22 problems

Let $\theta_n$, for $n\geq 2$, be the unique root in $(0,1)$ of $-1+x+x^n$, and call a real algebraic integer greater than $1$ a Salem number when it has the usual Salem-number pro…

Let $f:X\to X$ be a morphism of a smooth projective variety over $\overline{\mathbb{Q}}$, let $D$ be an ample eigendivisor satisfying $f^*D\sim\delta D$ for some $\delta>1$, and le…

David-Hindry's conjecture. For every integer $n\in\mathbb{N}$, there is a constant $c(A/K,\mathcal{L},n)>0$ such that, for every $n$-tuple $(P_1,\ldots,P_n)$ of points of infinite…

Let $E$ be an elliptic curve defined over a number field $K$. For a point $P\in E(\overline{K})$, let $\widehat{h}(P)$ denote the canonical height, and suppose that $P$ is non-tors…

Let $A$ be an abelian variety defined over a number field $k$, endowed with an ample symmetric line bundle $\mathcal{L}$. Let $V$ be a proper $k$-irreducible subvariety of $A$ that…

Let $n$ be a positive integer, let $W$ be a geometrically irreducible subvariety of $bmathbb{G}_m^n$ that is not contained in any torsion subvariety, and let $bmathbb{L}$ be an abe…

Let $n\geq 1$ be an integer, and let $\boldsymbol{\alpha}\in{\mathbb G}_m^n$ have multiplicatively independent coordinates. The obstruction index…

Abelian Lehmer problem. There is a constant $c(A/K,\mathcal{L})>0$ such that, for every point $P\in A(\overline{K})$ of infinite order modulo every proper abelian subvariety of $A$…

Let $A$ be an abelian variety of CM type over a number field $k$, endowed with an ample symmetric line bundle $\mathcal{L}$, and let $V$ be a proper $k$-irreducible algebraic subva…

For an integer $n\geqslant2$, let $\mathbb{Z}_{\!/2}^n$ denote the direct sum of $n$ copies of the cyclic group $\mathbb{Z}_{\!/2}$. Let $\lambda(\mathbb{Z}_{\!/2}^n)$ denote its L…

For an integer $n\geqslant2$, let $\rho(n)$ denote the smallest prime number that does not divide $n$. For the cyclic group $\mathbb{Z}_{\!/n}$, let $\lambda(\mathbb{Z}_{\!/n})$ de…

Dynamical strong Lehmer conjecture. If $4\beta<\tau_{10}4$ is a Perron number, then $4\beta^n4$ is a Parry number for only finitely many integers $4n4$.

Strong Lehmer conjecture. The optimal lower-bound constant is $c=\tau_{10}$.

Let $M(\alpha)$ denote the Mahler measure of an algebraic number, and let a Salem number be a real algebraic integer greater than $1$ whose conjugates lie in the closed unit disk,…

Generalized Lehmer conjecture. For every torsionfree group $G$, all four constants satisfy

Let $k$ be a number field, let $\alpha\in\overline{\mathbb{Q}}^\times$, and write … Here $k^{\mathrm{ab}}$ denotes the maximal abelian extension of $k$. Relative Lehmer conjecture.…

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 $\mathbb{D}=(\mathbb{G}_{a},\Phi)$ be a Drinfeld module, and let $\mathbb{D}(\overline{k})_{NT}$ denote the set of its non-torsion points. For…

Relative Lehmer conjecture. There is a real number $c>0$ such that every point $x\in G(\overline{\mathbf Q})$ not contained in any proper algebraic subgroup of $G$ satisfies

David's Lehmer conjecture. There is a real number $c>0$ such that every point $x\in G(\overline{\mathbf Q})$ not contained in any proper algebraic subgroup of $G$ satisfies

Let $\mathcal{U}_{m,p}$ be the completion of the vector space $\mathcal{U}$ of algebraic units with respect to the Mahler $p$-norm, and let … be the additive subgroup generated by…

Let $(X,f,\mathscr L)$ be a polarized dynamical system over a number field $K$, with canonical height $\hat h_{\mathscr L}$, and let $x\in X(\overline{\mathbb Q})$ be a non-preperi…