Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields

@misc{he2026finiteness,
  title = {{Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields}},
  author = {Zilong He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34604},
  eprint = {2610.34604},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34604},
}

TY  - UNPB
TI  - Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields
AU  - Zilong He
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34604
UR  - https://arxiv.org/abs/2610.34604
AB  - Motivated by Clark's work on Euclidean quadratic forms, we introduce trace Euclidean and house Euclidean quadratic lattices over totally real number fields, and relate these concepts through the p-norm Euclidean property. We prove that, as F varies over all totally real number fields of arbitrary degree, there are only finitely many positive definite trace Euclidean 𝒪_F-lattices of arbitrary rank n≥1, up to isometry. For every p∈[1,∞], the same finiteness holds for positive definite p-norm Euclidean lattices. We also classify all positive definite trace Euclidean lattices of rank one over real quadratic fields: there are exactly six isometry classes, and every such lattice is free. As a corollary, the trace Euclidean real quadratic fields are precisely ℚ(√2), ℚ(√5), and ℚ(√(13)).
ER  -