Elliptic curves

• 作者: Husemöller, Dale.
• 出版: New York : Springer ©2004.
• 版本: 2nd ed.
• 稽核項: 1 online resource (xxi, 487 pages) :illustrations.
• 叢書名: Graduate texts in mathematics ;111
• 標題: Curves, Elliptic. , Curves, Algebraic. , Group schemes (Mathematics) , GeometryAlgebraic. , Electronic books. , MATHEMATICS Geometry -- Algebraic. , MATHEMATICS
• ISBN: 0387215778 , 9780387215778
• ISBN: 0387954902 , 9780387954905 , 6610189773 , 9786610189779
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• 附註: Includes bibliographical references (pages 465-478) and index. Introduction to Rational Points on Plane Curves -- Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve -- Plane Algebraic Curves -- Factorial Rings and Elimination Theory -- Elliptic Curves and Their Isomorphism -- Families of Elliptic Curves and Geometric Properties of Torsion Points -- Reduction mod p and Torsion Points -- Proof of Mordell's Finite Generation Theorem -- Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields -- Descent and Galois Cohomology -- Elliptic and Hypergeometric Functions -- Theta Functions -- Modular Functions -- Endomorphisms of Elliptic Curves -- Elliptic Curves over Finite Fields -- Elliptic Curves over Local Fields -- Elliptic Curves over Global Fields and l-adic Representations -- L-Functions of an Elliptic Curve and Its Analytic Continuation -- Remarks on the Birch and Swinnerton-Dyer Conjecture -- Remarks on the Modular Curves Conjecture and Fermat's Last Theorem -- Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties -- Families of Elliptic Curves -- Appendix I: Calabi-Yau Manifolds and String Theory -- Appendix II: Elliptic Curves in Algorithmic Number Theory -- Appendix III: Guide to the Exercises.
• 摘要: This book is an introduction to the theory of elliptic curves, ranging from elementary topics to current research. The first chapters, which grew out of Tate's Haverford Lectures, cover the arithmetic theory of elliptic curves over the field of rational numbers. This theory is then recast into the powerful and more general language of Galois cohomology and descent theory. An analytic section of the book includes such topics as elliptic functions, theta functions, and modular functions. Next, the book discusses the theory of elliptic curves over finite and local fields and provides a survey of results in the global arithmetic theory, especially those related to the conjecture of Birch and Swinnerton-Dyer. This new edition contains three new chapters. The first is an outline of Wiles's proof of Fermat's Last Theorem. The two additional chapters concern higher-dimensional analogues of elliptic curves, including K3 surfaces and Calabi-Yau manifolds. Two new appendices explore recent applications of elliptic curves and their generalizations. The first, written by Stefan Theisen, examines the role of Calabi-Yau manifolds and elliptic curves in string theory, while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory. About the First Edition: "All in all the book is well written, and can serve as basis for a student seminar on the subject."--G. Faltings, Zentralblatt
• 電子資源: https://dbs.tnua.edu.tw/login?url=https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=108073
• 系統號: 005309176
• 資料類型: 電子書
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• 引用網址: 複製連結
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