Related papers: Hasse norm principle for Heisenberg extensions of …

We determine the structure of the obstruction group of the Hasse norm principle for a finite separable extension $K/k$ of a global field of degree $d$, where $d$ has a square-free prime factor $p$ and a $p$-Sylow subgroup of the Galois…

We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of…

Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$. Hoshi, Kanai and Yamasaki [HKY22] determined…

In this paper, we study the Hasse norm principle for some non-Galois extensions of number fields. Our main theorem is that for any square-free composite number $d$ which is divisible by at least one of $3$, $55$, $91$ or $95$, there exists…

Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$ where $\overline{k}$ is a fixed separable closure of…

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 show the validity of weak approximation for…

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…

Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as…

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 approximation for the norm one torus…

Let $L/k$ an Galois extension of number fields with Galois group isomorphic to a dihedral group of order $2n$. In this note, we give a general description of the Hasse norm principle for $L/k$ and the weak approximation for the norm one…

Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$…

Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$…

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…

Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where $A(T)$ is the kernel of…

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the…

We give an equivalent condition for the validity of the Hasse norm principle for finite separable extensions of prime squared degree of global fields. Our theorem recovers the result of Drakokhrust--Platonov, which claims that the Hasse…

Let $k$ be a global field, $K/k$ be a finite separable field extension and $L/k$ be the Galois closure of $K/k$ with Galois groups $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$. In 1931, Hasse proved that if $G$ is cyclic, then the Hasse…

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and $\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field.…

The aim of this paper is to revisit the question of local-global principles for embeddings of \'etale algebras with involution into central simple algebras with involution over global fields of characteristic not 2. A necessary and…

A finite extension of global fields $L/K$ satisfies the Hasse norm principle if any nonzero element of $K$ has the property that it is a norm locally if and only if it is a norm globally. In 1931, Hasse proved that any cyclic extension…