Standard norm varieties for Milnor symbols mod $p$

Abstract

We prove that the standard norm varieties for Milnor symbols mod $p$ of length $n$ are birationally isomorphic to Pfister quadrics when $p=2$, to Severi–Brauer varieties when $p>2$ and $n=2$, and to varieties defined by reduced norms of cyclic algebras when $p>2$ and $n=3$. In the case $p=2$ and the case $p>2$ and $n=2$, the results imply that the standard norm varieties for two equal Milnor symbols mod $p$ are birationally isomorphic, and we conjecture this in general.