Elliptic Curves. (MN-40), Volume 40 /
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| Author / Creator: | Knapp, Anthony W., author. |
|---|---|
| Imprint: | Princeton, NJ : Princeton University Press, [2018] ©1993 |
| Description: | 1 online resource |
| Language: | English |
| Series: | Mathematical Notes ; 40 |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/13918831 |
| Related Items: | Title is part of eBook package:
Princeton Mathematical Notes eBook-Package 1970-2016 Title is part of eBook package: Princeton University Press eBook-Package Archive 1927-1999 |
MARC
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| 100 | 1 | |a Knapp, Anthony W., |e author. |4 aut |4 http://id.loc.gov/vocabulary/relators/aut | |
| 245 | 1 | 0 | |a Elliptic Curves. (MN-40), Volume 40 / |c Anthony W. Knapp. |
| 264 | 1 | |a Princeton, NJ : |b Princeton University Press, |c [2018] | |
| 264 | 4 | |c ©1993 | |
| 300 | |a 1 online resource | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 347 | |a text file |b PDF |2 rda | ||
| 490 | 0 | |a Mathematical Notes ; |v 40 | |
| 505 | 0 | 0 | |t Frontmatter -- |t CONTENTS -- |t List of Figures -- |t List of Tables -- |t Preface -- |t Standard Notation -- |t Elliptic Curves -- |t I. Overview -- |t II. Curves in Projective Space -- |t III. Cubic Curves in Weierstrass Form -- |t IV. Mordell's Theorem -- |t V. Torsion Subgroup of E(Q) -- |t VI. Complex Points -- |t VII. Dirichlet's Theorem -- |t VIII. Modular Forms for SL(2,ℤ) -- |t IX. Modular Forms for Hecke Subgroups -- |t X. L Function of an Elliptic Curve -- |t XI. Eichler-Shimura Theory -- |t XII. Taniyama-Weil Conjecture -- |t Notes -- |t References -- |t Index of Notation -- |t Index |
| 520 | |a 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. | ||
| 546 | |a In English. | ||
| 588 | 0 | |a Description based on online resource; title from PDF title page (publisher's Web site, viewed 29. Jun 2022) | |
| 650 | 0 | |a Curves, Elliptic. | |
| 650 | 7 | |a MATHEMATICS / Geometry / Algebraic. |2 bisacsh | |
| 653 | |a Affine plane (incidence geometry). | ||
| 653 | |a Affine space. | ||
| 653 | |a Affine variety. | ||
| 653 | |a Algebra homomorphism. | ||
| 653 | |a Algebraic extension. | ||
| 653 | |a Algebraic geometry. | ||
| 653 | |a Algebraic integer. | ||
| 653 | |a Algebraic number theory. | ||
| 653 | |a Algebraic number. | ||
| 653 | |a Analytic continuation. | ||
| 653 | |a Analytic function. | ||
| 653 | |a Associative algebra. | ||
| 653 | |a Automorphism. | ||
| 653 | |a Big O notation. | ||
| 653 | |a Binary quadratic form. | ||
| 653 | |a Birch and Swinnerton-Dyer conjecture. | ||
| 653 | |a Bounded set (topological vector space). | ||
| 653 | |a Change of variables. | ||
| 653 | |a Characteristic polynomial. | ||
| 653 | |a Coefficient. | ||
| 653 | |a Compactification (mathematics). | ||
| 653 | |a Complex conjugate. | ||
| 653 | |a Complex manifold. | ||
| 653 | |a Complex number. | ||
| 653 | |a Conjecture. | ||
| 653 | |a Coprime integers. | ||
| 653 | |a Cusp form. | ||
| 653 | |a Cyclic group. | ||
| 653 | |a Degeneracy (mathematics). | ||
| 653 | |a Dimension (vector space). | ||
| 653 | |a Dirichlet character. | ||
| 653 | |a Dirichlet series. | ||
| 653 | |a Division algebra. | ||
| 653 | |a Divisor. | ||
| 653 | |a Eigenform. | ||
| 653 | |a Eigenvalues and eigenvectors. | ||
| 653 | |a Elementary symmetric polynomial. | ||
| 653 | |a Elliptic curve. | ||
| 653 | |a Elliptic function. | ||
| 653 | |a Elliptic integral. | ||
| 653 | |a Equation. | ||
| 653 | |a Euler product. | ||
| 653 | |a Finitely generated abelian group. | ||
| 653 | |a Fourier analysis. | ||
| 653 | |a Function (mathematics). | ||
| 653 | |a Functional equation. | ||
| 653 | |a General linear group. | ||
| 653 | |a Group homomorphism. | ||
| 653 | |a Group isomorphism. | ||
| 653 | |a Hecke operator. | ||
| 653 | |a Holomorphic function. | ||
| 653 | |a Homomorphism. | ||
| 653 | |a Ideal (ring theory). | ||
| 653 | |a Integer matrix. | ||
| 653 | |a Integer. | ||
| 653 | |a Integral domain. | ||
| 653 | |a Intersection (set theory). | ||
| 653 | |a Inverse function theorem. | ||
| 653 | |a Invertible matrix. | ||
| 653 | |a Irreducible polynomial. | ||
| 653 | |a Isogeny. | ||
| 653 | |a J-invariant. | ||
| 653 | |a Linear fractional transformation. | ||
| 653 | |a Linear map. | ||
| 653 | |a Liouville's theorem (complex analysis). | ||
| 653 | |a Mathematical induction. | ||
| 653 | |a Meromorphic function. | ||
| 653 | |a Minimal polynomial (field theory). | ||
| 653 | |a Modular form. | ||
| 653 | |a Monic polynomial. | ||
| 653 | |a Möbius transformation. | ||
| 653 | |a Number theory. | ||
| 653 | |a P-adic number. | ||
| 653 | |a Polynomial ring. | ||
| 653 | |a Power series. | ||
| 653 | |a Prime factor. | ||
| 653 | |a Prime number theorem. | ||
| 653 | |a Prime number. | ||
| 653 | |a Principal axis theorem. | ||
| 653 | |a Principal ideal domain. | ||
| 653 | |a Principal ideal. | ||
| 653 | |a Projective line. | ||
| 653 | |a Projective variety. | ||
| 653 | |a Quadratic equation. | ||
| 653 | |a Quadratic function. | ||
| 653 | |a Quadratic reciprocity. | ||
| 653 | |a Riemann surface. | ||
| 653 | |a Riemann zeta function. | ||
| 653 | |a Simultaneous equations. | ||
| 653 | |a Special case. | ||
| 653 | |a Summation. | ||
| 653 | |a Taylor series. | ||
| 653 | |a Theorem. | ||
| 653 | |a Torsion subgroup. | ||
| 653 | |a Transcendence degree. | ||
| 653 | |a Uniformization theorem. | ||
| 653 | |a Unique factorization domain. | ||
| 653 | |a Variable (mathematics). | ||
| 653 | |a Weierstrass's elliptic functions. | ||
| 653 | |a Weil conjecture. | ||
| 773 | 0 | 8 | |i Title is part of eBook package: |d De Gruyter |t Princeton Mathematical Notes eBook-Package 1970-2016 |z 9783110494921 |o ZDB-23-PMN |
| 773 | 0 | 8 | |i Title is part of eBook package: |d De Gruyter |t Princeton University Press eBook-Package Archive 1927-1999 |z 9783110442496 |
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