Images of Galois representations in mod p Hecke algebras.

Let (𝕋 f , 𝔪 f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f , which is a commutative algebra over some finite field 𝔽 q of characteristic p and with residue field 𝔽 q . By a result of Carayol we know that, if the residual Galois representation ρ ¯ f : G ℚ → GL 2 (𝔽 q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρ f : G ℚ → GL 2 (𝕋 f) that is a lift of ρ ¯ f . We will show how one can determine the image of ρ f under the assumptions that (i) the image of the residual representation contains SL 2 (𝔽 q) , (ii) 𝔪 f 2 = 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. [ABSTRACT FROM AUTHOR]