Extremal Lattices
Extremal Lattices
复制标题
极值格子
DOI：
--
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
R. Schulze
中科院分区：
文献类型：
--
作者：
Rudolf Scharlau;R. Schulze
本文研究欧几里得向量空间的离散子群，等价有限生成的自由阿贝尔群（对某n∈n同构于Z）及其正定二次型。这样的结构简称为晶格，通常用L，M，…表示。，值(v, w)∈R的双线性形式（其中v， w∈v、L，包络的R向量空间）。本报告的一般背景是由球体填充问题（具有大最小值的格的构造）、模形式理论和有限矩阵群理论提供的。几乎所有晶格对提到的地区之一是“代数”甚至“理性”,我们需要理性的价值观对他们意味着形式:v (v, w)∈Q, w∈l .尺度改变之后,也就是说,相乘的形式与一些积极的积分常数α,一个理性的晶格成为积分:所有v (v, w)∈Z, w∈l .新格将被定义,用l .点阵是积分当且仅当它是包含在它的对偶晶格
This paper deals with discrete subgroups of euclidean vector spaces, equivalently finitely generated free abelian groups (isomorphic to Z for some n ∈ N) together with a positive definite quadratic form. Such a structure will be called a lattice for short, typically denoted by L,M, . . . , with values (v, w) ∈ R of the bilinear form (where v, w ∈ V ⊃ L, the enveloping R-vector space). The general background of this report is provided by the sphere packing problem (construction of lattices with large minimum), by the theory of modular forms, and by the theory of finite matrix groups. Almost all lattices of interest for one of the mentioned areas are “algebraic” or even “rational”, by which we mean that the form takes rational values on them: (v, w) ∈ Q for v, w ∈ L. After rescaling, that is, multiplying the form with some positive integral constant α, a rational lattice becomes integral : (v, w) ∈ Z for all v, w ∈ L. The rescaled lattice will be denoted by L. By definition, a lattice is integral if and only if it is contained in its dual lattice