26 problems

Let $U_m$ be the affine Markoff surface … where $m$ is an integer, and let $U_m(\mathbb{A}_{\mathbb{Z}})$ denote its integral adelic points. Ghosh–Sarnak conjecture. The number of…

Let $k$ be a global field and let $X$ be a smooth, projective, geometrically integral variety over $k$. Write $CH_0(X)$ for the Chow group of zero-cycles modulo rational equivalenc…

Let $X$ be a smooth, proper, geometrically irreducible variety over a number field $k$. A zero-cycle of degree one on $X$ is a formal integer linear combination of closed points wh…

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…

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…

Colliot-Thélène's conjecture. Let $X$ be a rationally connected smooth variety over $k$. Then $X(k)$ is dense in $X(k_{\Omega})^{\operatorname{Br}_{\mathrm{nr}}}$.

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…

Colliot-Thélène's conjecture. The complex above is exact.

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).

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

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

Conjecture Fconst. The subset

Conjecture F+. The subset

Let $k$ be a number field and let $X/k$ be a variety with $X(k)=\emptyset$. A stratification of $X$ is a decomposition … For each stratum, let $\cl{X_i}$ be its closure and let…

Let $k$ be a number field and let $V$ be a proper smooth geometrically integral variety over $k$. Consider the complex … where the modified Chow groups are the usual Chow groups at…

Let $K$ be a number field, let $X$ be a smooth projective geometrically integral variety over $K$, and let $P_K$ be the set of places of $K$. Write $X_{K_v}$ for the base change to…

Let $k$ be a number field, and let $\{X_\omega\}_\omega$ be the family of all smooth, proper, geometrically integral varieties over $k$. The Brauer–Manin obstruction to weak approx…

Let $X$ be a smooth proper variety over a number field $k$. Let $\Omega_f$ be the set of finite places of $k$ and $\Omega_\infty$ the set of infinite places. For each infinite plac…

Let $k$ be a number field and let $X$ be a Kummer surface over $k$. A 2-covering associated to $X$ means one of the 2-coverings of the abelian surfaces associated to $X$. Kummer-su…

Integral Brauer–Manin conjecture. If

Harari–Voloch conjecture. The diagonal map

Colliot-Thélène's approximation conjecture. If $\{z_v\}\in\prod_{v\in\Omega_k}Z_0(X_v)$ is a degree-$\delta$ family orthogonal to $Br(X)$, then for every positive integer $m$ there…