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In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular. We will first remind the reader of the content of these results and then briefly outline the method of proof. If N is a positive integer then we let Γ1(N) denote the subgroup of SL2(Z) consisting of matrices that modulo N are of the form ( 1 ∗ 0 1 ) .

On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises