On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Let E/F be a modular elliptic curve defined over a totally real number field F and let φ be its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when [F : ℚ] is even and φ not new at any prime.