The Arithmetic of Elliptic Curves

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The Arithmetic of Elliptic Curves is written by Joseph H. Silverman and published by Springer Science & Business Media. It's available with International Standard Book Number or ISBN identification 0387094946 (ISBN 10) and 9780387094946 (ISBN 13).

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.