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Strong modularity of reducible Galois representations
Number Theory · Submitted 5 Apr 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $\rho$.
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The abstract, sentence by sentence
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 $(N,k)=(1,2)$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.60 | 0.59 | 0.10 | asserted 0.88 |
| s1 | 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 $\rho$. | result 0.62 | 0.95 | 0.82 | asserted 1.00 |
| s2 | This generalizes a result of Mazur on the case $(N,k)=(1,2)$. | interpretation 0.69 | 0.44 | 0.06 | asserted 1.00 |
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Original 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 $(N,k)=(1,2)$.
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