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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

Let K be the function field of a curve C over a p-adic field k. We prove that, for each n,d 1 and for each hypersurface Z in Kn of degree d with d2 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 Kn of degree d with d n.

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
Received: 14 December 2022
Revised: 28 March 2023
Accepted: 29 May 2023
Published: 26 February 2024
Diego Izquierdo
Centre de Mathématiques Laurent Schwartz
École polytechnique
Giancarlo Lucchini Arteche
Departamento de Matemáticas, Facultad de Ciencias
Universidad de Chile

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