32 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…

Let $k$ be a field, let $d\geq 1$, and let $D$ be a central division algebra of degree $d$ over $k$. For a positive rational number $r$, write $\operatorname{Mod}_{D,r}$ for the fu…

Generation conjecture. The algebra $A$ is generated by $X$ and $Y$.

A central simple algebra is a finite-dimensional associative algebra whose center is the base field and which becomes a matrix algebra after extending scalars. Its degree is the sq…

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

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 $F$ be a quadratic field extension of $F_0$, let $A$ be a central simple $F$-algebra with a unitary involution $\tau$ such that $F_0=\{x\in F\mid\tau(x)=x\}$. Write $SK_1(A)$ f…

Let $K$ be a field and let $A$ be a central simple algebra over $K$. Write $\operatorname{per}(A)$ and $\operatorname{ind}(A)$ for the period and index of $A$, respectively. A fiel…

Let $K/F$ be a totally positive field extension of formally real fields, and let $A$ be a central simple algebra over $F$ of exponent $2$. The Pythagorean index conjecture asserts…

Global Azumaya algebra conjecture. There exists a central simple algebra $D$ over $\mathbb Q(\pi)$ such that, for every prime $p$,

Let $K$ be a complete discrete valued field of characteristic $p$ with residue field $k$, and let $A$ be a $p$-algebra over $K$. A maximal subfield of $A$ is totally ramified cycli…

Let $D$ be a central simple algebra and let $G$ and $G'$ be the groups in the Guo–Jacquet comparison, with test functions $f_D=1_{O_D^\times}$ and $f'_D$ as defined from the standa…

Merkurjev–Suslin conjecture. There is an exact sequence

Let $F$ be a field, let $ell$ be a prime, and let $D$ be a central division $F$-algebra of $ell$-power degree. The Rost kernel is the subgroup of $F^*$ consisting of those $lambda$…

Bayer-Fluckiger et al.'s conjecture. The following statements are equivalent:

Let $n\geq 2$. The totally positive elements of the universal division algebra $\operatorname{USA}_n$ are those elements $a$ satisfying $a(X)\succeq 0$ for every…

Let $E$ be a field with cohomological dimension $cd(E)$, and let $A$ be a central simple algebra over $E$. Denote its reduced Whitehead group by $SK_1(A)$. Suslin's conjecture. If…

Let $E$ be a field and let $A$ be a central simple algebra over $E$. Its reduced Whitehead group is denoted by $SK_1(A)$. Triviality conjecture. The group $SK_1(A)$ is trivial for…

Let $F$ be a field of characteristic $p$, let $A=[\alpha,\beta)_{p,F}$ be a symbol $p$-algebra of prime degree $p$, and let a $p$-central subspace mean an $F$-vector subspace of…

Maximality conjecture. The $p^k$-central space $V$ is maximal with respect to inclusion.

Finite representability conjecture. The Clifford algebra of any given form is finitely representable.

Let $A$ be a central simple algebra over a field $F$, and let ${\mathcal F}_{A,o}$ denote the function field associated with $A$ and orthogonal involutions. An orthogonal involutio…

Universal-element conjecture. All central simple Lie algebras of the same type can be assembled from the same universal elements, which are Lie algebra structures over $\mathbb Z$.

Let $D$ be a division algebra over $F$ and let $E/F$ be a field extension such that $D_E$ remains a division algebra. Consider the Severi–Brauer varieties of $D$ and their motivic…

Let the remaining types not covered by the preceding theorem be the types $\mathsf C_{2^k}$ and $\mathsf D_{2^k}$ of central simple Lie algebras. Exclusion conjecture. The types…