Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Diego Izquierdo, Giancarlo Lucchini Arteche

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 $d \leq n$.

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

Abstract

Let be the function field of a curve over a -adic field . We prove that, for each and for each hypersurface in of degree with , the second Milnor -theory group of is spanned by the images of the norms coming from finite extensions of over which has a rational point. When the curve has a point in the maximal unramified extension of , we generalize this result to hypersurfaces in of degree with .
Paper Structure (18 sections, 22 theorems, 131 equations)

This paper contains 18 sections, 22 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. On the $C_2^2$-property for $p$-adic function fields
  4. Proof of Theorem \ref{['corint']}
  5. Step 0: Interpreting norms in Milnor $K$-theory in terms of motivic cohomology
  6. Step 1: Reducing to curves with residual index $1$
  7. Step 2: Solving the problem locally
  8. Step 3: Globalizing local field extensions
  9. Step 4: Computation of a Tate-Shafarevich group
  10. Step 5: Proof of Theorem \ref{['corint']} for smooth proper varieties
  11. Step 6: Proof of Theorem \ref{['corint']}
  12. Proof of Main Theorem \ref{['MTA']}
  13. On the $C_1^2$ property for $p$-adic function fields
  14. Proof of Theorem \ref{['extell']}
  15. Step 1: Solving the local problem
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $l/k$ be a finite unramified extension and set $L:=lK$. Let $Z$ be a proper $K$-variety. Then the quotient: is $\chi_K(Z,E)^2$-torsion for each coherent sheaf $E$ on $Z$.

Theorems & Definitions (47)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 37 more