Ben Howard

Heegner points, Iwasawa theory, and the Gross-Zagier theorem

The two best-known theorems concerning Heegner points on elliptic curves are the Gross-Zagier theorem, which relates the Neron-Tate heights of Heegner points to central derivatives of L-functions, and the theorem of Kolyvagin, which shows that the nonvanishing of a Heegner point implies bounds on the Shaferavich-Tate group and Mordell-Weil rank. Shortly thereafter, Perrin-Riou begin a program to formulate and prove p-adic and Iwasawa-theoretic versions of the theorems of Gross-Zagier and Kolyvagin. We will give a survey of the main results and conjectures in the subject.