On the notion of genus for division algebras and algebraic groups

Let $D$ be a central division algebra of degree $n$ over a field $K$. One defines the genus gen$(D)$ of $D$ as the set of classes $[D']$ in the Brauer group Br$(K)$ where $D'$ is a central division $K$-algebra of degree $n$ having the same isomorphism classes of maximal subfields as $D$. I will review the results on gen$(D)$ obtained in the last several years, in particular the finiteness theorem for gen$(D)$ when $K$ is finitely generated of characteristic not dividing $n$. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic $K$-groups using maximal $K$-tori in place of maximal subfields, and report on some recent progress in this direction. (Joint work with V. Chernousov and I. Rapinchuk.)

Date

Affiliation

University of Virginia

Speakers

Andrei Rapinchuk

Files & Media