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Supersolvable descent for rational points

Yonatan Harpaz, Olivier Wittenberg

Abstract

We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer-Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the existence of supersolvable Galois extensions of number fields with prescribed norms, generalising work of Frei-Loughran-Newton.

Supersolvable descent for rational points

Abstract

We construct an analogue of the classical descent theory of Colliot-Thélène and Sansuc in which algebraic tori are replaced with finite supersolvable groups. As an application, we show that rational points are dense in the Brauer-Manin set for smooth compactifications of certain quotients of homogeneous spaces by finite supersolvable groups. For suitably chosen homogeneous spaces, this implies the existence of supersolvable Galois extensions of number fields with prescribed norms, generalising work of Frei-Loughran-Newton.
Paper Structure (18 sections, 29 theorems, 31 equations)

This paper contains 18 sections, 29 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Acknowledgment
  4. Finite descent types
  5. Finite descent types and torsors
  6. Supersolvability
  7. Supersolvable descent
  8. Fibrations over tori
  9. Cyclic descent
  10. Proof of Theorem \ref{['th:main']} in the general case
  11. Applications
  12. Homogeneous spaces of linear algebraic groups
  13. Outer Galois actions and $\sigma$-algebraic maps
  14. Statement
  15. Proof of Theorem \ref{['th:genthb']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $X$ be a smooth, proper and geometrically irreducible variety over a number field $k$ such that $\mathrm{Pic}(X_{\bar{k}})$ is torsion-free. Let $T$ be an algebraic torus over $k$. Let $\lambda \in H^1_{\mathrm{\acute et}}(X_{\bar{k}},T_{\bar{k}})^{\mathrm{Gal}(\bar{k}/k)}$. Then where $f:Y\to X$ ranges over the isomorphism classes of torsors $Y \to X$ of type $\lambda$. In particular, if $Y(

Theorems & Definitions (59)

  • Conjecture 1.1: ctbudapest
  • Theorem 1.2: Colliot-Thélène--Sansuc ctsandescent2
  • Theorem 1.3: hwzceh
  • Theorem 1.4: supersolvable descent, see Theorem \ref{['th:main']}
  • Corollary 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • ...and 49 more