Heights

Skoruppa H.-P.

Table of contents :
1.1 The height of a rational number……Page 7
1.2 The Mahler measure of a polynomial……Page 8
1.3 Pisot and Salem numbers……Page 14
1.4 The height of an algebraic number……Page 16
1.5 Two easy Lehmer type theorems……Page 20
1.6 Numbers with conjugates outside a given set……Page 21
1.7 Transfinite diameters……Page 25
1.8 Heights of non-reciprocal numbers……Page 28
1.9 Proof of Smyth’s theorem……Page 29
1.10 Remarks……Page 34
2.1 Heights on affine plane curves……Page 37
2.2 Heights on projective space……Page 40
2.3 Plane curves as diophantine equations……Page 42
2.5 Heights on elliptic curves……Page 46
2.6 Infinite descent on elliptic curves……Page 52
2.7 The Mordell-Weil theorem……Page 53
2.8 Supplements……Page 58
2.9 Local decomposition……Page 59
2.9.1 The Green’s function of an elliptic curve……Page 60
2.9.2 The Néron functions associated to places……Page 65
2.9.3 The decomposition formula……Page 67
3.1 Mesure de Mahler de polynômes en plusieurs variables……Page 69
3.2 Calcul rapide de l’hauteur canonique……Page 70
3.3 Fonctions de Néron……Page 71
Bibliography……Page 73