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Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
Abstract
Let F be an algebraic number field and 𝒪F the ring of integers of F. Let L be an integral 𝒪F-lattice of rank n≥1 on an anisotropic quadratic space V. We prove that if L is norm Euclidean, then L is 𝒪F-maximal of class number one, and the class number of F equals the class number of L⊥H, where H is the hyperbolic lattice with quadratic form xy. Under the same assumption, F has class number one if n≥3 and at most two if n=1,2. We also prove finiteness up to isometry for totally definite norm Euclidean lattices over totally real number fields in ranks at least three as the field varies, in rank two for fixed field degree, and in rank one over a fixed field. This confirms Clark's finiteness conjecture in ranks at least three.
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BibTeX
@misc{he2026class,
title = {{Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields}},
author = {Zilong He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37377},
eprint = {2610.37377},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37377},
}
RIS
TY - UNPB
TI - Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
AU - Zilong He
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37377
UR - https://arxiv.org/abs/2610.37377
AB - Let F be an algebraic number field and 𝒪_F the ring of integers of F. Let L be an integral 𝒪_F-lattice of rank n≥1 on an anisotropic quadratic space V. We prove that if L is norm Euclidean, then L is 𝒪_F-maximal of class number one, and the class number of F equals the class number of L⊥H, where H is the hyperbolic lattice with quadratic form xy. Under the same assumption, F has class number one if n≥3 and at most two if n=1,2. We also prove finiteness up to isometry for totally definite norm Euclidean lattices over totally real number fields in ranks at least three as the field varies, in rank two for fixed field degree, and in rank one over a fixed field. This confirms Clark's finiteness conjecture in ranks at least three.
ER -