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
15 - A New Application of Diophantine Approximations
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
The method of diophantine approximation has yielded many finiteness results, as the theorems of Thue and Siegel or the theory of rational points on subvarieties of abelian schemes. Its main drawback is non-effectiveness. In the present overview I first recall some progress made in the last decade, and the remaining problems. After that I explain how to extend the known methods to some new cases, proving finiteness of integral points on certain affine schemes.
Known results
Before stating them we have to introduce some terminology. Recall that for a rational point χ ∈ ℙn(ℚ) in projective n-space we define its height as follows:
Represent x = (x0 : … : xn) as a vector with integers xi such that their greatest common divisor is 1. Then the (big) height H(x) is the length of this vector, and the (little) height h(x) its logarithm.
This definition can be made more sophisticated using Arakelov theory, and extends to number fields. The height measures the arithmetic complexity of the point x, and for a given bound c the number of points x with H(x) ≤ c is finite.
By restriction we get a height function on the rational points of any subvariety X ⊂ ℙn. Up to bounded functions it only depends on the ample line bundle ℒ = O (1) on X. Also one can define heights for subvarieties Z ⊂ X, or for effective algebraic cycles (see Faltings 1991).
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- Information
- A Panorama of Number Theory or The View from Baker's Garden , pp. 231 - 246Publisher: Cambridge University PressPrint publication year: 2002
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