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OverviewThe book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture. Full Product DetailsAuthor: Henri DarmonPublisher: American Mathematical Society Imprint: American Mathematical Society Volume: No. 101 Weight: 0.268kg ISBN: 9780821828687ISBN 10: 0821828681 Pages: 136 Publication Date: 30 December 2003 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Temporarily unavailable  The supplier advises that this item is temporarily unavailable. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out to you. Table of ContentsElliptic curves Modular forms Heegner points on $X_0(N)$ Heegner points on Shimura curves Rigid analytic modular forms Rigid analytic modular parametrisations Totally real fields ATR points Integration on $\mathcal{H}_p\times\mathcal{H}$ Kolyvagin's theorem Bibliography.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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