The Hasse norm principle for $A_4$-quartic extensions of global function fields

Abstract

For a finite extension of global fields $K/k$, the norm map $N_{K/k}:K^{\times }\to k^{\times }$ extends to a map on idèle groups. The Hasse norm principle holds if every element of $k^{\times }$ that is a norm everywhere locally is also a norm globally. In this paper, we study the statistics of the Hasse norm principle in a setting that is out of reach in the number field context, namely that of $A_4$-quartic extensions. We show that failures of the Hasse norm principle are generally rare for $A_4$-quartic extensions of global function fields ${\mathbb{F}}_q(t)$. We achieve this by introducing a decorated Hurwitz space parametrising the failures of the Hasse norm principle and then using the Chebotarev density theorem to estimate their frequency.

25 pages, comments welcome!