Elliptic Curves: Diophantine AnalysisSpringer Science & Business Media, 1978年11月1日 - 264 頁 It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points. |
內容
II | 3 |
III | 6 |
IV | 10 |
V | 13 |
VI | 17 |
VII | 19 |
VIII | 23 |
IX | 26 |
XLII | 148 |
XLIII | 151 |
XLIV | 155 |
XLV | 159 |
XLVII | 162 |
XLVIII | 164 |
XLIX | 166 |
L | 169 |
X | 33 |
XII | 37 |
XIII | 43 |
XIV | 47 |
XV | 48 |
XVI | 54 |
XVII | 55 |
XVIII | 62 |
XIX | 68 |
XX | 73 |
XXI | 77 |
XXIII | 84 |
XXIV | 85 |
XXV | 88 |
XXVI | 93 |
XXVII | 98 |
XXVIII | 101 |
XXX | 105 |
XXXI | 107 |
XXXII | 109 |
XXXIII | 115 |
XXXIV | 128 |
XXXV | 129 |
XXXVI | 137 |
XXXVII | 140 |
XXXVIII | 142 |
XXXIX | 144 |
XL | 146 |
XLI | 147 |
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常見字詞
a₁ abelian varieties absolute value algebraic integers algebraic number analytic apply assume b₁ B₁u₁ C₁ Chapter coefficients complex multiplication complex numbers concludes the proof coordinates degree denominator denote derivatives diophantine diophantine approximation elements elliptic curve elliptic curve defined elliptic function estimate exists factor finite extension finite number follows formula Galois group give given height Hence homomorphism induction inequality integral points isomorphism Kummer theory lattice linear combinations linear equations linearly independent log max logarithms lower bound main lemma Math Néron function notation number field number of equations obtained p-adic P₁ parameter pole positive integer prime number rational numbers reduction roots of unity satisfying Siegel subgroup sufficiently large Suppose Tate curve Theorem 1.2 torsion points u₁ u₂ units variables w₁ w₂ Weierstrass equation Weierstrass form Weierstrass function whence write x₁ zero