8 problems

Let $K$ be a number field, and let $G$ be either $\mathbb{G}_m^t$ for some $t\in\mathbb{Z}_{\geq 0}$ or an abelian variety over $K$. Let $\Gamma\subset G(\overline{K})$ be a subgro…

Let $K$ be a finitely generated field, let $A$ be the coefficient ring of a Drinfeld module, and let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module. For a point…

Let $\Gamma$ be a finitely generated subgroup of $\overline{\mathbb{Q}}^*$ and let … be its division group. Rémond's generalized Lehmer conjecture. There is a positive constant $c$…

Weak Lehmer conjecture. For each $s\in\mathbb N$, there exists $\varepsilon(s)>0$ such that for any irreducible monic polynomial $P$ with integer coefficients and $s(P)\leq s$, eit…

Generalized Lehmer conjecture. There exists a positive constant $c=c(A,K,\mathcal L)$ such that

Bounded-coefficient limit-ratio conjecture. If the limit ratio of such a sequence is strictly below some constant $c(N)>0$, then its limit ratio is equal to $0$.

Let $F$ be a number field, let $L_1$ be the logarithmic image of $\mathcal{O}_F^*$, and for each $k$ define $\mu_{1,k}(F)$ as the minimum of the $L^1$-norms of wedges…

Let $F$ be a number field of degree $n$, let $I$ be the set of embeddings $F\to\mathbb{C}$, and let $V$ be the real vector space with basis $[\sigma]$ for $\sigma\in I$. Define the…