## STNB2018 (32nd edition)

### Heegner points and the Kolyvagin argument.

#### Presenters

Carlos de Vera Piquero

#### Abstract

In this talk, we introduce the classical theory of Heegner points. These points are images of complex multiplication points on modular and Shimura curves through the modular parametrization. Moreover we will describe one of the most important applications of this theory: Kolywagin´s strategy to prove the Birch and Swinnerton-Dyer conjecture in case of elliptic curves $E$ over $\Q$ of rank 1 over a quadratic imaginary field satisfying the Heegner Hypothesis. Such strategy relies on the construction of an Euler system of Heegner points that controls the size of the Selmer group.