Arbitrary potential modularity for elliptic curves over totally real number fields.

*(English)*Zbl 1277.11046Author’s introduction: It is conjectured that an elliptic curve \(E\) defined over a totally real number
field \(F\) is modular i.e. the associated \(l\)-adic representation \(\rho_E:=\rho_{E,l}\) of \(\Gamma_F :=\mathrm{Gal}(\overline F/F)\), for some rational prime \(l\), is isomorphic to the \(l\)-adic representation \(\rho_\pi :=\rho_{\pi,l}\) of \(\Gamma_F\) associated to some automorphic representation \(\pi\) of \(\mathrm{GL}(2)=F\). This conjecture was proved when \(F = \mathbb Q\) (see [C. Breuil et al., J. Am. Math. Soc. 14, No. 4, 843–939 (2001; Zbl 0982.11033)], [A. Wiles, Ann. Math. (2) 141, 443–551 (1995; Zbl 0823.11029)]).

In this paper, we prove the following result:

Theorem 1.1. Let \(E\) be an elliptic curve defined over a totally real number field \(F\). Then there exist a totally real number field \(F''\), which contains \(F\), and rational primes \(l\) and \(p\) that are totally split in \(F''\) such that \(E/F'\) is modular for any totally real number field \(F'\) which contains \(F''\) and has the property that \(l\) and \(p\) split completely in \(F'\).

In this paper, we prove the following result:

Theorem 1.1. Let \(E\) be an elliptic curve defined over a totally real number field \(F\). Then there exist a totally real number field \(F''\), which contains \(F\), and rational primes \(l\) and \(p\) that are totally split in \(F''\) such that \(E/F'\) is modular for any totally real number field \(F'\) which contains \(F''\) and has the property that \(l\) and \(p\) split completely in \(F'\).

##### MSC:

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11F80 | Galois representations |

11G05 | Elliptic curves over global fields |

11R80 | Totally real fields |

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\textit{C. Virdol}, Funct. Approximatio, Comment. Math. 45, No. 2, 265--269 (2011; Zbl 1277.11046)

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##### References:

[1] | C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over \(\mathbbQ\): wild 3-adic exercises , J. Amer. Math. Soc. 14 (2001), 843-939. · Zbl 0982.11033 |

[2] | C. Khare, M. Larsen and R. Ramakrishna, Transcendental \(l\)-adic Galois representations , Math. Res. Lett. 12 (2005), no. 5-6, 685-699. · Zbl 1134.11023 |

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[4] | J-P. Serre, Abelian \(l\)-adic representations and elliptic curves . Revised preprint of the 1968 edition, A.K Peters, Ltd., Wellesley, MA, 1998. · Zbl 0902.14016 |

[5] | C. Skinner and A. Wiles, Nearly ordinary deformations of irreducible residual representations , Ann. Fac. Sci. Toulouse MATH.(6) 10 (2001), no. 1, 185-215. · Zbl 1024.11036 |

[6] | R. Taylor, On Galois representations associated to Hilbert modular forms , Invent. Math. 98 , 1989, 265-280. · Zbl 0705.11031 |

[7] | R. Taylor, Remarks on a conjecture of Fontaine and Mazur , Journal of the Institute of Mathematics of Jussieu 1 (2002), 125-143. · Zbl 1047.11051 |

[8] | A. Wiles, Modular elliptic curves and Fermat’s last theorem , Annals of Mathematics 141 (1995), 443-551. · Zbl 0823.11029 |

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