Strong modularity of reducible Galois representations

Nicolas Billerey; Ricardo Menares

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arXiv · 1604.01173

Strong modularity of reducible Galois representations

Nicolas Billerey·Ricardo Menares

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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 $l\textgreater{}k+1$, 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 $ρ$ 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 $ρ$. This generalizes a result of Mazur on the case $(N,k)=(1,2)$.

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BibTeX RIS

Nicolas Billerey, Ricardo Menares. 2016-05-25. Strong modularity of reducible Galois representations. https://arxiv.org/abs/1604.01173

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