On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves

Abstract

Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n,d\ge 1$ and for each hypersurface $Z$ in ${\mathbb{P}}_K^n$ of degree $d$ with $d^2\le n$, the second Milnor $K$-theory group of $K$ is spanned by the images of the norms coming from finite extensions $L$ of $K$ over which $Z$ has a rational point. When the curve $C$ has a point in the maximal unramified extension of $k$, we generalize this result to hypersurfaces $Z$ in ${\mathbb{P}}_K^n$ of degree $d$ with $d\le n$.

34 pages. Final version