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Elliptic Curves: Diophantine Analysis

Elliptic Curves: Diophantine Analysis
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Field name Details
Dewey Class 515
Title Elliptic Curves ([EBook]) : Diophantine Analysis / by Serge Lang.
Author Lang, Serge. , 1927-2005
Other name(s) SpringerLink (Online service)
Publication Berlin, Heidelberg : Springer , 1978.
Physical Details XII, 264 pages : online resource.
Series Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 0072-7830 ; ; 231
ISBN 9783662070109
Summary Note 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.:
Contents note I. General Algebraic Theory -- I. Elliptic Functions -- II. The Division Equation -- III. p-Adic Addition -- IV. Heights -- V. Kummer Theory -- V1. Integral Points -- II. Approximation of Logarithms -- VII. Auxiliary Results -- VIII. The Baker—Feldman Theorem -- IX. Linear Combinations of Elliptic Logarithms -- X. The Baker—Tijdeman Theorem -- XI. Refined Inequalities.
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