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Arbitrary potential modularity for elliptic curves over totally real number fields. (English) Zbl 1277.11046
Author’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 \(F'\).
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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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
[3] R. P. Langlands, Base change for GL\(_2\) , Ann. of Math. Studies 96, Princeton University Press, 1980. · Zbl 0444.22007
[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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