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

Abstract

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 $\mathcal{O}_{F}$-lattices of arbitrary rank $n\ge1$, up to isometry. For every $p\in[1,\infty]$, 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 $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{13})$.