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Images of Galois representations in mod p Hecke algebras

    Let (𝕋f,𝔪f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f, which is a commutative algebra over some finite field 𝔽q of characteristic p and with residue field 𝔽q. By a result of Carayol we know that, if the residual Galois representation ρ¯f:GGL2(𝔽q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρf:GGL2(𝕋f) 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(𝔽q), (ii) 𝔪f2=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.

    AMSC: 11F80, 11F33, 20E34


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