Let E/F be a modular elliptic curve defined over a totally real number field F and let f be its associated eigenform. This paper presents a new method, inspired by the 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 Logachev, that is, when [F:Q] is even and f not new at any prime.
|Titolo:||On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields|
|Parole Chiave:||Birch and swinnerton-dyer conjecture; Congruences between hilbert modular forms; Elliptic curves; Shimura varieties|
|Data di pubblicazione:||2006|
|Digital Object Identifier (DOI):||10.5802/aif.2197|
|Appare nelle tipologie:||01 - Articolo su periodico|