41 problems

Colliot-Thélène's conjecture. Weak approximation with Brauer–Manin obstruction is the only obstruction to weak approximation for smooth rationally connected varieties over number f…

A K3 surface is a smooth, simply connected projective surface with trivial canonical bundle. For a variety $X$ over a number field $k$, write $X(\mathbb{A}_k)$ for its adelic point…

Stoll's conjecture. The set $C(K)$ is dense in

Let $C$ be a pointless curve of genus at least $2$ over a number field $k$, let $J$ be its Jacobian, and let $C({\mathbb A}_k)^{\operatorname{PS}}$ denote the Poonen–Scharaschkin s…

Colliot-Thélène–Sansuc–Kato–Saito conjecture. The complex

Harari–Voloch conjecture. The Brauer–Manin obstruction is the only obstruction to the existence of $S$-integral points on $U$.

Let $K$ be a global field, and let $V/K$ be a smooth projective variety such that its base change $V_{\overline K}=V\operatorname*{\otimes}\overline K$ is rational and has Picard g…

Adelic Mordell–Lang conjecture. For any closed $K$-subscheme $X$ of any $A$,

Let $k$ be a number field, and let $C$ be a curve over $k$ of genus greater than $1$ with $C(k)=\emptyset$. A method of Scharaschkin, equivalent to the Brauer–Manin obstruction in…

Brauer–Manin zero-cycle conjecture. If $X({\bf A}_k)^{\operatorname{Br}}$ is non-empty, then $I(X)=1$.

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…

Let $k$ be a number field and $M\in \mathrm{M}_n(\mathcal{O}_k)$. The triangularization variety $X_M$ is the variety associated with triangularizing $M$, and $X_M(k)$ denotes its r…

Let $k$ be a global field, let $A$ be an abelian variety over $k$, and let $X\subset A$ be a closed subvariety. Assume that $A_{\overline{k}}$ has no positive-dimensional isotrivia…

Let $k$ be a global function field, let $A$ be an abelian variety over $k$, and let $X$ be a closed subvariety of $A$. Write $\overline{X(k)}$ and $\overline{A(k)}$ for the respect…

Let $V$ be a smooth weak Fano variety over a number field $F$, let $\Gamma_F$ be the absolute Galois group, and write…

Let $X$ be a smooth, projective, geometrically integral, weak Fano variety over $\mathbb{Q}$. Let $U \subseteq X$ be an open subset and let…

Colliot-Thélène–Sansuc analogue. The surface $X$ verifies (BM).

Brauer-adjusted heuristic conjecture. The main heuristic holds with

Let $(X,D_\epsilon)$ be the Campana orbifold, let $L$ be the underlying big divisor, and let $F$ and $S$ be as above. Let $\alpha((X,D_\epsilon),L)$, $a((X,D_\epsilon),L)$, and…

Descent conjecture. Under these assumptions, the set $X(k)$ is dense in

Colliot-Thélène's exactness conjecture. The complex

Let $X$ be a smooth projective geometrically connected variety over a number field $F$. For each place $v$ of $F$, let $X_v$ be the base change of $X$ to $F_v$, and write…

Let $X$ be a smooth, projective, geometrically integral variety over a number field, and let $d$ be an integer. Zero-cycle Brauer–Manin conjecture. The Brauer–Manin obstruction to…

Conjecture Fconst. The subset

Conjecture F+. The subset