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Hasse norm principle for metacyclic extensions with trivial Schur multiplier

Akinari Hoshi, Aiichi Yamasaki

TL;DR

The paper establishes that for K/k with L/k its Galois closure and G = Gal(L/k), if G is metacyclic and its Schur multiplier M(G) vanishes, then the stabilizer H = Gal(L/K) is cyclic and the Hasse norm principle holds for K/k. The core mechanism is a cohomological analysis of Sha^2_ω(G,J_{G/H}) and its relation to M(G); under the stated hypotheses, Sha^2_ω(G,J_{G/H}) = 0, which via Ono’s correspondence implies Sha(T) = 0 for the norm one torus T = R^{(1)}_{K/k}(G_m,K) and thus the Hasse principle for all T-torsors. The authors provide extensive families of metacyclic groups with M(G) = 0 (including dihedral, quasidihedral, modular, generalized quaternion, Z-groups, and extraspecial groups) and perform GAP computations to catalog transitive G ≤ S_n (2 ≤ n ≤ 30) with [G:H] = n that satisfy the hypotheses, yielding many new non-Galois examples where the Hasse norm principle holds. Collectively, the work connects group cohomology, norm one tori, and the Hasse principle to produce a broad class of extensions K/k for which the HNP is guaranteed by the vanishing Schur multiplier of the metacyclic Galois closure.

Abstract

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 norm principle holds for $K/k$. We show that if $G$ is metacyclic with trivial Schur multiplier $M(G)=0$, then $H$ is cyclic and the Hasse norm principle holds for $K/k$. Some examples of metacyclic, dihedral, quasidihedral, modular, generalized quaternion, extraspecial groups and $Z$-groups $G$ with trivial Schur multiplier $M(G)=0$ are given. These provide new examples which the Hasse norm principle hold for non-Galois extensions $K/k$ whose Galois closure is $L/k$ with metacyclic $G={\rm Gal}(L/k)$ and $M(G)=0$.

Hasse norm principle for metacyclic extensions with trivial Schur multiplier

TL;DR

The paper establishes that for K/k with L/k its Galois closure and G = Gal(L/k), if G is metacyclic and its Schur multiplier M(G) vanishes, then the stabilizer H = Gal(L/K) is cyclic and the Hasse norm principle holds for K/k. The core mechanism is a cohomological analysis of Sha^2_ω(G,J_{G/H}) and its relation to M(G); under the stated hypotheses, Sha^2_ω(G,J_{G/H}) = 0, which via Ono’s correspondence implies Sha(T) = 0 for the norm one torus T = R^{(1)}_{K/k}(G_m,K) and thus the Hasse principle for all T-torsors. The authors provide extensive families of metacyclic groups with M(G) = 0 (including dihedral, quasidihedral, modular, generalized quaternion, Z-groups, and extraspecial groups) and perform GAP computations to catalog transitive G ≤ S_n (2 ≤ n ≤ 30) with [G:H] = n that satisfy the hypotheses, yielding many new non-Galois examples where the Hasse norm principle holds. Collectively, the work connects group cohomology, norm one tori, and the Hasse principle to produce a broad class of extensions K/k for which the HNP is guaranteed by the vanishing Schur multiplier of the metacyclic Galois closure.

Abstract

Let be a global field, be a finite separable field extension and be the Galois closure of with Galois groups and . In 1931, Hasse proved that if is cyclic, then the Hasse norm principle holds for . We show that if is metacyclic with trivial Schur multiplier , then is cyclic and the Hasse norm principle holds for . Some examples of metacyclic, dihedral, quasidihedral, modular, generalized quaternion, extraspecial groups and -groups with trivial Schur multiplier are given. These provide new examples which the Hasse norm principle hold for non-Galois extensions whose Galois closure is with metacyclic and .

Paper Structure

This paper contains 14 sections, 20 theorems, 39 equations.

Table of Contents

  1. Introduction: Main theorem (Theorem \ref{['thmain0']})
  2. Hasse norm principle and norm one tori
  3. Proof of Theorem \ref{['thmain0']} (Theorem \ref{['thmain']})
  4. Examples of Theorem \ref{['thmain0']} (Theorem \ref{['thmain']}): Metacyclic groups $G$ with $M(G)=0$
  5. Metacyclic groups $G$ with $M(G)=0$
  6. The $Z$-groups $G$ with $M(G)=0$
  7. Dihedral $D_n$, quasidihedral $QD_{8m}$, modular $M_{16m}$ and generalized quaternion $Q_{4m}$ groups
  8. The metacyclic $p$-groups $G$ with $M(G)=0$
  9. The extraspecial group $G=E_{p^2}(p^3)$ of order $p^3$ with exponent $p^2$ with $M(G)=0$
  10. Metacyclic groups $G=nTm\leq S_n$ $(2\leq n\leq 30)$ and $H\leq G$ with $[G:H]=n$
  11. Examples of Proposition \ref{['prop4.1']}: Not metacyclic groups $G=nTm\leq S_n$ $(2\leq n\leq 19)$ and $H\leq G$ with $[G:H]=n$
  12. Appendix: Examples of groups $G$ with $M(G)=0$: Galois cases with $H=\{1\}$
  13. GAP computations: Schur multipliers $M(G)$ and $M(H)$ as in Table 1, Table 2, Table 3 and Table 4
  14. GAP computations: Schur multipliers $M(G)$ as in Remark \ref{['rem8.2']}, Remark \ref{['rem8.4']} and Remark \ref{['rem8.6']}

Key Result

Theorem 1.1

Let $k$ be a global field, $K/k$ be a finite Galois extension with Galois group $G={\rm Gal}(K/k)$. Let $V_k$ be the set of all places of $k$ and $G_v$ be the decomposition group of $G$ at $v\in V_k$. Then where $\widehat{H}$ is the Tate cohomology. In particular, the Hasse norm principle holds for $K/k$ if and only if the restriction map $H^3(G,\mathbbm{Z})\xrightarrow{\rm res}\bigoplus_{v\in V_

Theorems & Definitions (40)

  • Theorem 1.1: Tate Tat67
  • Theorem 1.2: Hasse norm principle for metacyclic extensions with trivial Schur multiplier $M(G)=0$, see Theorem \ref{['thmain']} for the precise statement
  • Theorem 2.1: Ono Ono63, see also Platonov Pla82, Kunyavskii Kun84, Platonov and Rapinchuk PR94
  • Theorem 2.2: Voskresenskii Vos69, Vos70, see also Vos98
  • Theorem 2.3: Voskresenskii Vos70, Colliot-Thélène and Sansuc CTS77
  • Theorem 2.4: Colliot-Thélène and Sansuc CTS87, see also San81, Vos98, CTHS05, BP20
  • Proposition 3.1
  • proof
  • Theorem 3.2: Hasse norm principle for metacyclic extensions with trivial Schur multiplier $M(G)=0$: the precise statement of Theorem \ref{['thmain0']}
  • proof
  • ...and 30 more