Let  be the modular elliptic curve with equation

 

and let  be any elliptic curve defined over  which is a quadratic twist of . Let  be the group of rational points of ,  its Tate-Shafarevich group, and  its complex -series. The aim of my four lectures will be to discuss in some detail the proof of the following theorem.

 

Theorem 0.1. We have if and only if both  and the 2-primary subgroup of  are finite. When these equivalent conditions hold, the full Birch-Swinnerton-Dyer conjecture is valid for E.

 

To my knowledge, a result like this is not konown for the family of all quadratic twists of any other elliptic curve defined over . The proof involves earlier work of K. Rubin and C. Gonzalez-Aviles, as well as some joint recent work by Y. Kezuka, Y. Li, Y. Tian and myself. If time allows, I will also explain a partial generalization to a large family of quadratic twists of the Gross curves , with complex multiplication by the ring of integers of an imaginary quadratic field  , where  is any prime which is congruent to 7 modulo 8.