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ON SERRE’S MODULARITY CONJECTURE FOR 2DIMENSIONAL MOD p REPRESENTATIONS OF ... Unramified Outside p
, 2005
"... We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint wor ..."
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We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint work with JP. Wintenberger, together with a new method of “weight reduction”.
Serre’s modularity conjecture (II)
, 2007
"... We provide proofs of Theorems 4.1 and 5.1 of [30]. ..."
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, 2008
"... We first prove the existence of minimally ramified padic lifts of 2dimensional mod p representations ¯ρ, that are odd and irreducible, of the absolute Galois group of Q, in many cases. This is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight ..."
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We first prove the existence of minimally ramified padic lifts of 2dimensional mod p representations ¯ρ, that are odd and irreducible, of the absolute Galois group of Q, in many cases. This is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre’s conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a ¯ρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Deltafunction. In the last part of the paper we present arguments that reduce Serre’s conjecture on modularity of representations ¯ρ as above to proving modularity lifting theorems of the type pioneered by Wiles. While these modularity lifting results are not known as yet they might be relatively accessible because of the basic method of Wiles and TaylorWiles and recent developments in the padic Langlands programme initiated by Breuil.