39 problems

Period-index conjecture.

Let $F$ be a field, and let $A$ and $B$ be central simple algebras over $F$. Write $SB(A)$ and $SB(B)$ for their associated Severi–Brauer varieties, and let $[A]$ and $[B]$ denote…

Skorobogatov's conjecture. The Brauer–Manin obstruction is the only obstruction to the Hasse principle for algebraic K3 surfaces over number fields.

Strong uniform boundedness conjecture. If $X$ is a K3 surface over a number field of bounded degree $d$, then

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…

Várilly-Alvarado's conjecture. The order

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

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

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 $F$ be an infinite field, and let $A$ and $B$ be central simple algebras over $F$ of the same degree. For each $r<\deg(A)$, write $F_r(A)$ and $F_r(B)$ for the function fields…

Let $X$ be a smooth, proper, geometrically connected surface over a finite field of characteristic $p$. Write … for its Brauer group, let $\operatorname{Br}(X)_{\nd}$ be the quotie…

Strong uniform boundedness conjecture. There is a constant $B=B(d,L)$, independent of $X$, such that

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…

Product Severi–Brauer conjecture. The following are equivalent:

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

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

Brauer spectrum conjecture. There is an equivalence

Let $\pi:\mathcal{X}\to B$ be a smooth del Pezzo fibration. Fix an intersection profile $\lambda$, let $N_\lambda$ be the affine subset of $N_1(\mathcal{X})$ consisting of curve cl…

Let $X$ be a smooth projective curve in the setting above, and let the SY torsor of $X$ parametrize the choices used to construct its SY bundles. Each element of this torsor determ…

Fix a positive integer $n$ and a primitive lattice embedding … Let $X$ be a K3 surface over a number field of degree $n$ such that the geometric Néron–Severi lattice…

Dominant-families conjecture. There is some translate $\mathcal{T}$ of $\mathrm{Nef}_{\lambda}$ in $N_{\lambda}$ such that every numerical class in $\mathcal{T}_{\mathbb{Z}}$ is re…

Topological Period-Index Conjecture. One has

The $\ell$-adic Tate conjecture for divisors. This map should be surjective. The conjecture predicts that divisor classes account for all cohomology classes fixed by the absolute G…