Titlebook: Rational Points on Elliptic Curves; Joseph H. Silverman,John Tate Textbook 19921st edition Springer Science+Business Media New York 1992 A

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Complex Multiplication,ere we mean points of finite order with arbitrary complex coordinates, not just the ones with rational coordinates that we studied in Chapter II. So we will need to use some basic theorems about extension fields and Galois theory, but nothing very fancy. We will start by reminding you of most of the

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Points of Finite Order,n our study of points of finite order on cubic curves by looking at points of order two and order three. As usual, we will assume that our non-singular cubic curve is given by a Weierstrass equation ., and that the point at infinity . is taken to be the zero element for the group law.

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Integer Points on Cubic Curves,y), then the set of all rational points on . forms a finitely generated abelian group. So we can get every rational point on . by starting from some finite set and adding points using the geometrically defined group law.

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Introduction,The theory of Diophantine equations is that branch of number theory which deals with the solution of polynomial equations in either integers or rational numbers. The subject itself is named after one of the greatest of the ancient Greek algebraists, Diophantus of Alexandria,. who formulated and solved many such problems.

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