Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields

Source abstract

Let $F$ be an algebraic number field and $\mathcal{O}_{F}$ the ring of integers of $F$. Let $L$ be an integral $\mathcal{O}_{F}$-lattice of rank $n\ge1$ on an anisotropic quadratic space $V$. We prove that if $L$ is norm Euclidean, then $L$ is $\mathcal{O}_{F}$-maximal of class number one, and the class number of $F$ equals the class number of $L\perp\mathbf{H}$, where $\mathbf{H}$ is the hyperbolic lattice with quadratic form $xy$. Under the same assumption, $F$ has class number one if $n\ge3$ 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.