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PreprintCC BY 4.0arXiv

The Hasse norm principle for A4-quartic extensions of global function fields

  • Anand Deopurkar
  • Rachel Newton
  • Vaidehee Thatte
  • Rosa Winter
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Abstract

For a finite extension of global fields K/k, the norm map NK/k : K× → k× extends to a map on idèle groups. The Hasse norm principle holds if every element of k× 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 A4-quartic extensions. We show that failures of the Hasse norm principle are generally rare for A4-quartic extensions of global function fields 𝔽q(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.

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Cite

BibTeXRIS
BibTeX
@misc{deopurkar2026hasse,
  title = {{The Hasse norm principle for $A_4$-quartic extensions of global function fields}},
  author = {Anand Deopurkar and Rachel Newton and Vaidehee Thatte and Rosa Winter},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02444},
  eprint = {2609.02444},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02444},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Hasse norm principle for A_4-quartic extensions of global function fields
AU  - Anand Deopurkar
AU  - Rachel Newton
AU  - Vaidehee Thatte
AU  - Rosa Winter
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02444
UR  - https://arxiv.org/abs/2609.02444
AB  - For a finite extension of global fields K/k, the norm map N_K/k : K^× → k^× extends to a map on idèle groups. The Hasse norm principle holds if every element of k^× 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 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.
ER  -