By (author) Anthony W. Knapp
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Elliptic Curves by Anthony W. Knapp
Book DescriptionAn 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.
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Book DetailsISBN: 9780691085593
(229mm x 152mm x 25mm)
Imprint: Princeton University Press
Publisher: Princeton University Press
Publish Date: 5-Oct-1992
Country of Publication: United States
Books By Author Anthony W. Knapp
Advanced Algebra, Hardback (November 2007)
Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Many examples and hundreds of problems are included, along with hints or complete solutions.
Lie Groups Beyond an Introduction, Hardback (August 2002)
Takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis, this title uses Lie-theoretic methods to develop a theory having wide applications in mathematics and physics. It includes many problems, tables, and bibliographical notes.
Representation Theory of Semisimple Groups, Paperback (September 2001)
Offers a survey of representation theory of semisimple Lie groups. Suitable for both graduate students and researchers, this book states the theorems precisely, and gives many illustrative examples or classes of examples. It includes for the reader a useful 300-item bibliography and an extensive section of notes.
Cohomological Induction and Unitary Representations, Hardback (May 1995)» View all books by Anthony W. Knapp
Offers a systematic treatment of the development and use of cohomological induction to construct unitary representations. This book develops the necessary background in representation theory and includes an introductory chapter of motivation, a treatment of the "translation principle"ï¿½', and four appendices on algebra and analysis.
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