On Kato and Kuzumaki’s properties for the Milnor K2 of function fields of p-adic curves

Izquierdo, Diego; Lucchini Arteche, Giancarlo

doi:10.2140/ant.2024.18.815

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On Kato and Kuzumaki's properties for the Milnor $\mathrm{K}_2$ of function fields of $p$-adic curves

Diego Izquierdo and Giancarlo Lucchini Arteche

Vol. 18 (2024), No. 4, 815–846

DOI: 10.2140/ant.2024.18.815

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 $ℙ_{K}^{n}$ 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 $ℙ_{K}^{n}$ of degree $d$ with $d \leq n$ .

Keywords

Milnor $K$-theory, zero-cycles, Fano hypersurfaces, p-adic function fields, $C_i$ property, Galois cohomology, cohomological dimension

Mathematical Subject Classification

Primary: 11E76, 12G05, 14G27, 14J70, 19C99

Secondary: 11G20, 12G10, 14G05, 14J45, 19F05

Milestones

Received: 14 December 2022

Revised: 28 March 2023

Accepted: 29 May 2023

Published: 26 February 2024

Authors

Diego Izquierdo
Centre de Mathématiques Laurent Schwartz École polytechnique Palaiseau France

Giancarlo Lucchini Arteche
Departamento de Matemáticas, Facultad de Ciencias Universidad de Chile Santiago Chile

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Vol. 18, No. 4, 2024