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Zbl 0752.14034
Silverman, Joseph H.; Tate, John
Rational points on elliptic curves.
(English)
[B] Undergraduate Texts in Mathematics. New York: Springer-Verlag. x, 281 p. (1992). ISBN 0-387-97825-9/hbk

The book gives a good introduction for students which are interested in Diophantine equations and arithmetic geometry. It is based on lectures of J. Tate from 1961. It contains a lot of exercises. Often further developments and applications are explained, for instance Lenstra's algorithm for factorisation of integers using elliptic curves.\par The book starts with the geometry and group structure of elliptic curves. It contains the Nagell-Lutz theorem describing points if finite order, the Mordell-Weil theorem on the finite generation of the group of rational points, the Thue-Siegel theorem on the finiteness of the set of integer points. Also points over finite fields are considered. --- At the end one finds complex multiplication and Galois representations associated to torsion points.\par The algebraic geometry needed for the purpose of the book (for instance Bezout's theorem) is presented in an appendix.
[G.Pfister (Berlin)]
MSC 2000:
*14H52 Elliptic curves
14G05 Rationality questions, rational points
14-01 Textbooks (algebraic geometry)
14-02 Research monographs (algebraic geometry)
14G15 Finite ground fields
11G05 Elliptic curves over global fields

Keywords: Diophantine equations; elliptic curves; Nagell-Lutz theorem; Mordell-Weil theorem; rational points; Thue-Siegel theorem; integer points; finite fields; complex multiplication; torsion points

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