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 function performed by way of modularity in laying off mild on those heavily comparable concerns. the most subject of the publication is the speculation of advanced multiplication, Heegner issues, and a few conjectural versions. the 1st 3 chapters introduce the heritage and conditions: elliptic curves, modular varieties and the Shimura-Taniyama-Weil conjecture, complicated multiplication and the Heegner aspect development. the following 3 chapters introduce versions of modular parametrizations within which modular curves are changed via Shimura curves connected 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 complicated multiplication to events the place the bottom box isn't a CM box. bankruptcy 10 explains the evidence of Kolyvagin's theorem, which relates Heegner issues to the mathematics of elliptic curves and ends up in the simplest proof to date for the Birch and Swinnerton-Dyer conjecture.
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