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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
##### Keywords:
elliptic curves; potential modularity
##### Citations:
Zbl 0982.11033; Zbl 0823.11029
Full Text:
##### References:
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