On the modularity of reducible mod l Galois representations
Abstract
We prove that every odd semisimple reducible (2-dimensional) mod l Galois representation arises from a cuspidal eigenform. In addition, we investigate the possible different types (level, weight, character) of such a modular form. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime level. The bound is valid in a subset of the primes with natural (lower) density at least one half.
Keywords
Cite
@article{arxiv.1309.3717,
title = {On the modularity of reducible mod l Galois representations},
author = {Nicolas Billerey and Ricardo Menares},
journal= {arXiv preprint arXiv:1309.3717},
year = {2017}
}
Comments
Minor modifications. Accepted for publication in Mathematical Research Letters