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Number fields with prescribed norms

Christopher Frei, Daniel Loughran, Rachel Newton, Yonatan Harpaz, Olivier Wittenberg

Abstract

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

Number fields with prescribed norms

Abstract

We study the distribution of extensions of a number field with fixed abelian Galois group , from which a given finite set of elements of are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for of -extensions of , when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.

Paper Structure

This paper contains 34 sections, 50 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Methodology and structure of the paper
  3. Notation and conventions
  4. Frobenian functions
  5. Counting with local conditions
  6. Statement of the result
  7. The set of places $S$
  8. Dirichlet series
  9. Möbius inversion
  10. Class field theory
  11. Harmonic analysis
  12. Fourier transforms
  13. The local Fourier transforms
  14. Poisson summation
  15. Analytic continuation of the Fourier transforms
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $k$ be a number field, $G$ a finite abelian group and $\mathcal{A} \subset k^*$ a finitely generated subgroup. Then there exists an abelian extension $K/k$ with Galois group $G$ such that every element of $\mathcal{A}$ is a norm from $K$.

Theorems & Definitions (96)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 86 more