# The Hasse norm principle for abelian extensions

@article{Frei2015TheHN, title={The Hasse norm principle for abelian extensions}, author={C. 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

#### 17 Citations

The Hasse norm principle for A-extensions

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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.

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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
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- Mathematics
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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

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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

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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

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- 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

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We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group f P HpK,T q with discriminant-like invariant invpfq ď X as X Ñ 8. Specifically, Poisson summation… Expand

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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

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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

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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

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