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Ordered algebraic structures --- Algebraic geometry --- Complex analysis

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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.

Curves, Elliptic --- Group schemes (Mathematics) --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Geometry, algebraic. --- Curves, Elliptic. --- Curves, Algebraic. --- Algebraic geometry --- Geometry --- Geometry, Algebraic.

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Geometry, Algebraic. --- Arithmetical algebraic geometry. --- Homology theory. --- Cohomologie p-adique. --- Géométrie algébrique arithmétique. --- Cohomologie cristalline.

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Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions. The second edition contains five new chapters which present some of the most important recent result on the subject. Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture.

Abelian varieties. --- Riemann surfaces --- Riemann, surfaces de --- Riemann surfaces. --- 512.74 --- Surfaces, Riemann --- Functions --- Varieties, Abelian --- Geometry, Algebraic --- Algebraic groups. Abelian varieties --- 512.74 Algebraic groups. Abelian varieties --- Abelian varieties --- Algebraic geometry. --- Number theory. --- Functions of complex variables. --- Algebraic Geometry. --- Number Theory. --- Several Complex Variables and Analytic Spaces. --- Complex variables --- Elliptic functions --- Functions of real variables --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry --- Geometry

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Algebraic geometry --- 514.1 --- General geometry --- 514.1 General geometry --- Vector bundles. --- Rings (Algebra) --- Homology theory. --- Fibrés vectoriels --- Anneaux (algèbre) --- Homologie --- Fibrés vectoriels. --- Homologie.