Strong modularity of reducible Galois representations
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- by Nicolas Billerey and Ricardo Menares PDF
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Abstract:
Let $\rho \colon \mathrm {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \mathrm {GL}_2(\overline {\mathbf {F}}_{l})$ be an odd, semi-simple Galois representation. Here, $l\geq 5$ is prime and $\overline {\mathbf {F}}_{l}$ is an algebraic closure of the finite field $\mathbf {Z}/l\mathbf {Z}$. When the representation is irreducible, the strongest form of Serre’s original modularity conjecture (which is now proved) asserts that $\rho$ arises from a cuspidal eigenform of type $(N,k,\varepsilon )$ over $\overline {\mathbf {F}}_{l}$, where $N$, $k$ and $\varepsilon$ are, respectively, the level, weight and character attached to $\rho$ by Serre.
In this paper we characterize, under the assumption $l>k+1$, reducible semi-simple representations, that we call strongly modular, such that the same result holds. This characterization generalizes a classical theorem of Ribet pertaining to the case $N=1$. When the representation is not strongly modular, we give a necessary and sufficient condition on primes $p$ not dividing $Nl$ for which $\rho$ arises in level $Np$, hence generalizing a classical theorem of Mazur concerning the case $(N,k)=(1,2)$.
The proofs rely on the classical analytic theory of Eisenstein series and on local properties of automorphic representations attached to newforms.
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Additional Information
- Nicolas Billerey
- Affiliation: Université Clermont Auvergne, Université Blaise Pascal, Laboratoire de Mathé- matiques, BP 10448, F-63000 Clermont-Ferrand, France – and – CNRS, UMR 6620, LM, F-63171 Aubière, France
- MR Author ID: 823614
- Email: Nicolas.Billerey@uca.fr
- Ricardo Menares
- Affiliation: Pontificia Universidad Católica de Valparaíso, Instituto de Matemáticas, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
- MR Author ID: 880333
- Email: ricardo.menares@pucv.cl
- Received by editor(s): April 11, 2016
- Received by editor(s) in revised form: May 12, 2016
- Published electronically: August 15, 2017
- Additional Notes: The first author was partially supported by CNRS and ANR-14-CE-25-0015 Gardio.
The second author was partially supported by PUCV grant 037.469/2015 - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 967-986
- MSC (2010): Primary 11F80, 11F33; Secondary 11F70
- DOI: https://doi.org/10.1090/tran/6979
- MathSciNet review: 3729493