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

TL;DR

This paper investigates the structure of the second Milnor K-theory group of function fields of p-adic curves, showing it is generated by norm images from certain finite extensions, with generalizations under specific conditions.

Contribution

It proves that the second Milnor K-theory group is spanned by norms from extensions where hypersurfaces have rational points, extending previous results to broader cases.

Findings

K-theory group spanned by norms from extensions with rational points

Generalization to hypersurfaces of degree d ≤ n when curve has a point in unramified extension

Results apply to function fields of p-adic curves with specific geometric conditions

Abstract

Let $K$ be the function field of a curve $C$ over a $p$-adic field $k$. We prove that, for each $n, d \geq 1$ and for each hypersurface $Z$ in $\mathbb{P}^n_{K}$ of degree $d$ with $d^2 \leq n$, 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 $\mathbb{P}^n_{K}$ of degree $d$ with $d \leq n$.