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PreprintCC0arXiv

A trace lower bound for totally positive units in ℚ(ζp)+ and its application to fusion category

  • Yue Meng
  • Zhiqiang Yu
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Abstract

Let p = 4q + 1 be a prime integer, where q is an odd prime. Let L = ℚ(ζp)+ denote the maximal totally real subfield of the p-th cyclotomic field ℚ(ζp). In this paper, we prove that if UL+=UL2 (i.e., every totally positive unit is a square in L), then for any non-trivial subfield K ⊆ L and any non-trivial totally positive unit ε ∈ UK+, the absolute trace satisfies the sharp lower bound TrK/ℚ(ε) ≥ p − 4. As an application, we show that a spherical fusion category C with global dimension dim(𝒞)=p=4q+1 is pointed, where $2$ is a primitive root modulo q, or p=4q+1<222 and p≠ 29.

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Cite

BibTeXRIS
BibTeX
@misc{meng2026trace,
  title = {{A trace lower bound for totally positive units in $\mathbbQ(ζ_p)^+$ and its application to fusion category}},
  author = {Yue Meng and Zhiqiang Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15643},
  eprint = {2610.15643},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15643},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - A trace lower bound for totally positive units in ℚ(ζ_p)^+ and its application to fusion category
AU  - Yue Meng
AU  - Zhiqiang Yu
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15643
UR  - https://arxiv.org/abs/2610.15643
AB  - Let p = 4q + 1 be a prime integer, where q is an odd prime. Let L = ℚ(ζ_p)^+ denote the maximal totally real subfield of the p-th cyclotomic field ℚ(ζ_p). In this paper, we prove that if U_L^+=U_L^2 (i.e., every totally positive unit is a square in L), then for any non-trivial subfield K ⊆ L and any non-trivial totally positive unit ε ∈ U_K^+, the absolute trace satisfies the sharp lower bound Tr_K/ℚ(ε) ≥ p − 4. As an application, we show that a spherical fusion category C with global dimension dim(𝒞)=p=4q+1 is pointed, where $2$ is a primitive root modulo q, or p=4q+1<2^22 and p≠ 29.
ER  -