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

Abstract

Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and $\mathrm{Pic}\overline{X}$ be the Picard group of $\overline{X}=X{\times }_k\overline{k}$. Hoshi, Kanai and Yamasaki [HKY22] determined $H^1(k,\mathrm{Pic}\overline{X})$ for norm one tori $T=R_{K/k}^{(1)}(G_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]=n\le 15$ and $n\ne 12$. In this paper, we determine $64$ cases with $H^1(k,\mathrm{Pic}\overline{X})\ne 0$ and give a necessary and sufficient condition for the Hasse norm principle for $K/k$ where $[K:k]=12$.

To appear in J. Number Theory, 148 pages

References in corpus (3)

• The Hasse norm principle for $A_n$-extensions
• Norm one tori and Hasse norm principle
• Rationality problem for norm one tori

Cited by in corpus (1)

• Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensions