APPENDIX: POTENTIAL MODULARITY OF ELLIPTIC CURVES OVER TOTALLY REAL FIELDS

The following theorem is well known to experts. Theorem 0.1. Let E an elliptic curve over a totally real number field F. Then there exists a totally real number field F ′ ⊃ F such that EF ′ is modular. We explain what we mean by “modular”. Let F ′ be a totally real number field (a finite extension of Q). Let π be a cuspidal automorphic represen-tation of GL2(AF ′). We shall suppose that the archimedean components of π are such that π corresponds to a Hilbert modular form of parallel weight 2. Taylor has associated to π a compatible system (ρπ,λ) of representations of the Galois group GF ′ ([12]). There is a conductor n, which is an ideal of the rings of integers of F ′, a Hecke algebra T with Hecke operators Tq ∈ T, q prime of F ′ prime to n, and a morphism θ: T → C. The subfield L of C generated by the image of θ is a finite extension of Q. For each prime λ of L, the Galois representation ρπ,λ: GF ′ → GL2(Lλ) is absolutely irre-ducible (prop. 2.1. of [14]), unramified outside the primes dividing n and the rational prime ℓ below λ, and is characterized by: tr(ρπ,λ(Frobq)) = θ(Tq), for every prime q of F ′ which is prime to nℓ. When we say that E is modular over F ′, we mean that there exists such a π such that, for any prime λ of L, the Galois representation ρπ,λ is isomorphic to the Galois representation ρ(E)ℓ given by the action of GF ′ on the Tate module Vℓ(E) = Qℓ ⊗Zℓ limnE(Q)[ℓn] (ℓ is the characteristic of λ). By compatibility of the Galois representations attached to π and E and the absolute irreducibility of the Galois representations attached to π, it suffices to check the isomorphism ρπ,λ ≃ ρ(E)ℓ for one λ. Of course, it is believed that one can take F ′ = F in the theorem. The following proposition is much weaker, but it is useful (see [5] cor. 12.2.10 and def. 12.11.3, and [6] thm. 1). Proposition 0.2. Let T be a finite set of primes of F such that E has good reduction at all q ∈ T. One can then impose that F ′/F is unramified at T. Remark. Let N be a finite extension of F. One can furthermore impose that N and F ′ are linearly disjoint extensions of F (prop. 2.1. of [2]). Let us give a proof of the theorem and the proposition