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PreprintCC BY-NC-SA 4.0arXiv

The density of discriminants of A4-quartic fields

  • Xiaoheng Wang
  • Stanley Yao Xiao
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Abstract

Let κA4 = (35 π)/(1458 √3) ∏p ≡ 1 pmod3 (1 − 1/p)3 (1 + 3/p ) ∏p ≡ 2 pmod3 (1 − 1/p2 )2 and NA4(X) be the number of A4-quartic fields of discriminant up to X. We establish the asymptotic formula NA4(X) = κA4 X1/2 log X + Oε (X1/2 (log log X)3+ε ). As part of our proof, we also establish that the average size of the $2$-torsion of the class group of cyclic cubic fields is bounded. As an application of our theorem, we obtain the consequence that the number of tetrahedral holomorphic cuspidal newforms of weight 1 of level N is Oε(Nε) on average.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026density,
  title = {{The density of discriminants of $A_4$-quartic fields}},
  author = {Xiaoheng Wang and Stanley Yao Xiao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26510},
  eprint = {2610.26510},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26510},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - The density of discriminants of A_4-quartic fields
AU  - Xiaoheng Wang
AU  - Stanley Yao Xiao
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.26510
UR  - https://arxiv.org/abs/2610.26510
AB  - Let κ_A_4 = (35 π)/(1458 √3) ∏_p ≡ 1 pmod3 (1 − 1/p)^3 (1 + 3/p ) ∏_p ≡ 2 pmod3 (1 − 1/p^2 )^2 and N_A_4(X) be the number of A_4-quartic fields of discriminant up to X. We establish the asymptotic formula N_A_4(X) = κ_A_4 X^1/2 log X + O_ε (X^1/2 (log log X)^3+ε ). As part of our proof, we also establish that the average size of the $2$-torsion of the class group of cyclic cubic fields is bounded. As an application of our theorem, we obtain the consequence that the number of tetrahedral holomorphic cuspidal newforms of weight 1 of level N is O_ε(N^ε) on average.
ER  -