Elliptic Curves (eBook)

Anthony W. Knapp (Autor)

eBook Download: PDF

2018
448 Seiten
Princeton University Press (Verlag)
978-0-691-18690-0 (ISBN)

Infos

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.

Autor

Anthony W. Knapp is Professor of Mathematics at the University of New York, Stony Brook. He is the author of Representation Theory of Semisimple Groups: An Overview Based on Examples and Lie Groups, Lie Algebras, and Cohomology (both published by Princeton University Press).

Produkt-Details

• Erscheint lt. Verlag: 5.6.2018
• Reihe/Serie: Mathematical Notes
• Reihe/Serie: Mathematical Notes
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: Affine plane (incidence geometry) • Affine space • affine variety • Algebra homomorphism • algebraic extension • Algebraic Geometry • Algebraic integer • algebraic number • algebraic number theory • analytic continuation • analytic function • Associative algebra • automorphism • Big O notation • Binary quadratic form • Birch and Swinnerton-Dyer conjecture • Bounded set (topological vector space) • Change of variables • Characteristic Polynomial • coefficient • Compactification (mathematics) • Complex conjugate • complex manifold • complex number • conjecture • Coprime integers • cusp form • Cyclic group • Degeneracy (mathematics) • Dimension (vector space) • Dirichlet character • Dirichlet series • division algebra • Divisor • Eigenform • Eigenvalues and Eigenvectors • Elementary symmetric polynomial • Elliptic Curve • elliptic function • Elliptic Integral • Equation • Euler product • Finitely generated abelian group • fourier analysis • functional equation • Function (mathematics) • general linear group • Group homomorphism • Group isomorphism • Hecke Operator • holomorphic function • Homomorphism • Ideal (ring theory) • Integer • Integer matrix • integral domain • Intersection (set theory) • inverse function theorem • Invertible matrix • irreducible polynomial • isogeny • J-invariant • Linear fractional transformation • Linear map • Liouville's theorem (complex analysis) • Mathematical Induction • Meromorphic Function • Minimal polynomial (field theory) • Möbius transformation • modular form • monic polynomial • Number Theory • P-adic number • polynomial ring • power series • prime factor • Prime number • prime number theorem • Principal axis theorem • Principal ideal • Principal ideal domain • Projective line • projective variety • Quadratic equation • quadratic function • Quadratic Reciprocity • Riemann surface • Riemann zeta function • Simultaneous Equations • Special case • Summation • Taylor series • Theorem • Torsion subgroup • transcendence degree • uniformization theorem • Unique factorization domain • Variable (mathematics) • Weierstrass's elliptic functions • Weil conjecture
• ISBN-10: 0-691-18690-1 / 0691186901
• ISBN-13: 978-0-691-18690-0 / 9780691186900