The Main Conjecture and Annihilation of Class Groups
In the mid 1980s, Thaine and Kolyvagin invented new techniques for constructing relations in ideal class groups. These methods have had profound consequences. Not only is it now possible to give a fairly elementary proof of the Main Conjecture, but these ideas also allowed Rubin to give the first examples of finite Tate-Shafarevich groups for elliptic curves.
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