# The Hasse norm principle for abelian extensions

@article{Frei2015TheHN, title={The Hasse norm principle for abelian extensions}, author={Christopher Frei and Christopher Daniel Rachel Loughran and Christopher Daniel Rachel Newton}, journal={American Journal of Mathematics}, year={2015}, volume={140}, pages={1639 - 1685} }

Abstract:We study the distribution of abelian extensions of bounded discriminant of a number field $k$ which fail the Hasse norm principle. For example, we classify those finite abelian groups $G$ for which a positive proportion of $G$-extensions of $k$ fail the Hasse norm principle. We obtain a similar classification for the failure of weak approximation for the associated norm one tori. These results involve counting abelian extensions of bounded discriminant with infinitely many local… Expand

#### 18 Citations

The Hasse norm principle for A-extensions

- Mathematics
- 2018

We prove that, for every $n \geq 5$, the Hasse norm principle holds for a degree $n$ extension $K/k$ of number fields with normal closure $F$ such that $\operatorname{Gal}(F/k) \cong A_n$. We also… Expand

Number fields with prescribed norms.

- Mathematics
- 2018

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… Expand

Explicit methods for the Hasse norm principle and applications to A
n
and S
n
extensions

- Mathematics
- 2019

Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak… Expand

Nonabelian Cohen–Lenstra moments

- Mathematics
- Duke Mathematical Journal
- 2019

In this paper we give a conjecture for the average number of unramified $G$-extensions of a quadratic field for any finite group $G$. The Cohen-Lenstra heuristics are the specialization of our… Expand

Average bounds for the $$\ell $$ℓ-torsion in class groups of cyclic extensions

- Mathematics
- Research in Number Theory
- 2018

For all positive integers $$\ell $$ℓ, we prove non-trivial bounds for the $$\ell $$ℓ-torsion in the class group of K, which hold for almost all number fields K in certain families of cyclic… Expand

The Hasse Norm Principle in Global Function Fields

- Mathematics
- 2020

Let $L$ be a finite extension of $\mathbb{F}_q(t)$. We calculate the proportion of polynomials of degree $d$ in $\mathbb{F}_q[t]$ that are everywhere locally norms from $L/\mathbb{F}_q(t)$ which fail… Expand

Harmonic analysis and statistics of the first Galois cohomology group

- Mathematics
- Research in the Mathematical Sciences
- 2021

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $$f\in H^1(K,T)$$
f
∈
H
1
(
K
,
T
)
with discriminant-like invariant $$\text… Expand

Statistics of the First Galois Cohomology Group: A Refinement of Malle's Conjecture

- Mathematics
- 2019

Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$… Expand

On Galois extensions with prescribed decomposition groups

- Mathematics
- 2019

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition… Expand

On ℓ-torsion in class groups of number fields

- Mathematics
- 2016

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

On the Hasse norm principle.

- Mathematics
- 1978

Recently D. Garbanati [5], [6] has found computable criteria to determine when the Hasse norm principle holds for certain non-cyclic extensions of the rational field. His methods employ some rather… Expand

On the probabilities of local behaviors in abelian field extensions

- Mathematics
- Compositio Mathematica
- 2009

Abstract For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In… Expand

ON THE DENSITY OF DISCRIMINANTS OF ABELIAN EXTENSIONS OF A NUMBER FIELD

- Mathematics
- 2010

Let G = Cl × Cl denote the product of two cyclic groups of prime order l, and let k be an algebraic number field. Let N(k, G, m) denote the number of abelian extensions K of k with Galois group… Expand

The density of discriminants of quartic rings and fields

- Mathematics
- 2005

We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a… Expand

Homological algebra with locally compact abelian groups

- Mathematics
- 2007

In this article we study locally compact abelian groups using the language of derived categories. We define a derived Hom-functor on the bounded derived category of LCA groups with values in the… Expand

The geometric sieve and the density of squarefree values of invariant polynomials

- Mathematics
- 2014

We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The… Expand

The Hasse norm principle for elementary abelian extensions

- Mathematics
- 1993

Let K/k be an elementary abelian extension of finite algebraic number fields. The Hasse norm principle for K/k and its relation to the Hasse norm principles for all proper subextensions of K/k will… Expand

The density of abelian cubic fields

- Mathematics
- 1954

In the following note we show that the abelian cubic fields are rare in relation to all cubic fields over the rationals. This is no surprise since an irreducible cubic equation generates an abelian… Expand

Enumerating Quartic Dihedral Extensions of ℚ

- Mathematics
- Compositio Mathematica
- 2002

We give an explicit Dirichlet series for the generating function of the discriminants of quartic dihedral extensions of ℚ. From this series we deduce an asymptotic formula for the number of… Expand

Fonctions ZÊta Des Hauteurs Des Espaces Fibrés

- Mathematics
- 2000

In this paper we study the compatibility of Manin’s conjectures concerning asymptotics of rational points on algebraic varieties with certain natural geometric constructions. More precisely, we… Expand