By Henri Darmon
The e-book surveys a few fresh 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 the most important function performed by means of modularity in laying off mild on those heavily similar matters. the most topic of the publication is the speculation of advanced multiplication, Heegner issues, and a few conjectural variations. the 1st 3 chapters introduce the historical past and stipulations: elliptic curves, modular varieties and the Shimura-Taniyama-Weil conjecture, advanced multiplication and the Heegner element building. the subsequent 3 chapters introduce variations of modular parametrizations during which modular curves are changed by means of Shimura curves hooked up to yes 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 events the place the bottom box isn't a CM box. bankruptcy 10 explains the facts of Kolyvagin's theorem, which relates Heegner issues to the mathematics of elliptic curves and ends up in the easiest facts to this point for the Birch and Swinnerton-Dyer conjecture.
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