12 problems

Schneider–Lang–Ramachandra's exponential conjecture. Under these assumptions, the determinant of $M$ is nonzero.

The torus analogue of Mazur's conjecture. The connected component of the identity of the closure of the image of $F$ in $\Gamma\backslash S({\mathbb R})$ equals the connected compo…

Let $Y$ be a rigid affine algebraic variety over ${\mathbb K}$. The connected component $\operatorname{Aut}^0(Y)$ is an algebraic torus of rank at most $\dim Y$. Perepechko–Zaidenb…

Malle-type conjecture for algebraic tori. For every $n\geq 1$ and every finite subgroup $1\neq H\leq\operatorname{GL}_n(\mathbb{Z})$, there are positive integers $a(H),b(H)$ and a…

Diminishing-action conjecture. If

Let $V\subseteq (\mathbb{C}^{\times})^n$ be an algebraic variety. Atypical subvarieties are the components of intersections with algebraic subtori having larger-than-expected dimen…

Growth conjecture.

Let $f\in\mathbb{C}[X_1,\ldots,X_n]$, let $Z(f)\subset\mathbb{G}_{\mathrm{m}}^n$ be its zero set, and let $\Delta$ be the Newton polytope of $f$. An isolated torsion point of…

Let $\mathcal{Y}$ be an irreducible subvariety of ${\bf G}_m^n$ defined over ${\bf C}$. For algebraic subgroups $H\subset {\bf G}_m^n$ satisfying … consider the union of the inters…

Let ${\mathbb G}_{\rm m}^N=(\overline{\mathbb Q}^{\times})^N$ be an algebraic torus, and let $W$ be a subvariety of ${\mathbb G}_{\rm m}^N$. For an algebraic subgroup, call it prop…

Let $K$ be a totally real number field and let $T$ be a torus over $K$ that is split by a totally real field. Let $X(T)=\operatorname{Hom}(\mathbb{G}_m,T)$ be the cocharacter group…

Let $K$ be the function field in the preceding setting, and let $T=T_0\times_{\mathbb F_q}K$, where $T_0$ is a fixed $\mathbb F_q$-torus. Put $d=\dim T$,…