By Henri Darmon
The booklet surveys a few contemporary advancements within the mathematics of modular elliptic curves. It areas a different emphasis at the building of rational issues on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the an important position performed via modularity in laying off gentle on those heavily similar matters. the most subject matter of the e-book is the speculation of advanced multiplication, Heegner issues, and a few conjectural versions. the 1st 3 chapters introduce the historical past and conditions: elliptic curves, modular types and the Shimura-Taniyama-Weil conjecture, advanced multiplication and the Heegner aspect building. the subsequent 3 chapters introduce versions of modular parametrizations within which modular curves are changed by means of Shimura curves hooked up to sure indefinite quaternion algebras. the most new contributions are present in Chapters 7-9, which survey the author's makes an attempt to increase the idea of Heegner issues and intricate multiplication to occasions the place the bottom box isn't really a CM box. bankruptcy 10 explains the evidence of Kolyvagin's theorem, which relates Heegner issues to the mathematics of elliptic curves and results in the easiest proof to date for the Birch and Swinnerton-Dyer conjecture.
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