Some Computations with Heeke Rings and Deformation Rings

In order to prove that semistable elliptic curves E over Q are modular, A. Wiles [Wiles 95] considers the Galois representation ρ : Gal(Q/Q) −→ GL2(F3) provided by the 3-torsion points E[3] of a semistable elliptic curve E. He proves that certain rather restricted deformations of ρ are modular. It follows then that, in particular, the deformation provided by the Galois representation ρE : Gal(Q/Q) −→ GL2(Z3) associated to the 3-adic Tate module of E is modular. This implies that the curve E is modular. Wiles proceeds in two steps. First, he considers certain minimal deformations of ρ. Using the Langlands-Tunnell Theorem [Gelbart 97, Theorem 1.3] and some so-called “level lowering theorems” [Edixhoven 97], he constructs a normalized eigenform of weight 2 and minimal level N whose associated Galois representation