Images of Galois representations in mod p Hecke algebras

Let (f, f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f, which is a commutative algebra over some finite field q of characteristic p and with residue field q. By a result of Carayol we know that, if the residual Galois representation ρ¯f: G →GL2(q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρf: G →GL2(f) that is a lift of ρ¯f. We will show how one can determine the image of ρf under the assumptions that (i) the image of the residual representation contains SL2(q), (ii) f2 = 0 and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain p-elementary abelian extensions of big non-solvable number fields.