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Ideal lattices. (English) Zbl 1043.11057

Wüstholz, Gisbert (ed.), A panorama in number theory or The view from Baker’s garden. Based on a conference in honor of Alan Baker’s 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press (ISBN 0-521-80799-9/hbk). 168-184 (2002).
This paper presents a number of examples and results concerning ideal lattices. An {ideal lattice} is a pair \((I,b)\) where \(I\) is a (fractional) ideal of a number field \(K\) and \(b\) is a lattice (a non-degenerate symmetric bilinear form \(I \times I \to {\mathbb R}\) satisfying \(b(\lambda x, y) = b(x,\bar\lambda y)\) for all \(x, y \in I, \lambda \in {\mathcal O}\), the ring of algebraic integers of \(K\), where \(\bar{ }\) denotes an \({\mathbb R}\)-linear involution on the étale \({\mathbb R}\)-algebra \(K_{\mathbb R} = K \otimes_{\mathbb Q} {\mathbb R})\).
The paper begins by considering integral ideal lattices (where \(b(x,y) \in {\mathbb Z}\)), and gives a number of specific results and examples for quadratic and cyclotomic fields. Next a generalization of integral ideal lattice from number fields to finite-dimensional \({\mathbb Q}\)-algebras with involution is given, with examples presented showing places where they naturally occur. The article then considers embeddings of ideal lattices in Euclidean space and concludes with considerations of certain invariants of the lattice, so-called Arakelov invariants.
For the entire collection see [Zbl 0997.00017].

MSC:

11H99 Geometry of numbers
11R99 Algebraic number theory: global fields
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