17 problems

Unramified period-index conjecture. There exists a positive integer $e$ such that, for all $b1 \in \operatorname{Br}(X)$,

Global period-index conjecture. For any such Brauer class $\alpha$,

Let $K$ be a $C_d$ field, meaning that every homogeneous form of degree $e$ in more than $e^d$ variables over $K$ has a nontrivial zero, and let $\alpha\in\operatorname{Br}(K)$. Th…

Let $K$ be a $C_d$ field, and let $\alpha\in \operatorname{Br}(K)$ be a Brauer class. The period $\operatorname{per}(\alpha)$ is its order in the Brauer group, and the index…

Let $\mathcal{K}$ be a complete discrete valued field of characteristic zero with residual characteristic $p$, and let $\kappa$ be its residue field. For a field $E$ of characteris…

Unramified period-index conjecture. For every $\alpha \in \operatorname{Br}(X)$,

Period-index conjecture. For every $\alpha \in \operatorname{Br}(K)$,

Let $F$ be a field, let $d$ be an integer, and let $\ell$ be the prime appearing in the cohomology group. The diophantine dimension $\operatorname{dd}(F)$ is the smallest integer…

Lang's conjecture. The function field $\mathbb R(X)$ is $C_i$.

Antieau-Williams' conjecture.

Antieau-Williams' conjecture.

The period-index conjecture. For every $\alpha\in\operatorname{Br}(K)$,

Let $p$ be prime, let $e<f$ be positive integers, and set $m=p^e$, $n=p^f$ as above. With $n=km$, consider the symmetric-function expression … where the sum runs over partitions fi…

Effective-index conjecture. One should have

Period–Index Conjecture for Function Fields. For every $\alpha\in\operatorname{Br}(k(X))$,

Étale-index conjecture.

Period-index conjecture.