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The differential geometry of curves and surfaces in Euclidean space has fascinated mathematicians since the time of Newton. Here the authors cast the theory into a new light, that of singularity theory. This second edition has been thoroughly revised throughout and includes a multitude of new exercises and examples. A new final chapter has been added which covers recently developed techniques in the classification of functions of several variables, a subject central to many applications of singularity theory. Also in this second edition are new sections on the Morse lemma and the classification of plane curve singularities. The only prerequisites for students to follow this textbook are a familiarity with linear algebra and advanced calculus. Thus it will be invaluable for anyone who would like an introduction to modern singularity theory.

Singularities (Mathematics) --- Curves.

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Algebraic geometry --- Curves, Elliptic --- Diophantine analysis --- Courbes elliptiques --- Analyse diophantienne --- Rational points (Geometry) --- Curves, Elliptic. --- Diophantine analysis. --- Rational points (Geometry). --- 511 --- 511 Number theory --- Number theory --- Indeterminate analysis --- Forms, Quadratic --- Points, Rational (Geometry) --- Arithmetical algebraic geometry --- Elliptic curves --- Curves, Algebraic

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Boundary value problems. --- Curves, Elliptic. --- Fluid dynamics --- Mathematical models.

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An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem.Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways.Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

Curves, Elliptic --- 512.74 --- 511.3 --- Elliptic curves --- Curves, Algebraic --- Algebraic groups. Abelian varieties --- Analytical, additive and other number-theory problems. Diophantine approximations --- Curves, Elliptic. --- 511.3 Analytical, additive and other number-theory problems. Diophantine approximations --- 512.74 Algebraic groups. Abelian varieties

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Computer-aided design --- Curves, Algebraic --- Surfaces --- Conception assistée par ordinateur --- Courbes algébriques --- Data processing --- Data processing --- Informatique