Hasse norm principle for $M_{11}$ extensions

Akinari Hoshi, Kazuki Kanai, Aiichi Yamasaki

Published 2022-10-17Version 1

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 the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, ${\rm Pic}\,\overline{X}$ is the Picard group of $\overline{X}=X\times_k\overline{k}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K/k$. We determine $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ when the Galois group ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ is isomorphic to the Mathieu group $M_{11}$ of degree $11$. We also give a necessary and sufficient condition for the Hasse norm principle for such extensions $K/k$ with ${\rm Gal}(L/k)\simeq M_{11}$.