Book contents
- Frontmatter
- Contents
- Contributors
- Introduction
- 1 One Century of Logarithmic Forms
- 2 Report on p-adic Logarithmic Forms
- 3 Recent Progress on Linear Forms in Elliptic Logarithms
- 4 Solving Diophantine Equations by Baker's Theory
- 5 Baker's Method and Modular Curves
- 6 Application of the André–Oort Conjecture to some Questions in Transcendence
- 7 Regular Dessins, Endomorphisms of Jacobians, and Transcendence
- 8 Maass Cusp Forms with Integer Coefficients
- 9 Modular Forms, Elliptic Curves and the ABC-Conjecture
- 10 On the Algebraic Independence of Numbers
- 11 Ideal Lattices
- 12 Integral Points and Mordell–Weil Lattices
- 13 Forty Years of Effective Results in Diophantine Theory
- 14 Points on Subvarieties of Tori
- 15 A New Application of Diophantine Approximations
- 16 Search Bounds for Diophantine Equations
- 17 Regular Systems, Ubiquity and Diophantine Approximation
- 18 Diophantine Approximation, Lattices and Flows on Homogeneous Spaces
- 19 On Linear Ternary Equations with Prime Variables – Baker's Constant and Vinogradov's Bound
- 20 Powers in Arithmetic Progression
- 21 On the Greatest Common Divisor of Two Univariate Polynomials, I
- 22 Heilbronn's Exponential Sum and Transcendence Theory
11 - Ideal Lattices
Published online by Cambridge University Press: 20 August 2009
- Frontmatter
- Contents
- Contributors
- Introduction
- 1 One Century of Logarithmic Forms
- 2 Report on p-adic Logarithmic Forms
- 3 Recent Progress on Linear Forms in Elliptic Logarithms
- 4 Solving Diophantine Equations by Baker's Theory
- 5 Baker's Method and Modular Curves
- 6 Application of the André–Oort Conjecture to some Questions in Transcendence
- 7 Regular Dessins, Endomorphisms of Jacobians, and Transcendence
- 8 Maass Cusp Forms with Integer Coefficients
- 9 Modular Forms, Elliptic Curves and the ABC-Conjecture
- 10 On the Algebraic Independence of Numbers
- 11 Ideal Lattices
- 12 Integral Points and Mordell–Weil Lattices
- 13 Forty Years of Effective Results in Diophantine Theory
- 14 Points on Subvarieties of Tori
- 15 A New Application of Diophantine Approximations
- 16 Search Bounds for Diophantine Equations
- 17 Regular Systems, Ubiquity and Diophantine Approximation
- 18 Diophantine Approximation, Lattices and Flows on Homogeneous Spaces
- 19 On Linear Ternary Equations with Prime Variables – Baker's Constant and Vinogradov's Bound
- 20 Powers in Arithmetic Progression
- 21 On the Greatest Common Divisor of Two Univariate Polynomials, I
- 22 Heilbronn's Exponential Sum and Transcendence Theory
Summary
Introduction
An ideal lattice is a pair (I, b), where I is an ideal of a number field, and b is a lattice, satisfying an invariance relation (see §1 for the precise definition). Ideal lattices naturally occur in many parts of number theory, but also in other areas. They have been studied in special cases, but, as yet, not much in general. In the special case of integral ideal lattices, the survey paper Bayer-Fluckiger (1999) collects and slightly extends the known results.
The first part of the paper (see §2) concerns integral ideal lattices, and states some classification problems. In §3, a more general notion of ideal lattices is introduced, as well as some examples in which this notion occurs.
The aim of §4 is to define twisted embeddings, generalising the canonical embedding of a number field. This section, as well as the subsequent one, is devoted to positive definite ideal lattices with respect to the canonical involution of the real étale algebra generated by the number field. These are also called Arakelov divisors of the number field. The aim of §5 is to study invariants of ideals and also of the number field derived from Hermite type invariants of the sphere packings associated to ideal lattices. This again gives rise to several open questions.
Definitions, notation and basic facts
A lattice is a pair (L, b), where L is a free Z-module of finite rank, and b : L × L → R is a non-degenerate symmetric bilinear form.
- Type
- Chapter
- Information
- A Panorama of Number Theory or The View from Baker's Garden , pp. 168 - 184Publisher: Cambridge University PressPrint publication year: 2002
- 20
- Cited by