# Strong modularity of reducible Galois representations

Abstract : 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 $(N,k)=(1,2)$.
Keywords :
Type de document :
Pré-publication, Document de travail
Revised version. To appear in Trans. Amer. Math. Soc. 2016

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https://hal.archives-ouvertes.fr/hal-01295749
Contributeur : Nicolas Billerey <>
Soumis le : lundi 23 mai 2016 - 16:07:06
Dernière modification le : jeudi 26 mai 2016 - 01:06:12

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### Identifiants

• HAL Id : hal-01295749, version 2
• ARXIV : 1604.01173

### Citation

Nicolas Billerey, Ricardo Menares. Strong modularity of reducible Galois representations. Revised version. To appear in Trans. Amer. Math. Soc. 2016. 〈hal-01295749v2〉

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