Strong modularity of reducible Galois representations

TL;DR

This paper characterizes strongly modular reducible Galois representations and provides conditions for their level raising, extending classical results by Ribet and Mazur in the context of Serre's conjecture.

Contribution

It offers a precise characterization of strongly modular reducible Galois representations under certain conditions, generalizing Ribet's and Mazur's classical theorems.

Findings

Characterization of strongly modular representations when l > k+1.

Necessary and sufficient conditions for level raising of non-strongly modular representations.

Extension of classical theorems to broader cases with 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 $l\textgreater{}k+1$, 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 $\rho$ 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 $(N,k)=(1,2)$.