Globalizing prescribed local Brauer classes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A global field $K$ is a number field or a one-variable function field over a finite field. For a place $v$, write $K_v$ for its completion. The Brauer group $\operatorname{Br}(K)$ consists of equivalence classes of finite-dimensional central simple $K$-algebras, with tensor product as addition; $\operatorname{Br}(K)[n]$ denotes its subgroup killed by $n$. Write $\operatorname{inv}_v$ for the local Brauer invariant, with its usual values in $\mathbb Q/\mathbb Z$ (and the real or complex subgroup at archimedean places).

Statement recorded in the supplied source.

Given $K$, a positive integer $n$, a finite set $S$ of places of $K$, and for each $v\in S$ a class $c_v\in\operatorname{Br}(K_v)[n]$ represented by a central simple algebra $A_v$, explicitly construct one or more classes $c\in\operatorname{Br}(K)$, represented by central simple $K$-algebras $A$, such that \[ \operatorname{inv}_v(c)=\operatorname{inv}_v(c_v)\quad(v\in S). \] Equivalently the localizations $A\otimes_KK_v$ have the prescribed Brauer classes. In particular, address the case in which the local splitting fields are dihedral extensions. [S2]

Short English statement

Explicitly globalize finitely many prescribed local Brauer classes, especially those split by dihedral extensions.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800009 (AMR-087-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gerhard Frey, Duality in Arithmetic Geometry and Applications to Discrete Logarithms, IPAM Number Theory and Cryptography: Open Problems workshop (9--13 October 2006), Problem 7, PDF pages 45--49. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6219.pdf.