# 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 :
Document type :
Preprints, Working Papers, ...

Cited literature [15 references]

https://hal.archives-ouvertes.fr/hal-01295749
Contributor : Nicolas Billerey <>
Submitted on : Monday, May 23, 2016 - 4:07:06 PM
Last modification on : Thursday, July 26, 2018 - 12:16:02 PM

### Files

minimal_level.pdf
Files produced by the author(s)

### Identifiers

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

### Citation

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

Record views