Mathematical Research Letters

Volume 23 (2016)

Number 1

On the modularity of reducible mod $l$ Galois representations

Pages: 15 – 41



Nicolas Billerey (Laboratoire de Mathématiques, Université Clermont Auvergne, Université Blaise Pascal, Clermont-Ferrand, France; and CNRS, UMR 6620, LM, Aubière, France)

Ricardo Menares (Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Chile)


Given an odd, semisimple, reducible, 2-dimensional $\mathrm{mod} \: l$ Galois representation, we investigate the possible levels of the modular forms giving rise to it. When the representation is the direct sum of the trivial character and a power of the $\mathrm{mod} \: l$ 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 $3/4$.

2010 Mathematics Subject Classification

Primary 11F33, 11F80. Secondary 11N25.

Accepted 16 September 2014

Published 25 May 2016