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# Mathematical Research Letters

## Volume 23 (2016)

### Number 1

### On the modularity of reducible mod $l$ Galois representations

Pages: 15 – 41

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n1.a2

#### Authors

#### Abstract

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.

Published 25 May 2016