Ebook: Elliptic Curves

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

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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

---

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

Content:
Front Matter....Pages i-xxi
Introduction to Rational Points on Plane Curves....Pages 1-22
Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve....Pages 23-43
Plane Algebraic Curves....Pages 45-63
Elliptic Curves and Their Isomorphisms....Pages 65-84
Families of Elliptic Curves and Geometric Properties of Torsion Points....Pages 85-102
Reduction mod p and Torsion Points....Pages 103-123
Proof of Mordell’s Finite Generation Theorem....Pages 125-142
Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields....Pages 143-156
Descent and Galois Cohomology....Pages 157-166
Elliptic and Hypergeometric Functions....Pages 167-187
Theta Functions....Pages 189-207
Modular Functions....Pages 209-231
Endomorphisms of Elliptic Curves....Pages 233-252
Elliptic Curves over Finite Fields....Pages 253-273
Elliptic Curves over Local Fields....Pages 275-289
Elliptic Curves over Global Fields and ?-Adic Representations....Pages 291-308
Remarks on the Birch and Swinnerton-Dyer Conjecture....Pages 309-324
Remarks on the Modular Elliptic Curves Conjecture and Fermat’s Last Theorem....Pages 325-332
Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties....Pages 333-343
Back Matter....Pages 345-381
Families of Elliptic Curves....Pages 403-487
....Pages 383-401