Even unimodular 8-dimensional quadratic forms over $$\mathbb{Q}\left( {\sqrt 2 } \right)$$

Abstract

Note added in proof. A theoretical proof of the uniqueness of the empty root lattice is also possible based on a method similar to that used in [3] for the Q(I/5) case.

Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields

“…[40, §16.6, pp133-134], [35, §5.16, pp33-34]) or small class number, however almost all actual computational results have been limited to either the case F = Q or to unimodular lattices over real quadratic fields. When [F : Q] > 1, the author is only aware of the papers [11,22,21,23,31].…”

Even positive definite unimodular quadratic forms over ${\bf Q}(\sqrt 3)$

“…These properties remain true for even unimodular lattices over Q(\/5) in dimensions up to 12, and over Q(V2) in dimensions up to 8 (see [4,7]), but no longer hold over Q(Vï) when the dimension is 8. This calculation shows that the only lattices which admit vectors of quadratic norm 2 are exactly those previously obtained in our constructions.…”

“…In the case of real quadratic fields, Maass had determined in [8] all 4-and 8-dimensional even unimodular forms over the ring of integers of Q(\/5). Subsequently, all such forms were classified up to dimension 12 over Q(\/5) and up to dimension 8 over <Q>(\/2) (see [4,14,6,7]). In this paper we will investigate positive definite even unimodular forms over the ring of integers of Q{\/3).…”