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Rational Points on Modular Elliptic Curves
Henri Darmon, McGill University, Montreal, QC, Canada
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
2004; 129 pp; softcover
Number: 101
ISBN-10: 0-8218-2868-1
ISBN-13: 978-0-8218-2868-7
List Price: US$32
Member Price: US$25.60
All Individuals: US$25.60
Order Code: CBMS/101
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The 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.

Readership

Graduate students and research mathematicians interested in number theory and arithmetic algebraic geometry.

Reviews

"The book is well- written, and would be a good text to run a graduate seminar on, or for a graduate student to make independent study of, as the author has tried his best to make the material accessible."

-- Chandrashekhar Khare for Mathematical Reviews

Table of Contents

  • Elliptic 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
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