Norm one tori and Hasse norm principle, II: Degree $12$ case

TL;DR

This paper investigates the Hasse norm principle for degree 12 extensions of number fields by analyzing the cohomology of Picard groups of associated algebraic tori, extending previous work up to degree 15.

Contribution

It determines 64 cases where the cohomology is nontrivial and establishes a necessary and sufficient condition for the Hasse norm principle specifically for degree 12 extensions.

Findings

Identified 64 cases with nontrivial cohomology for degree 12.

Provided a criterion for the Hasse norm principle in degree 12 extensions.

Extended previous classifications up to degree 15 to include degree 12.

Abstract

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 $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]=n\leq 15$ and $n\neq 12$. In this paper, we determine $64$ cases with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ and give a necessary and sufficient condition for the Hasse norm principle for $K/k$ where $[K:k]=12$.