Dmitry Logachev
Simon Bolivar University, Venezuela
Review of the Birch and Swinnerton-Dyer conjecture: theorems of Gross and Zagier, Kolyvagin, Shouwu Zhang....
Abstract: The Birch and Swinnerton-Dyer conjecture relates the analytic and algebraic properties of elliptic curves over number fields. I shall formulate the conjecture giving definitions of all involved objects: the L-function of an elliptic curve, the height pairing on it, the Tate-Shafarevich group etc. Further, I review a significant progress made in the proof of the conjecture in 1990-ies. Namely, I define the Heegner points on elliptic curves, the Gross and Zagier formula for their height and its generalization to the quaternionic modular curves made by Zhang. Kolyvagin's theorem on Tate-Shafarevich group and Wiles' modularity theorem give us the proof of the Birch and Swinnerton-Dyer conjecture for all elliptic curves over Q of rank 0 and 1. The case of rank greater than 1 remains misterious.