42 problems

Let $k$ be a field and let $T$ be an algebraic $k$-torus. The torus $T$ is stably $k$-rational if its function field becomes purely transcendental over $k$ after adjoining finitely…

Let $C$ be an irreducible algebraic curve in $(\mathbb{C}^{\times})^n$. A cyclotomic point of $C$ is a point $(z_1,\ldots,z_n)\in C$ whose coordinates are all roots of unity. Lang'…

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…

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…

Volume-form preservation conjecture. Every automorphism of $({\Bbb C}^*)^2$ preserves this form. This conjecture arose after an earlier claimed proof for automorphisms extending to…

Let $({\Bbb C}^*)^2$ be the product of two punctured complex planes. Consider the automorphisms obtained from arbitrary holomorphic functions $f$, integers $n_1,n_2$, and complex n…

Le Bruyn's conjecture. The torus $T_n$ is never $k$-rational, except possibly when $n=6$.

Mellin–solution commutation conjecture. There exists a canonical isomorphism

Constructibility conjecture. The complex $\mathcal{S}ol^{<0}(\mathcal{M})$ has $\mathbb{C}$-constructible cohomology sheaves.

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…

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 $n$ be a positive integer, let $G$ be a finite group, and let $\operatorname{symrank}(\mathbb{Z}^n,G)$ denote the minimal size of a $G$-stable generating set of $\mathbb{Z}^n$.…

Let $T$ be a torus over $\mathbb{Q}$, and let $\mathcal{L}\subset T(\mathbb{Q})$ be a finitely generated torsion-free abelian group of rank $m$. For every finite prime $p$, let…

Let $X$ be a rigid affine variety, meaning that it admits no effective $\mathbb G_{\mathrm a}$-action. The rigid-affine automorphism-torus conjecture. The identity component…

Let $C_1,C_2\trianglelefteq\mathbb{G}_m^N(\mathbb{C})$ be irreducible closed algebraic curves, with $N\geq 3$. Suppose there does not exist an algebraic subgroup…

Bogomolov conjecture. Rational points should exhibit “sparsity” when compared with the density of algebraic points on the variety.

Chai's perfect-residue-field conjecture. The assumption that $\kappa$ be perfect cannot be dropped: without it, the asserted conclusions of Chai's conjecture fail, and the base cha…

Let $K/k$ be a finite separable extension with Galois closure $L/k$, let $T=R^{(1)}_{K/k}(\mathbbm{G}_m)$ be its norm one torus, and write … For an integer $d\geq 1$, consider…

Let $K/k$ be a CM extension, let $L/k$ be a finite abelian extension with group $G$, let $H$ be the one-dimensional torus over $k$ split by $K$, and let $I$ be the augmentation ide…

Let $X=\{(x,y)\in \mathbb G_m^2(\overline{\mathbb F}_2):x+y=1\}$, and define … Here $\operatorname{ord}(\mathbf{x})$ denotes the order of the torsion point $\mathbf{x}$, and $T$ te…

Let $\mathbf V\subset\mathbb C^n$ be an irreducible degree-$D$ hypersurface, and consider its torsion points, meaning points all of whose coordinates are roots of unity. Ruppert–Al…

Let $F$ define an irreducible degree-$D$ hypersurface … and let $\mathbf V_{\mathrm{tor}}$ be its locus of torsion points, namely points whose coordinates are roots of unity. A tor…

For $n\geq 1$, let $N_n^{\operatorname{tor}}(X)$ denote the number of isomorphism classes of $n$-dimensional algebraic tori over $\mathbb{Q}$ with Artin conductor $C(T)\leq X$. The…