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Strong modularity of reducible Galois representations


In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l>k+1$, we give a precise characterization of strongly modular representations, hence generalizing a classical theorem of Ribet pertaining to the case of conductor $1$. When the representation $\rho$ is not strongly modular, we give a necessary and sufficient condition on the primes $p$ not dividing $Nl$ for which it arises in level $Np$, where $N$ denotes the conductor of $\rho$. This generalizes a result of Mazur on the case $(N,k)=(1,2)$.
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Dates and versions

hal-01295749 , version 1 (31-03-2016)
hal-01295749 , version 2 (23-05-2016)



Nicolas Billerey, Ricardo Menares. Strong modularity of reducible Galois representations. 2016. ⟨hal-01295749v2⟩
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