# Fundamentals of Diophantine Geometry

@inproceedings{Lang1983FundamentalsOD, title={Fundamentals of Diophantine Geometry}, author={Serge Lang}, year={1983} }

1 Absolute Values.- 2 Proper Sets of Absolute Values. Divisors and Units.- 3 Heights.- 4 Geometric Properties of Heights.- 5 Heights on Abelian Varieties.- 6 The Mordell-Weil Theorem.- 7 The Thue-Siegel-Roth Theorem.- 8 Siegel's Theorem and Integral Points.- 9 Hilbert's Irreducibility Theorem.- 10 Weil Functions and Neron Divisors.- 11 Neron Functions on Abelian Varieties.- 12 Algebraic Families of Neron Functions.- 13 Neron Functions Over the Complex Numbers.- Review of S. Lang's Diophantine…

## 353 Citations

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This result is inspired by an analogous theorem for elliptic curves over number fields due to J. Cheon and S. Hahn [ChH, Theorem], generalizing a result of J. Silverman [Si2, Proposition 10], and by…

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- MathematicsCompositio Mathematica
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Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous…

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We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor,…

### The Mordell-Weil Theorem

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In this chapter we prove the celebrated theorem of Mordell—Weil for elliptic curves defined over the field of rational numbers. Our treatment is elementary in the sense that no sophisticated results…

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- MathematicsMath. Comput.
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The regulator of A(k), which appears in the statement of the conjecture of Birch and Swinnerton-Dyer, is defined in terms of the canonical height and thus the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank is needed.

### A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing

- Mathematics
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In 1929 Siegel proved a celebrated theorem on finiteness for integral solutions of certain diophantine equations. This theorem applies to systems of polynomial equations which either (a) describe an…