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Strong modularity of reducible Galois representations


Authors: Nicolas Billerey and Ricardo Menares
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F80, 11F33; Secondary 11F70
DOI: https://doi.org/10.1090/tran/6979
Published electronically: August 15, 2017
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Abstract: Let $ \rho \colon \mathrm {Gal}(\overline {\mathbf {Q}}/\mathbf {Q}) \rightarrow \textup {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
Email: Nicolas.Billerey@uca.fr

Ricardo Menares
Affiliation: Pontificia Universidad Católica de Valparaíso, Instituto de Matemáticas, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
Email: ricardo.menares@pucv.cl

DOI: https://doi.org/10.1090/tran/6979
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
Article copyright: © Copyright 2017 American Mathematical Society