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A note on the Hasse norm principle

Peter Koymans, Nick Rome

Abstract

Let $A$ be a finite, abelian group. We show that the density of $A$-extensions satisfying the Hasse norm principle exists, when the extensions are ordered by discriminant. This strengthens earlier work of Frei--Loughran--Newton \cite{FLN}, who obtained a density result under the additional assumption that $A/A[\ell]$ is cyclic with $\ell$ denoting the smallest prime divisor of $\# A$.

A note on the Hasse norm principle

Abstract

Let be a finite, abelian group. We show that the density of -extensions satisfying the Hasse norm principle exists, when the extensions are ordered by discriminant. This strengthens earlier work of Frei--Loughran--Newton \cite{FLN}, who obtained a density result under the additional assumption that is cyclic with denoting the smallest prime divisor of .
Paper Structure (5 sections, 8 theorems, 31 equations)

This paper contains 5 sections, 8 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Ingredients
  3. Proof that limit exists
  4. Value of limit
  5. Value of limit II

Key Result

Theorem 1.1

Let $k$ be a number field. Let $A$ be a non-trivial, finite, abelian group. Then the limit exists.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark
  • Lemma 2.1
  • proof
  • Theorem 2.2: Wright
  • Theorem 2.3: Wood
  • Lemma 3.1
  • Remark
  • ...and 3 more