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Arithmetics of homogeneous spaces over $p$-adic function fields

Nguyen Manh Linh

Abstract

Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of $\mathbb{Q}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak approximation problems for homogeneous spaces of $\textrm{SL}_{n,K}$ with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot-Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over $K$, rediscovering the positive answer to the abelian case of the inverse Galois problem over $\mathbb{Q}_p(t)$. In the case where the curve is defined over a higher-dimensional local field instead of a finite extension of $\mathbb{Q}_p$, coarser results are also given.

Arithmetics of homogeneous spaces over $p$-adic function fields

Abstract

Let be the function field of a smooth projective geometrically integral curve over a finite extension of . Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local-global and weak approximation problems for homogeneous spaces of with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot-Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over , rediscovering the positive answer to the abelian case of the inverse Galois problem over . In the case where the curve is defined over a higher-dimensional local field instead of a finite extension of , coarser results are also given.
Paper Structure (6 sections, 8 theorems, 27 equations)

This paper contains 6 sections, 8 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Context of the problem
  3. Statement of the main results
  4. Notations and conventions
  5. Reciprocity obstruction
  6. Arithmetic duality theorems

Key Result

Theorem A

Let $K$ be the function field of a smooth projective geometrically integral curve $\Omega$ over a $p$-adic field, $X$ a homogeneous space of $\operatorname{SL}_{n,K}$ whose geometric stabilizers are extensions of a group of multiplicative type by a unipotent group. Then the unramified first obstruct

Theorems & Definitions (11)

  • Theorem A: Theorems \ref{['thm:Hasse']} and \ref{['thm:HasseModified']}
  • Theorem B: Theorems \ref{['thm:Weak']} and \ref{['thm:WeakModified']}
  • Theorem C: Theorem \ref{['thm:Descent']}
  • Theorem D: Theorem \ref{['thm:HigherHasse']}
  • Theorem E: Theorem \ref{['thm:HigherWeak']}
  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3: Generalized Weil reciprocity law
  • ...and 1 more