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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 \(F'\).
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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