Quaternary quadratic lattices over number fields

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space [Formula: see text] with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of [Formula: see text]. This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in [Formula: see text].

On indecomposable quadratic lattices over global function fields

In this paper, we prove that there exist indecomposable lattices of ranks 5 and 6 over a Hasse domain of any global function field in which [Formula: see text] is not a square, which solves a problem proposed by Gerstein.

Group schemes and local densities of quadratic lattices in residue characteristic 2

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.