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

Matteo Longo[1]

  • [1] Université Louis Pasteur IRMA 7, rue René Descartes 67084 Strasbourg (France)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 3, page 689-733
  • ISSN: 0373-0956

Abstract

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Let E / F be a modular elliptic curve defined over a totally real number field F and let φ be its associated eigenform. This paper 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.

How to cite

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Longo, Matteo. "On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields." Annales de l’institut Fourier 56.3 (2006): 689-733. <http://eudml.org/doc/10161>.

@article{Longo2006,
abstract = {Let $E/F$ be a modular elliptic curve defined over a totally real number field $F$ and let $\phi $ be its associated eigenform. This paper 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 $[F:\mathbb\{Q\}]$ is even and $\phi $ not new at any prime.},
affiliation = {Université Louis Pasteur IRMA 7, rue René Descartes 67084 Strasbourg (France)},
author = {Longo, Matteo},
journal = {Annales de l’institut Fourier},
keywords = {Elliptic Curves; Birch and Swinnerton-Dyer Conjecture; Shimura Varieties; Congruences between Hilbert Modular Forms; elliptic curves; Birch and Swinnerton-Dyer conjecture; Shimura variety; congruences between Hilbert modular forms},
language = {eng},
number = {3},
pages = {689-733},
publisher = {Association des Annales de l’institut Fourier},
title = {On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields},
url = {http://eudml.org/doc/10161},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Longo, Matteo
TI - On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 3
SP - 689
EP - 733
AB - Let $E/F$ be a modular elliptic curve defined over a totally real number field $F$ and let $\phi $ be its associated eigenform. This paper 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 $[F:\mathbb{Q}]$ is even and $\phi $ not new at any prime.
LA - eng
KW - Elliptic Curves; Birch and Swinnerton-Dyer Conjecture; Shimura Varieties; Congruences between Hilbert Modular Forms; elliptic curves; Birch and Swinnerton-Dyer conjecture; Shimura variety; congruences between Hilbert modular forms
UR - http://eudml.org/doc/10161
ER -

References

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