25 problems

Grothendieck–Serre conjecture. Every étale $G$-torsor over $A$ that becomes trivial after base change to $K$ is already trivial over $A$.

Let $R$ be a regular local ring of dimension $d\geq 1$, let $m$ be its maximal ideal, and let $u\in m\setminus m^2$. Let $G$ be a strictly isotropic reductive group over $R$. Nisne…

Let $P$ be the algebra of periods and let $T$ be one of the corresponding torsors of isomorphisms, with the period construction giving a map from the period algebra to $T$. In the…

Wittenberg's descent conjecture. Let $X$ be a smooth $k$-variety and let $G$ be a linear algebraic $k$-group. Let $f:Y\to X$ be a $G$-torsor with $Y$ rationally connected. Assume t…

Assume the divisor conjecture and let $pi:{rm Sh}{K1}(G,X)E$ be the generic-fiber $T{mathcal G}(mathbb Fp)$-torsor,…

Serre–Grothendieck conjecture. This map has trivial kernel; equivalently, every $G$-torsor over $R$ that becomes trivial over $K$ is already trivial over $R$. This conjecture is a…

Let $k$ be a number field, and let $\mathcal X$, $\mathcal Y$, and $\mathcal Z$ be algebraic $k$-stacks. Let $F$ and $G$ be linear $k$-groups, and let … be torsors. Torsor covering…

Let $G$ be a semisimple simply connected algebraic group over a perfect field $K$ of cohomological dimension at most $2$. Serre's Conjecture II. Every $G$-torsor over $K$ is trivia…

Refined secure-torsor counting conjecture. If $F$ is a number field, there exists $A>0$ such that

Secure-torsor counting conjecture. If $F$ is a number field, there exists $C>0$ such that

Let $X$ be a smooth variety over a field $k$, let $D\subset X$ be a $k$-smooth divisor, and let $G$ be a reductive group scheme over $X$. A $G$-torsor on $X\setminus D$ is generica…

Let $R$ be a regular ring, let $d\geq 0$, and let $G$ be a totally isotropic reductive $R$-group scheme. A $G$-torsor is Zariski locally trivial if it is trivial on a Zariski open…

Let $R$ be a regular local ring, let $K=\operatorname{Frac}(R)$ be its fraction field, and let $G$ be a reductive $R$-group scheme. Colliot-Thélène--Sansuc purity conjecture. In th…

Let $R$ be a regular local ring, let $K=\operatorname{Frac}(R)$ be its fraction field, and let $G$ be a reductive $R$-group scheme. A $G$-torsor over $R$ is generically trivial if…

Let $k$ be an algebraically closed field, let $G$ be a reductive group over $k$, and let $X$ be a smooth algebraic variety over $k$. A $G$-torsor over $X$ is generically trivial if…

Let $E$ be a $p$-adic field with perfect residue field, let $C$ be a perfect non-archimedean field of characteristic $p$ with an open and bounded valuation subring $C^+$, and set…

Let $k$ be a field, let $G$ be a connected reductive group over $k$, and let $X$ be an irreducible smooth $k$-scheme with function field $K$. Purity for $G$-torsors over $X$ means…

Let $k$ be an algebraically closed field, let $G$ be a connected reductive linear algebraic group over $k$, let $R$ be a regular local ring containing $k$, and let … be a $G$-torso…

Let $k$ be a field of characteristic $0$, let $R=k[x_1^{\pm 1},\ldots,x_n^{\pm 1}]$, and let $G$ be a loop reductive group over $R$, meaning that $G$ contains a maximal $R$-torus.…

Let $R$ be the base ring, let $K$ be its fraction field, and let $\mathbb{A}^n_R$ be $n$-dimensional affine space with origin $0$. Write $pi^{\text{qf}}(\mathbb{A}^n_R,0)$ for the…

Reduction of the parahoric torsor conjecture. The ${\mathcal G}$-torsor $\pi^\circ$ should admit a reduction to a ${\mathcal G}^\circ$-torsor.

Let $\mathfrak G$ be a loop reductive group scheme over $R_n$, and suppose that all semisimple quotients of $\mathfrak G$ are isotropic. Isotropic loop reductive torsor conjecture.…

Let $k$ be the ground field, let $G$ be a non-special semisimple $k$-group scheme, and let $X$ be a smooth affine $k$-variety. Purity failure conjecture. There exists a smooth affi…

Let $X$ be the spectrum of a regular noetherian integral semi-local ring, let $G$ be a smooth reductive algebraic $X$-group scheme, and let $K$ be the function field of $X$. Purity…

Let $S$ be the spectrum of a complete discrete valuation ring $R$ with algebraically closed residue field and fraction field $K$. Let $X$ be an integral, regular curve separated an…