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.
On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields / Matteo Longo. - In: ANNALES DE L'INSTITUT FOURIER. - ISSN 0373-0956. - 56:3(2006), pp. 689-733. [10.5802/aif.2197]
On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
M. Longo
2006
Abstract
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.
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