The Kolyvagin System of Heegner Points Pt I

Heegner points were introduced in the ‘70s by Birch, inspired by earlier work of Heegner. They are points on rational elliptic curves defined over ring class fields of imaginary quadratic fields and they are a key ingredient to prove much of what is nowadays known about the Birch and Swinnerton-Dyer conjecture. In these lectures, we will recall their construction and the main results involving them. In particular, we will focus on Kolyvagin’s work, which provides a bound on Selmer groups of elliptic curves. If time permits we will discuss Iwasawa theoretic analogues of this statement (leading to one divisibility in Perrin-Riou’s main conjecture) and possible generalisations of the theory of Heegner points.