Leseprobe
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Elliptic Curves. (MN-40), Volume 40
Anthony W. Knapp
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Integral domain, Big O notation, Elliptic function, Coefficient, Number theory, Summation, Projective line, Binary quadratic form, Algebraic integer, Monic polynomial, Equation, Weierstrass's elliptic functions, Group isomorphism, Liouville's theorem (complex analysis), Change of variables, Algebraic extension, Birch and Swinnerton-Dyer conjecture, Mathematical induction, Holomorphic function, Polynomial ring, Elementary symmetric polynomial, Analytic function, Hecke operator, Quadratic function, Prime number theorem, Algebra homomorphism, Theorem, Conjecture, Principal axis theorem, J-invariant, Eigenvalues and eigenvectors, Integer matrix, Variable (mathematics), Ideal (ring theory), Fourier analysis, Quadratic equation, Projective variety, Division algebra, Riemann surface, Isogeny, P-adic number, Automorphism, Minimal polynomial (field theory), Analytic continuation, Coprime integers, Function (mathematics), Characteristic polynomial, Cusp form, Modular form, Meromorphic function, Elliptic curve, Algebraic number theory, Eigenform, Quadratic reciprocity, Algebraic number, Complex number, Elliptic integral, Group homomorphism, Associative algebra, Riemann zeta function, General linear group, Linear fractional transformation, Functional equation, Dimension (vector space), Algebraic geometry, Prime number, Simultaneous equations, Inverse function theorem, Dirichlet series, Unique factorization domain