of an algebraic subgrou…

Let $g(x)$ be a polynomial over the rationals of degree $d$, and let $G$ be a complex multiplicative group of rank $r$. Uniform boundedness conjecture. There is a function $b(d,r)$…

Distribution conjecture. For each integer $p$ with $|p|>1$,