Rational Points on Modular Elliptic Curves

Released on by Henri Darmon
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  • Rational Points on Modular Elliptic Curves Book Detail

  • Author : Henri Darmon
  • Release Date : 2004
  • Publisher : American Mathematical Soc.
  • Genre : Mathematics
  • Pages : 146
  • ISBN 13 : 0821828681
  • File Size : 24,24 MB
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Rational Points on Modular Elliptic Curves by Henri Darmon PDF Summary

Book Description: 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.

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Rational Points on Modular Elliptic Curves

Rational Points on Modular Elliptic Curves

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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 e

Rational Points on Elliptic Curves

Rational Points on Elliptic Curves

File Size : 41,41 MB
Total View : 7156 Views
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The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic the