Let be a Dedekind domain with field of fractions and a finite group. We show that, if is a ring of -adic integers, then the Witt decomposition map between the Grothendieck-Witt group of bilinear -modules and the one of finite bilinear -modules is surjective. For number fields is also surjective, if is a nilpotent group of odd order, but there are counterexamples for groups of even order.
This article classifies the strongly modular lattices with longest and second longest possible shadow.
This paper classifies the strongly perfect lattices in dimension . There are up to similarity two such lattices, and its dual lattice.
S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.
General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.
A lattice is called dual strongly perfect if both, the lattice and its dual, are strongly perfect. We show that there are no dual strongly perfect lattices of dimension 13 and 15.
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