Elliptic Curves

1. Front Matter

   Pages i-xi

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2. General Algebraic Theory

   1. Front Matter

      Pages 1-1

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   2. Elliptic Functions

      • Serge Lang

      Pages 3-32

   3. The Division Equation

      • Serge Lang

      Pages 33-46

   4. p-Adic Addition

      • Serge Lang

      Pages 47-76

   5. Heights

      • Serge Lang

      Pages 77-100

   6. Kummer Theory

      • Serge Lang

      Pages 101-127

   7. Integral Points

      • Serge Lang

      Pages 128-153

3. Approximation of Logarithms

   1. Front Matter

      Pages 155-158

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   2. Auxiliary Results

      • Serge Lang

      Pages 159-180

   3. The Baker—Feldman Theorem

      • Serge Lang

      Pages 181-192

   4. Linear Combinations of Elliptic Logarithms

      • Serge Lang

      Pages 193-217

   5. The Baker—Tijdeman Theorem

      • Serge Lang

      Pages 218-233

   6. Refined Inequalities

      • Serge Lang

      Pages 234-252

4. Back Matter

   Pages 253-264

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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.

• Algebra
• Arithmetic
• Curves
• Diophantische Approximation
• Diophantische Ungleichung
• Elliptische Kurve
• equation
• function
• theorem

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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