Hasse norm principle for $M_{11}$ and $J_1$ extensions

TL;DR

This paper establishes a criterion for the Hasse norm principle in certain Galois extensions with Galois groups isomorphic to Mathieu or Janko groups, advancing understanding of this principle for sporadic simple groups.

Contribution

It provides a necessary and sufficient condition for the Hasse norm principle for extensions with Galois groups M11 or J1, linking cohomological properties to norm principles.

Findings

Hasse norm principle characterized for M11 and J1 Galois groups

Determined cohomology groups for associated norm one tori

Progress towards understanding the principle for all 26 sporadic groups

Abstract

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 order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,{\rm Pic}\, \overline{X})=0$ or $\mathbb{Z}/2\mathbb{Z}$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ with a smooth $k$-compactification $X$ and $\overline{X}=X\times_k\overline{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups.