77 problems

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 K3 surfaces.

Artin–Tate conjecture.

Várilly-Alvarado's conjecture. The order

Grothendieck's conjecture. The natural map

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…

Huybrechts's conjecture. For every hyper-Kähler variety $X$, the divisibility

Let $K$ be a field, let $p$ be a prime unequal to the characteristic of $K$, and let $L/K$ be the extension appearing in the relative Brauer group … Write $B_p$ for the $p$-part of…

Let $X$ be a smooth projective variety over a finitely generated field $k$, let $k^s$ be a separable closure, and write $G_k=\operatorname{Gal}(k^s/k)$. Skorobogatov–Zarkhin's fini…

Let $G$ be a finite simple group, and let $B_0(G)\subset H^2(G,\mathbb{Q}/\mathbb{Z})$ be the subgroup consisting of cohomology classes that restrict trivially to every abelian sub…

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

Let $X=M/G$, where $M$ is a quasi-projective variety of dimension $n$ and $G$ is finite, and let $\mathcal{X}$ be the associated stack. Let $Y$ be the component of the $G$-Hilbert…

Brauer–Witt conjecture. Every projective Schur algebra is Brauer equivalent to a radical abelian algebra.

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…

Twisted McKay correspondence. In the situation described above, for every $\alpha\in B_G(U)$ there exists a derived equivalence

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 complex variety and let $H^2(X,\mathcal{O}_X^*)$ classify holomorphic gerbes. The Brauer group consists of the elements arising from holomorphic locally free twisted s…

Let $X$ be a smooth, connected, projective variety over an algebraically closed field $k$, and let $lpha \in \operatorname{Br}(X)$. Write $\operatorname{per}(\alpha)$ for the order…

Let $X$ be a smooth, proper surface over a finite field. Its zeta function has a functional equation with a central point, and $\operatorname{Br}(X)$ denotes the Brauer group of…

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…

Period-index conjecture. The index should divide the period raised to the power one less than the dimension of $B_0$:

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

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…

Generalized Artin conjecture. If $K$ is a $C_d$-field, then